Log24

Monday, December 11, 2017

The Diamond Theorem at SASTRA

Filed under: General,Geometry — Tags: — m759 @ 12:35 PM

The following IEEE paper is behind a paywall,
but the first page is now available for free
at deepdyve.com

For further details on the diamond theorem, see
finitegeometry.org/sc/ or the archived version at . . .

DOI

Friday, October 28, 2016

Diamond-Theorem Application

Filed under: General,Geometry — Tags: — m759 @ 1:06 PM
 

Abstract:

"Protection of digital content from being tapped by intruders is a crucial task in the present generation of Internet world. In this paper, we proposed an implementation of new visual secret sharing scheme for gray level images using diamond theorem correlation. A secret image has broken into 4 × 4 non overlapped blocks and patterns of diamond theorem are applied sequentially to ensure the secure image transmission. Separate diamond patterns are utilized to share the blocks of both odd and even sectors. Finally, the numerical results show that a novel secret shares are generated by using diamond theorem correlations. Histogram representations demonstrate the novelty of the proposed visual secret sharing scheme."

— "New visual secret sharing scheme for gray-level images using diamond theorem correlation pattern structure," by  V. Harish, N. Rajesh Kumar, and N. R. Raajan.

Published in: 2016 International Conference on Circuit, Power and Computing Technologies (ICCPCT).
Date of Conference: 18-19 March 2016. Publisher: IEEE.
Date Added to IEEE Xplore: 04 August 2016

Excerpts —

Related material — Posts tagged Diamond Theorem Correlation.

Monday, March 2, 2015

Elements of Design

Filed under: General — m759 @ 1:28 AM

From "How the Guggenheim Got Its Visual Identity,"
by Caitlin Dover, November 4, 2013 —


For the square and half-square in the above logo
as independent design elements, see 
the Cullinane diamond theorem.

For the circle and half-circle in the logo,
see Art Wars (July 22, 2012).

For a rectangular space that embodies the name of
the logo's design firm 2×4, see Octad in this journal.

Friday, August 1, 2014

The Diamond-Theorem Correlation

Filed under: General,Geometry — Tags: , — m759 @ 2:00 AM

Click image for a larger, clearer version.

IMAGE- The symplectic correlation underlying Rosenhain and Göpel tetrads

Tuesday, December 3, 2013

Diamond Space

Filed under: General,Geometry — Tags: — m759 @ 1:06 PM

A new website illustrates its URL.
See DiamondSpace.net.

IMAGE- Site with keywords 'Galois space, Galois geometry, finite geometry' at DiamondSpace.net

Monday, February 11, 2013

The Penrose Diamond

Filed under: General,Geometry — Tags: — m759 @ 2:01 PM

IMAGE- The Penrose Diamond

Related material:

(Click to enlarge.)

See also remarks on Penrose linked to in Sacerdotal Jargon.

(For a connection of these remarks to
the Penrose diamond, see April 1, 2012.)

Thursday, May 31, 2012

Black Diamond

Filed under: General — Tags: — m759 @ 12:26 PM

IMAGE- Four-elements-diamond test problem in the style of Raven's Progressive Matrices (answer: the black diamond)

“To say more is to say less.”
― Harlan Ellison, as quoted at goodreads.com

Saying less—

Saturday, March 17, 2012

The Purloined Diamond

Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

(Continued)

The diamond from the Chi-rho page
of the Book of Kells —

The diamond at the center of Euclid's
Proposition I, according to James Joyce
(i.e., the Diamond in the Mandorla) —

Geometry lesson: the vesica piscis in Finnegans Wake

The Diamond in the Football

Football-mandorla

“He pointed at the football
  on his desk. ‘There it is.’”
         – Glory Road
   

Wednesday, December 21, 2011

The Purloined Diamond

Filed under: General — Tags: , — m759 @ 9:48 AM

Stephen Rachman on "The Purloined Letter"

"Poe’s tale established the modern paradigm (which, as it happens, Dashiell Hammett and John Huston followed) of the hermetically sealed fiction of cross and double-cross in which spirited antagonists pursue a prized artifact of dubious or uncertain value."

For one such artifact, the diamond rhombus formed by two equilateral triangles, see Osserman in this journal.

Some background on the artifact is given by John T. Irwin's essay "Mysteries We Reread…" reprinted in Detecting Texts: The Metaphysical Detective Story from Poe to Postmodernism .

Related material—

Mathematics vulgarizer Robert Osserman died on St. Andrew's Day, 2011.

A Rhetorical Question

Osserman in 2004

"The past decade has been an exciting one in the world of mathematics and a fabulous one (in the literal sense) for mathematicians, who saw themselves transformed from the frogs of fairy tales— regarded with a who-would-want-to-kiss-that aversion, when they were noticed at all— into fascinating royalty, portrayed on stage and screen….

Who bestowed the magic kiss on the mathematical frog?"

A Rhetorical Answer

http://www.log24.com/log/pix11C/111130-SunshineCleaning.jpg

Above: Amy Adams in "Sunshine Cleaning"

Sunday, July 10, 2011

Wittgenstein’s Diamond

Filed under: General,Geometry — Tags: — m759 @ 9:29 AM

Philosophical Investigations  (1953)

97. Thought is surrounded by a halo.
—Its essence, logic, presents an order,
in fact the a priori order of the world:
that is, the order of possibilities * ,
which must be common to both world and thought.
But this order, it seems, must be
utterly simple . It is prior  to all experience,
must run through all experience;
no empirical cloudiness or uncertainty can be allowed to affect it
——It must rather be of the purest crystal.
But this crystal does not appear as an abstraction;
but as something concrete, indeed, as the most concrete,
as it were the hardest  thing there is
(Tractatus Logico-Philosophicus  No. 5.5563).

— Translation by G.E.M. Anscombe

5.5563

All propositions of our colloquial language
are actually, just as they are, logically completely in order.
That simple thing which we ought to give here is not
a model of the truth but the complete truth itself.

(Our problems are not abstract but perhaps
the most concrete that there are.)

97. Das Denken ist mit einem Nimbus umgeben.
—Sein Wesen, die Logik, stellt eine Ordnung dar,
und zwar die Ordnung a priori der Welt,
d.i. die Ordnung der Möglichkeiten ,
die Welt und Denken gemeinsam sein muß.
Diese Ordnung aber, scheint es, muß
höchst einfach  sein. Sie ist vor  aller Erfahrung;
muß sich durch die ganze Erfahrung hindurchziehen;
ihr selbst darf keine erfahrungsmäßige Trübe oder Unsicherheit anhaften.
——Sie muß vielmehr vom reinsten Kristall sein.
Dieser Kristall aber erscheint nicht als eine Abstraktion;
sondern als etwas Konkretes, ja als das Konkreteste,
gleichsam Härteste . (Log. Phil. Abh.  No. 5.5563.)

See also

Related language in Łukasiewicz (1937)—

http://www.log24.com/log/pix10B/101127-LukasiewiczAdamantine.jpg

* Updates of 9:29 PM ET July 10, 2011—

A  mnemonic  from a course titled "Galois Connections and Modal Logics"—

"Traditionally, there are two modalities, namely, possibility and necessity.
The basic modal operators are usually written box (square) for necessarily
and diamond (diamond) for possibly. Then, for example, diamondP  can be read as
'it is possibly the case that P .'"

See also Intensional Semantics , lecture notes by Kai von Fintel and Irene Heim, MIT, Spring 2007 edition—

"The diamond symbol for possibility is due to C.I. Lewis, first introduced in Lewis & Langford (1932), but he made no use of a symbol for the dual combination ¬¬. The dual symbol was later devised by F.B. Fitch and first appeared in print in 1946 in a paper by his doctoral student Barcan (1946). See footnote 425 of Hughes & Cresswell (1968). Another notation one finds is L for necessity and M for possibility, the latter from the German möglich  ‘possible.’"

Barcan, Ruth C.: 1946. “A Functional Calculus of First Order Based on Strict Implication.” Journal of Symbolic Logic, 11(1): 1–16. URL http://www.jstor.org/pss/2269159.

Hughes, G.E. & Cresswell, M.J.: 1968. An Introduction to Modal Logic. London: Methuen.

Lewis, Clarence Irving & Langford, Cooper Harold: 1932. Symbolic Logic. New York: Century.

Saturday, May 10, 2014

Test Patterns

Filed under: General,Geometry — m759 @ 11:00 AM

 Raven’s Progressive Matrices  intelligence test—
IMAGE- Raven's Progressive Matrices problem based on triangular half- and quarter-diamonds

Wechsler Adult Intelligence Scale  test—  

Related art —  (Click images for further details.)

Patterns suggesting those of the Raven test:

Patterns suggesting those of the Wechsler test:

The latter patterns were derived from the former.

Wednesday, May 7, 2014

Meanwhile, in São Paulo…

Filed under: General — m759 @ 9:38 AM

This post was suggested by last night’s posts on conceptual art
(in São Paulo) and on a quarter-to-three story.

From this journal on May 31, 2012:

Matrix Problem:
IMAGE- Raven's Progressive Matrices problem based on triangular half- and quarter-diamonds
Click image for some related material.

Meanwhile…

A game released on the above date:

IMAGE- 'Max Payne 3,' a PC game set in São Paulo

Tuesday, March 19, 2013

Mathematics and Narrative (continued)

Filed under: General,Geometry — Tags: , — m759 @ 10:18 AM

Angels & Demons meet Hudson Hawk

Dan Brown's four-elements diamond in Angels & Demons :

IMAGE- Illuminati Diamond, pp. 359-360 in 'Angels & Demons,' Simon & Schuster Pocket Books 2005, 448 pages, ISBN 0743412397

The Leonardo Crystal from Hudson Hawk :

Hudson:

Mathematics may be used to relate (very loosely)
Dan Brown's fanciful diamond figure to the fanciful
Leonardo Crystal from Hudson Hawk 

"Giving himself a head rub, Hawk bears down on
the three oddly malleable objects. He TANGLES 
and BENDS and with a loud SNAP, puts them together,
forming the Crystal from the opening scene."

— A screenplay of Hudson Hawk

Happy birthday to Bruce Willis.

Thursday, May 31, 2012

Matrix Problem

Filed under: General,Geometry — Tags: — m759 @ 7:20 PM

IMAGE- Charlize Theron as Ravenna with raven in poster for 'Snow White and the Huntsman'

Poster from Walpurgisnacht 2012

Raven’s Progressive Matrices problem:

IMAGE- Raven's Progressive Matrices problem based on triangular quarter- and half-diamonds

Click the problem for a related story.

For some related geometry, see Elements Diamond.
See also a post (Dream Time, May 3, 2010)
about geometry and an earlier Walpurgisnacht.

Wednesday, May 23, 2012

Now What?

Filed under: General — m759 @ 10:30 PM

(Rhetorical question on the NY Times  online front page, 
10:01 PM May 23, 2012, in teaser for "The Stone" column
about Philip K. Dick, "Sci-Fi Philosopher")

Where Entertainment Is God

Perhaps The Last Airbender ?

The NY Times  philosophy column "The Stone" is currently about gnosticism
and science fiction.

The Last Airbender  is about an avatar who is master of the four elements
air, water, earth, and fire. For a more sophisticated approach to gnosticism
and the four elements, see Irenaeus: Against Heresies.

See, too, Elements Diamond in this journal.

Thursday, July 1, 2010

Plato’s Code

Filed under: General,Geometry — m759 @ 12:00 PM

John Allen Paulos yesterday at Twitter

"Plato's code cracked? http://bit.ly/ad6k1S
Fascinating if not a hoax or hype."

The story that Paulos linked to is about a British
academic who claims to have found some
symbolism hidden in Plato's writings by
splitting each into 12 parts and correlating
the 12 parts with semitones of a musical scale.

I prefer a different approach to Plato that is
related to the following hoax and hype—

HOAX:

From Dan Brown's novel Angels & Demons  (2000)

IMAGE- Illuminati Diamond, pp. 359-360 in 'Angels & Demons,' Simon & Schuster Pocket Books 2005, 448 pages, ISBN 0743412397

HYPE:

Image-- From 'Alchemy,' by Holmyard, the diamond of Aristotle's 4 elements and 4 qualities

This  four-elements diamond summarizes the classical
four elements and four qualities neatly, but some scholars
might call the figure "hype" since it deals with an academically
disreputable subject, alchemy, and since its origin is unclear.

For the four elements' role in some literature more respectable
than Dan Brown's, see Poetry's Bones.

Although an author like Brown might spin the remarks
below into a narrative—  The Plato Code — they are
neither  hoax nor hype.

NOT  HOAX:

Image-- From the Diamond in Plato's Meno to Modern Finite Geometry

NOT  HYPE:

http://www.log24.com/log/pix10A/100626-CrossOnSocratesSm.gif

For related non-hoax, non-hype remarks, see
The Rational Enterprise: Logos in Plato's Theaetetus,
by Rosemary Desjardins.

Those who prefer  hoax and hype in their philosophy may consult
the writings of, say, Barbara Johnson, Rosalind Krauss, and—
in yesterday's NY Times's  "The Stone" columnNancy Bauer.

Image-- The Philosophers' Stone according to The New York Times

— The New York Times

Friday, September 28, 2018

ART WARS Midrash

Filed under: General — m759 @ 9:00 AM

"When times are mysterious
Serious numbers
Will always be heard."
— Paul Simon,
"When Numbers Get Serious"

"There is a pleasantly discursive treatment of
Pontius Pilate's unanswered question 'What is truth?'"

— H. S. M. Coxeter, introduction to Richard J. Trudeau's remarks
on the "story theory" of truth as opposed to the "diamond theory"
of truth in The Non-Euclidean Revolution  (1987)

The deaths of Roth and Grünbaum on September 14th,
The Feast of the Holy Cross, along with Douthat's column
today titled "Only the Truth Can Save Us Now," suggest a
review of

Elements of Number Theory, by Vinogradov .

Monday, August 27, 2018

Children of the Six Sides

Filed under: General,Geometry — Tags: — m759 @ 11:32 AM

http://www.log24.com/log/pix18/180827-Terminator-3-tx-arrival-publ-160917.jpg

http://www.log24.com/log/pix18/180827-Terminator-3-tx-arrival-publ-161018.jpg

From the former date above —

Saturday, September 17, 2016

A Box of Nothing

Filed under: Uncategorized — m759 @ 12:13 AM

(Continued)

"And six sides to bounce it all off of.

From the latter date above —

Tuesday, October 18, 2016

Parametrization

Filed under: Uncategorized — m759 @ 6:00 AM

The term "parametrization," as discussed in Wikipedia, seems useful for describing labelings that are not, at least at first glance, of a vector-space  nature.

Examples: The labelings of a 4×4 array by a blank space plus the 15 two-subsets of a six-set (Hudson, 1905) or by a blank plus the 5 elements and the 10 two-subsets of a five-set (derived in 2014 from a 1906 page by Whitehead), or by a blank plus the 15 line diagrams of the diamond theorem.

Thus "parametrization" is apparently more general than the word "coodinatization" used by Hermann Weyl —

“This is the relativity problem:  to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.”

— Hermann Weyl, The Classical Groups , Princeton University Press, 1946, p. 16

Note, however, that Weyl's definition of "coordinatization" is not limited to vector-space  coordinates. He describes it as simply a mapping to a set of reproducible symbols

(But Weyl does imply that these symbols should, like vector-space coordinates, admit a group of transformations among themselves that can be used to describe transformations of the point-space being coordinatized.)

From March 2018 —

http://www.log24.com/log/pix18/180827-MIT-Rubik-Robot.jpg

Monday, June 11, 2018

Glitter

Filed under: General,Geometry — Tags: — m759 @ 8:32 PM

A Scientific American  headline today —

Glittering Diamond Dust in Space
Might Solve a 20-Year-Old Mystery

Related art —

"Never underestimate the power of glitter."

Glitter by Josefine Lyche, as of diamond dust

Background:  "Diamond Dust" + Glitter in this journal.

Saturday, June 9, 2018

SASTRA paper

Filed under: General,Geometry — Tags: — m759 @ 11:14 PM

Now out from behind a paywall . . .

The diamond theorem at SASTRA —

Monday, December 18, 2017

Wheelwright and the Dance

Filed under: G-Notes,General,Geometry — m759 @ 1:00 PM

The page preceding that of yesterday's post  Wheelwright and the Wheel —

See also a Log24 search for 
"Four Quartets" + "Four Elements".

A graphic approach to this concept:

"The Bounded Space" —

'Space Cross' from the Cullinane diamond theorem

"The Fire, Air, Earth, and Water" —

Logo for 'Elements of Finite Geometry'

Friday, December 8, 2017

Logos (Continued)

Filed under: General — Tags: — m759 @ 3:00 PM

Nietzsche, 'law in becoming' and 'play in necessity'

"Denn die Welt braucht ewig die Wahrheit,
also braucht sie ewig Heraklit:
obschon er ihrer nicht bedarf.
Was geht ihn sein Ruhm an?
Der Ruhm bei »immer fortfließenden Sterblichen!«,
wie er höhnisch ausruft.
Sein Ruhm geht die Menschen etwas an, nicht ihn,
die Unsterblichkeit der Menschheit braucht ihn,
nicht er die Unsterblichkeit des Menschen Heraklit.
Das, was er schaute, die Lehre vom Gesetz im Werden
und vom
Spiel in der Notwendigkeit 
, muß von jetzt
ab ewig geschaut werden: er hat von diesem größten
Schauspiel den Vorhang aufgezogen."

Logos for Philosophers
(Suggested by Modal Logic) —

Nietzsche, 'law in becoming' and 'play in necessity'

Saturday, October 28, 2017

Lowell Brown at Vanity Fair

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 8:18 PM

A sequel to the post  CP  is for Consolation Prize  (Sept. 3, 2016)

An image from Log24 on this date last year:

A recent comment on a discussion of CP symmetry

Wednesday, November 23, 2016

Yogiism

Filed under: General,Geometry — Tags: , — m759 @ 12:31 PM

From the American Mathematical Society (AMS) webpage today —

From the current AMS Notices

Related material from a post of Aug. 6, 2014

http://www.log24.com/log/pix10B/100915-SteinbergOnChevalleyGroups.jpg

(Here "five point sets" should be "five-point sets.")

From Gotay and Isenberg, “The Symplectization of Science,”
Gazette des Mathématiciens  54, 59-79 (1992):

“… what is the origin of the unusual name ‘symplectic’? ….
Its mathematical usage is due to Hermann Weyl who,
in an effort to avoid a certain semantic confusion, renamed
the then obscure ‘line complex group’ the ‘symplectic group.’
… the adjective ‘symplectic’ means ‘plaited together’ or ‘woven.’
This is wonderfully apt….”

IMAGE- A symplectic structure -- i.e. a structure that is symplectic (meaning plaited or woven)

The above symplectic  structure* now appears in the figure
illustrating the diamond-theorem correlation in the webpage
Rosenhain and Göpel Tetrads in PG(3,2).

* The phrase as used here is a deliberate 
abuse of language .  For the real definition of 
“symplectic structure,” see (for instance) 
“Symplectic Geometry,” by Ana Cannas da Silva
(article written for Handbook of Differential
Geometry 
, Vol 2.) To establish that the above
figure is indeed symplectic , see the post 
Zero System of July 31, 2014.

Tuesday, October 18, 2016

Parametrization

Filed under: General,Geometry — Tags: — m759 @ 6:00 AM

The term "parametrization," as discussed in Wikipedia,
seems useful for describing labelings that are not, at least
at first glance, of a vector-space  nature.

Examples: The labelings of a 4×4 array by a blank space
plus the 15 two-subsets of a six-set (Hudson, 1905) or by a
blank plus the 5 elements and the 10 two-subsets of a five-set
(derived in 2014 from a 1906 page by Whitehead), or by 
a blank plus the 15 line diagrams of the diamond theorem.

Thus "parametrization" is apparently more general than
the word "coodinatization" used by Hermann Weyl —

“This is the relativity problem:  to fix objectively
a class of equivalent coordinatizations and to
ascertain the group of transformations S
mediating between them.”

— Hermann Weyl, The Classical Groups ,
Princeton University Press, 1946, p. 16

Note, however, that Weyl's definition of "coordinatization"
is not limited to vector-space  coordinates. He describes it
as simply a mapping to a set of reproducible symbols

(But Weyl does imply that these symbols should, like vector-space 
coordinates, admit a group of transformations among themselves
that can be used to describe transformations of the point-space
being coordinatized.)

Saturday, June 4, 2016

Icons

Filed under: General — Tags: — m759 @ 12:00 PM

From this morning's news, a  cultural icon —

From November 18, 2015, four  icons —

— the three favicons above, and the following:

Jack in the Box, by Natasha Wescoat

Sunday, May 15, 2016

One Ring

Filed under: General,Geometry — Tags: , — m759 @ 5:06 PM

(Continued from May 11 and May 15.)

Poem by Eleanor Wilner from 'Reversing the Spell' speaks of diamonds and 'glitter.' (Pbk. publ. Nov. 1, 1997)

Friday, April 29, 2016

Blackboard Jungle…

Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

Continues .

An older and wiser James Spader —

"Never underestimate the power of glitter."

Glitter by Josefine Lyche, as of diamond dust

Saturday, January 23, 2016

Hard

Filed under: General — Tags: — m759 @ 9:00 PM

"Hard Science Fiction in the era of short attention spans,
crowd-sourcing, and rapid obsolescence"

— May 26, 2012, Dragon Press Bookstore symposium

Related material:  Posts now tagged Black Diamond.

IMAGE- 'The Stars My Destination' (with cover slightly changed)

Friday, January 22, 2016

Story

Filed under: General — Tags: — m759 @ 10:21 AM

The New Yorker , April 12, 2004 —

Wednesday, November 18, 2015

Tuesday, November 17, 2015

Friday, November 13, 2015

A Connection between the 16 Dirac Matrices and the Large Mathieu Group



Note that the six anticommuting sets of Dirac matrices listed by Arfken
correspond exactly to the six spreads in the above complex of 15 projective
lines of PG(3,2) fixed under a symplectic polarity (the diamond theorem
correlation
 
). As I noted in 1986, this correlation underlies the Miracle
Octad Generator of R. T. Curtis, hence also the large Mathieu group.

References:

Arfken, George B., Mathematical Methods for Physicists , Third Edition,
Academic Press, 1985, pages 213-214

Cullinane, Steven H., Notes on Groups and Geometry, 1978-1986

Related material:

The 6-set in my 1986 note above also appears in a 1996 paper on
the sixteen Dirac matrices by David M. Goodmanson —

Background reading:

Ron Shaw on finite geometry, Clifford algebras, and Dirac groups 
(undated compilation of publications from roughly 1994-1995)—

Thursday, October 15, 2015

Contrapuntal Interweaving

Filed under: General,Geometry — Tags: — m759 @ 2:01 AM

The title is a phrase from R. D. Laing's book The Politics of Experience .
(Published in the psychedelic year 1967. The later "contrapuntal interweaving"
below is of a less psychedelic nature.)

An illustration of the "interweaving' part of the title —
The "deep structure" of the diamond theorem:

IMAGE- A symplectic structure -- i.e. a structure that is symplectic (meaning plaited or woven).

The word "symplectic" from the end of last Sunday's (Oct. 11) sermon
describes the "interwoven" nature of the above illustration.

An illustration of the "contrapuntal" part of the title (click to enlarge):

The diamond-theorem correlation

 

Saturday, July 4, 2015

Context

Filed under: General,Geometry — Tags: , — m759 @ 10:00 AM

Some context for yesterday's post on a symplectic polarity —

This 1986 note may or may not have inspired some remarks 
of Wolf Barth in his foreword to the 1990 reissue of Hudson's
1905 Kummer's Quartic Surface .

See also the diamond-theorem correlation.  

Saturday, June 13, 2015

Egg Tales

Filed under: General — Tags: , — m759 @ 12:00 PM

"And not all the king's men nor his horses
 Will resurrect his corpus."

Finnegans Wake

See as well Andy Weir's "The Egg" and Working Backward.

Saturday, April 11, 2015

The Starbird Manifesto

Filed under: General,Geometry — Tags: , — m759 @ 10:00 PM

"But what was supposed to be the source of a compound's
authority? Why, the same as that of all new religious movements:
direct access to the godhead, which in this case was Creativity."

— Tom Wolfe, From Bauhaus to Our House

"Creativity is not a matter of magical inspiration."

— Burger and Starbird, The 5 Elements of Effective Thinking  (2012) 

Video published on Oct 19, 2012

"In this fifth of five videos, mathematics professor
Michael Starbird talks about the fifth element
in his new book, The 5 Elements of Effective Thinking ,
co-authored with Williams College professor
Edward B. Burger." 

For more on the Starbird manifesto, see Princeton University Press.

An excerpt —

See also a post for Abel's Birthday, 2011 —  
Midnight in Oslo — and a four-elements image from
the Jan. 26, 2010, post Symbology —

Logo for 'Elements of Finite Geometry'.

Wednesday, February 25, 2015

Words and Images

Filed under: General,Geometry — Tags: — m759 @ 5:30 PM

The words:  "symplectic polarity"—

The images:

The Natural Symplectic Polarity in PG(3,2)

Symmetry Invariance in a Diamond Ring

The Diamond-Theorem Correlation

Picturing the Smallest Projective 3-Space

Quilt Block Designs

Saturday, February 21, 2015

High and Low Concepts

Filed under: General,Geometry — Tags: — m759 @ 4:30 PM

Steven Pressfield on April 25, 2012:

What exactly is High Concept?

Let’s start with its opposite, low concept.
Low concept stories are personal,
idiosyncratic, ambiguous, often European. 
“Well, it’s a sensitive fable about a Swedish
sardine fisherman whose wife and daughter
find themselves conflicted over … ”

ZZZZZZZZ.

Fans of Oslo artist Josefine Lyche know she has
valiantly struggled to find a high-concept approach
to the diamond theorem. Any such approach must,
unfortunately, reckon with the following low
(i.e., not easily summarized) concept —

The Diamond Theorem Correlation:

From left to right

http://www.log24.com/log/pix14B/140824-Diamond-Theorem-Correlation-1202w.jpg

http://www.log24.com/log/pix14B/140731-Diamond-Theorem-Correlation-747w.jpg

http://www.log24.com/log/pix14B/140824-Picturing_the_Smallest-1986.gif

http://www.log24.com/log/pix14B/140806-ProjPoints.gif

For some backstory, see ProjPoints.gif and "Symplectic Polarity" in this journal.

Sunday, August 24, 2014

Symplectic Structure…

Filed under: General,Geometry — Tags: , , — m759 @ 12:00 PM

In the Miracle Octad Generator (MOG):

The above details from a one-page note of April 26, 1986, refer to the
Miracle Octad Generator of R. T. Curtis, as it was published in 1976:

http://www.log24.com/log/pix10A/100514-Curtis1976MOG.jpg

From R. T. Curtis (1976). A new combinatorial approach to M24,
Mathematical Proceedings of the Cambridge Philosophical Society ,
79, pp 25-42. doi:10.1017/S0305004100052075.

The 1986 note assumed that the reader would be able to supply, from the
MOG itself, the missing top row of each heavy brick.

Note that the interchange of the two squares in the top row of each
heavy brick induces the diamond-theorem correlation.

Note also that the 20 pictured 3-subsets of a 6-set in the 1986 note
occur as paired complements  in two pictures, each showing 10 of the
3-subsets.

This pair of pictures corresponds to the 20 Rosenhain tetrads  among
the 35 lines of PG(3,2), while the picture showing the 2-subsets
corresponds to the 15 Göpel tetrads  among the 35 lines.

See Rosenhain and Göpel tetrads in PG(3,2). Some further background:

Wednesday, August 13, 2014

Symplectic Structure continued

Filed under: General,Geometry — Tags: , , , — m759 @ 12:00 PM

Some background for the part of the 2002 paper by Dolgachev and Keum
quoted here on January 17, 2014 —

Related material in this journal (click image for posts) —

Monday, August 11, 2014

Syntactic/Symplectic

(Continued from August 9, 2014.)

Syntactic:

Symplectic:

"Visual forms— lines, colors, proportions, etc.— are just as capable of
articulation , i.e. of complex combination, as words. But the laws that govern
this sort of articulation are altogether different from the laws of syntax that
govern language. The most radical difference is that visual forms are not
discursive 
. They do not present their constituents successively, but
simultaneously, so the relations determining a visual structure are grasped
in one act of vision."

– Susanne K. LangerPhilosophy in a New Key

For examples, see The Diamond-Theorem Correlation
in Rosenhain and Göpel Tetrads in PG(3,2).

This is a symplectic  correlation,* constructed using the following
visual structure:

IMAGE- A symplectic structure -- i.e. a structure that is symplectic (meaning plaited or woven).

* Defined in (for instance) Paul B. Yale, Geometry and Symmetry ,
Holden-Day, 1968, sections 6.9 and 6.10.

Wednesday, August 6, 2014

Symplectic Structure*

Filed under: General,Geometry — Tags: — m759 @ 1:00 PM

From Gotay and Isenberg, "The Symplectization of Science,"
Gazette des Mathématiciens  54, 59-79 (1992):

"… what is the origin of the unusual name 'symplectic'? ….
Its mathematical usage is due to Hermann Weyl who,
in an effort to avoid a certain semantic confusion, renamed
the then obscure 'line complex group' the 'symplectic group.'
… the adjective 'symplectic' means 'plaited together' or 'woven.'
This is wonderfully apt…."

IMAGE- A symplectic structure -- i.e. a structure that is symplectic (meaning plaited or woven)

The above symplectic  structure** now appears in the figure
illustrating the diamond-theorem correlation in the webpage
Rosenhain and Göpel Tetrads in PG(3,2).

Some related passages from the literature:

http://www.log24.com/log/pix10B/100915-SteinbergOnChevalleyGroups.jpg

* The title is a deliberate abuse of language .
For the real definition of "symplectic structure," see (for instance)
"Symplectic Geometry," by Ana Cannas da Silva (article written for
Handbook of Differential Geometry, vol 2.) To establish that the
above figure is indeed symplectic , see the post Zero System of
July 31, 2014.

** See Steven H. Cullinane, Inscapes III, 1986

Thursday, July 31, 2014

Zero System

Filed under: General,Geometry — Tags: , — m759 @ 6:11 PM

The title phrase (not to be confused with the film 'The Zero Theorem')
means, according to the Encyclopedia of Mathematics,
a null system , and

"A null system is also called null polarity,
a symplectic polarity or a symplectic correlation….
it is a polarity such that every point lies in its own
polar hyperplane."

See Reinhold Baer, "Null Systems in Projective Space,"
Bulletin of the American Mathematical Society, Vol. 51
(1945), pp. 903-906.

An example in PG(3,2), the projective 3-space over the
two-element Galois field GF(2):

IMAGE- The natural symplectic polarity in PG(3,2), illustrating a symplectic structure

See also the 10 AM ET post of Sunday, June 8, 2014, on this topic.

Sunday, June 8, 2014

Vide

Filed under: General,Geometry — Tags: , , — m759 @ 10:00 AM

Some background on the large Desargues configuration

“The relevance of a geometric theorem is determined by what the theorem
tells us about space, and not by the eventual difficulty of the proof.”

— Gian-Carlo Rota discussing the theorem of Desargues

What space  tells us about the theorem :  

In the simplest case of a projective space  (as opposed to a plane ),
there are 15 points and 35 lines: 15 Göpel  lines and 20 Rosenhain  lines.*
The theorem of Desargues in this simplest case is essentially a symmetry
within the set of 20 Rosenhain lines. The symmetry, a reflection
about the main diagonal in the square model of this space, interchanges
10 horizontally oriented (row-based) lines with 10 corresponding
vertically oriented (column-based) lines.

Vide  Classical Geometry in Light of Galois Geometry.

* Update of June 9: For a more traditional nomenclature, see (for instance)
R. Shaw, 1995.  The “simplest case” link above was added to point out that
the two types of lines named are derived from a natural symplectic polarity 
in the space. The square model of the space, apparently first described in
notes written in October and December, 1978, makes this polarity clearly visible:

A coordinate-free approach to symplectic structure

Friday, February 21, 2014

Night’s Hymn of the Rock

Filed under: General,Geometry — Tags: , , — m759 @ 3:33 AM

One way of interpreting the symbol  IMAGE- Modal Diamond in a square 
at the end of yesterday's post is via
the phrase "necessary possibility."

See that phrase in (for instance) a post
of July 24, 2013, The Broken Tablet .

The Tablet  post may be viewed in light
of a Tom Wolfe passage quoted here on
the preceding day, July 23, 2013—

IMAGE- Tom Wolfe in 'The Painted Word' on conceptual art

On that  day (July 23) another weblog had
a post titled

Wallace Stevens: Night's Hymn of the Rock.

Some related narrative —

IMAGE- The 2001 film 'The Discovery of Heaven'

I prefer the following narrative —

Part I:  Stevens's verse from "The Rock" (1954) —
"That in which space itself is contained"

Part II:  Mystery Box III: Inside, Outside (2014)

Thursday, February 20, 2014

Relativity Blues

Filed under: General,Geometry — Tags: — m759 @ 9:00 AM

(Continued

A review of this date in 2005 —

Modal Theology

"We symbolize logical necessity
with the box (box.gif (75 bytes))
and logical possibility
with the diamond (diamond.gif (82 bytes))."

— Keith Allen Korcz

And what do we  
   symbolize by   The image “http://www.log24.com/theory/images/Modal-diamondbox.gif” cannot be displayed, because it contains errors. ?

Wednesday, December 18, 2013

Bing Bang Theory

Filed under: General,Geometry — Tags: , — m759 @ 3:00 PM

Microsoft in 2009 on its new search engine name—

"We like Bing because it sounds off in our heads
when we think about that moment of discovery
and decision making— when you resolve those
important tasks."

A search on Bing today —

IMAGE- Top search result on Bing for 'diamond space' on Dec. 18, 2013

A colorful tale —

IMAGE- The Diamond 16 Puzzle, with commentary

"Bing bang, I saw the whole gang
Bobby Darin, 1958

Thursday, December 12, 2013

Outsider Art

Filed under: General,Geometry — Tags: — m759 @ 4:10 PM

"… Galois was a mathematical outsider…."

— Tony Mann, "head of the department of mathematical sciences,
University of Greenwich, and president, British Society for the
History of Mathematics," in a May 6, 2010, review of Duel at Dawn
in Times Higher Education.

Related art: 

(Click for a larger image.)

IMAGE- Google search for 'Diamond Space' + Galois

For a less outside  version of the central image
above, see Kunstkritikk  on Oct. 15, 2013.

Monday, August 12, 2013

Form

Filed under: General,Geometry — Tags: — m759 @ 12:00 PM

The Galois tesseract  appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977—

IMAGE- Hypercube and 4x4 matrix from the 1976 'Diamond Theory' preprint, as excerpted in 'Computer Graphics and Art'

The Galois tesseract is the basis for a representation of the smallest 
projective 3-space, PG(3,2), that differs from the representation at
Wolfram Demonstrations Project. For the latter, see yesterday's post.

The tesseract representation underlies the diamond theorem, illustrated
below in its earliest form, also from the above February 1977 article—

IMAGE- Steven H. Cullinane, diamond theorem, from 'Diamond Theory,' Computer Graphics and Art, Vol. 2 No. 1, Feb. 1977, pp. 5-7

As noted in a more recent version, the group described by
the diamond theorem is also the group of the 35 square
patterns within the 1976 Miracle Octad Generator  (MOG) of
R. T. Curtis.

Saturday, June 29, 2013

Thursday, June 27, 2013

Monday, June 10, 2013

Galois Coordinates

Filed under: General,Geometry — Tags: , — m759 @ 10:30 PM

Today's previous post on coordinate systems
suggests a look at the phrase "Galois coordinates."

A search shows that the phrase, though natural,
has apparently not been used before 2011* for solutions
to what Hermann Weyl called "the relativity problem."

A thorough historical essay on Galois coordinatization
in this sense would require more academic resources
than I have available. It would likely describe a number
of applications of Galois-field coordinates to square
(and perhaps to cubical) arrays that were studied before
1976, the date of my Diamond Theory  monograph.

But such a survey might not  find any such pre-1976
coordinatization of a 4×4 array  by the 16 elements
of the vector 4-space  over the Galois field with two
elements, GF(2).

Such coordinatizations are important because of their
close relationship to the Mathieu group 24 .

See a preprint by Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of Kummer
surfaces in the Mathieu group 24 ," with its remark
denying knowledge of any such coordinatization
prior to a 1989 paper by R. T. Curtis.

Related material: 

Some images related to Galois coordinates, excerpted
from a Google search today (click to enlarge)—

*  A rather abstract  2011 paper that uses the phrase
   "Galois coordinates" may have some implications 
   for the naive form of the relativity problem
   related to square and cubical arrays.

Tuesday, May 28, 2013

Codes

Filed under: General,Geometry — Tags: , , , — m759 @ 12:00 PM

The hypercube  model of the 4-space over the 2-element Galois field GF(2):

IMAGE- A hyperspace model of the 4D vector space over GF(2)

The phrase Galois tesseract  may be used to denote a different model
of the above 4-space: the 4×4 square.

MacWilliams and Sloane discussed the Miracle Octad Generator
(MOG) of R. T. Curtis further on in their book (see below), but did not
seem to realize in 1977 that the 4×4 structures within the MOG are
based on the Galois-tesseract model of the 4-space over GF(2).

IMAGE- Octads within the Curtis MOG, which uses a 4x4-array model of the 4D vector space over GF(2)

The thirty-five 4×4 structures within the MOG:

IMAGE- The 35 square patterns within the Curtis MOG

Curtis himself first described these 35 square MOG patterns
combinatorially, (as his title indicated) rather than
algebraically or geometrically:

IMAGE- R. T. Curtis's combinatorial construction of 4x4 patterns within the Miracle Octad Generator

A later book co-authored by Sloane, first published in 1988,
did  recognize the 4×4 MOG patterns as based on the 4×4
Galois-tesseract model.

Between the 1977 and 1988 Sloane books came the diamond theorem.

Update of May 29, 2013:

The Galois tesseract appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977
(the year the above MacWilliams-Sloane book was first published):

IMAGE- Hypercube and 4x4 matrix from the 1976 'Diamond Theory' preprint, as excerpted in 'Computer Graphics and Art'

Wednesday, February 13, 2013

Form:

Filed under: General,Geometry — Tags: , — m759 @ 9:29 PM

Story, Structure, and the Galois Tesseract

Recent Log24 posts have referred to the 
"Penrose diamond" and Minkowski space.

The Penrose diamond has nothing whatever
to do with my 1976 monograph "Diamond Theory,"
except for the diamond shape and the connection
of the Penrose diamond to the Klein quadric—

IMAGE- The Penrose diamond and the Klein quadric

The Klein quadric occurs in the five-dimensional projective space
over a field. If the field is the two-element Galois field GF(2), the
quadric helps explain certain remarkable symmetry properties 
of the R. T. Curtis Miracle Octad Generator  (MOG), hence of
the large Mathieu group M24. These properties are also 
relevant to the 1976 "Diamond Theory" monograph.

For some background on the quadric, see (for instance)

IMAGE- Stroppel on the Klein quadric, 2008

See also The Klein Correspondence,
Penrose Space-Time, and a Finite Model
.

Related material:

"… one might crudely distinguish between philosophical
and mathematical motivation. In the first case one tries
to convince with a telling conceptual story; in the second
one relies more on the elegance of some emergent
mathematical structure. If there is a tradition in logic
it favours the former, but I have a sneaking affection for
the latter. Of course the distinction is not so clear cut.
Elegant mathematics will of itself tell a tale, and one with
the merit of simplicity. This may carry philosophical
weight. But that cannot be guaranteed: in the end one
cannot escape the need to form a judgement of significance."

– J. M. E. Hyland. "Proof Theory in the Abstract." (pdf)
Annals of Pure and Applied Logic 114, 2002, 43-78.

Those who prefer story to structure may consult 

  1. today's previous post on the Penrose diamond
  2. the remarks of Scott Aaronson on August 17, 2012
  3. the remarks in this journal on that same date
  4. the geometry of the 4×4 array in the context of M24.

Transgressing the Boundary

Filed under: General — Tags: — m759 @ 12:00 PM

The title refers not to the 1996 Sokal hoax (which has
Boundaries , plural, in the title), but to the boundary
discussed in Monday's Penrose diamond post

"Science is a differential equation.
Religion is a boundary condition."

Alan Turing in the epigraph to the
first chapter of a book by Terence Tao

From the Tao book, page 170—

"Typically the transformed solution extends to the
boundary of the Penrose diamond and beyond…."

Transgressing the boundary between science
and religion is the topic of a 1991 paper available
at JSTOR for $29.

For the Pope on Ash Wednesday:

"Think you might have access 
to this content via your library?" —JSTOR

See also Durkheim at Harvard.

Friday, February 1, 2013

Get Quotes

Filed under: General — Tags: , — m759 @ 4:01 PM

For Tony Kushner fans:

For logic fans:

IMAGE- NY Times market quotes, American Express Gold Card ad, Kevin Spacey in 'House of Cards' ad

John Searle on Derrida:

On necessity, possibility, and 'necessary possibility'

In the box-diamond notation, the axiom Searle quotes is

.

"The euclidean property guarantees the truth of this." — Wikipedia

Linking to Euclid

Clicking on "euclidean" above yields another Wikipedia article

"In mathematics, Euclidean relations are a class of binary relations that satisfy a weakened form of transitivity that formalizes Euclid's 'Common Notion 1' in The Elements : things which equal the same thing also equal one another."

Verification: See, for instance, slides on modal logic at Carnegie Mellon University and modal logic at plato.stanford.edu.

Tuesday, January 22, 2013

Raven Light

Filed under: General — Tags: , — m759 @ 11:40 AM

"…a fundamental cognitive ability known as 'fluid' intelligence: the capacity to solve novel problems, to learn, to reason, to see connections and to get to the bottom of things. …

…matrices are considered the gold standard of fluid-intelligence tests. Anyone who has taken an intelligence test has seen matrices like those used in the Raven’s: three rows, with three graphic items in each row, made up of squares, circles, dots or the like. Do the squares get larger as they move from left to right? Do the circles inside the squares fill in, changing from white to gray to black, as they go downward? One of the nine items is missing from the matrix, and the challenge is to find the underlying patterns— up, down and across— from six possible choices. Initially the solutions are readily apparent to most people, but they get progressively harder to discern. By the end of the test, most test takers are baffled."

— Dan Hurley, "Can You Make Yourself Smarter?," NY Times , April 18, 2012

See also "Raven Steals the Light" in this  journal.

Related material:

Plan 9 from MIT and, perhaps exemplifying crystallized  rather than fluid  intelligence, Black Diamond.

Thursday, November 1, 2012

For All Saints’ Day

Filed under: General — Tags: , — m759 @ 5:31 AM

Conclusion of "The Storyteller," a story 
by Cynthia Zarin about author Madeleine L'Engle—

The New Yorker , April 12, 2004 —

Note the black diamond at the story's end.

Monday, September 17, 2012

The Count

Filed under: General — Tags: , — m759 @ 11:01 PM
 

… I saw a shadow
sliding around the ropes
to get at me. The referee
moved it back, and then
went over and picked up the count.
"One!" The fog was clearing.

I rose to a knee,
and at "nine" to my feet.

— Louis Simpson, "The Appointment"

Simpson reportedly died on Holy Cross Day.

That day in this journal—

IMAGE- Log24 posts 'Please Mister Please' and 'Plan 9'

Thursday, June 7, 2012

Trinity Riddle

Filed under: General — Tags: — m759 @ 3:00 PM

IMAGE- Raven's Progressive Matrices problem, Bertrand Russell, the Mad Hatter, and the Hatter's riddle

Click image for some background.

Saturday, May 26, 2012

Talk Amongst Yourselves

Filed under: General,Geometry — Tags: , — m759 @ 3:33 PM

Hard Science Fiction weekend at Dragon Press Bookstore

Saturday May 26:
11am-noon Playing with the net up:
Hard Science Fiction in the era of
short attention spans, crowd-sourcing,
and rapid obsolescence
( Greg Benford, James Cambias, Kathryn Cramer)
….
3pm-4:30 Technological optimism and pessimism;
utopia and dystopia; happy endings & sad endings:
what do these oppositions have to do with one another?
Are they all the same thing? How are they different
from one another? Group discussion.

My own interests in this area include…

(Click image for some context)

IMAGE- 'The Stars My Destination' (with cover slightly changed)

    The above was adapted from a 1996 cover

IMAGE- PyrE on the 1996 Vintage Books cover of 'The Stars My Destination'

 Vintage Books, July 1996. Cover: Evan Gaffney.

For the significance of the flames, 
see PyrE in the book. For the significance
of the cube in the altered cover, see
The 2×2×2 Cube and The Diamond Archetype.

Sunday, April 1, 2012

The Palpatine Dimension

Filed under: General,Geometry — Tags: — m759 @ 9:00 AM

A physics quote relayed at Peter Woit's weblog today—

"The relation between 4D N=4 SYM and the 6D (2, 0) theory
is just like that between Darth Vader and the Emperor.
You see Darth Vader and you think 'Isn’t he just great?
How can anyone be greater than that? No way.'
Then you meet the Emperor."

— Arkani-Hamed

Some related material from this  weblog—

(See Big Apple and Columbia Film Theory)

http://www.log24.com/log/pix12/120108-Space_Time_Penrose_Hawking.jpg

The Meno Embedding:

http://www.log24.com/log/pix10B/101128-TheEmbedding.gif

Some related material from the Web—

IMAGE- The Penrose diamond and the Klein quadric

See also uses of the word triality  in mathematics. For instance…

A discussion of triality by Edward Witten

Triality is in some sense the last of the exceptional isomorphisms,
and the role of triality for n = 6  thus makes it plausible that n = 6
is the maximum dimension for superconformal symmetry,
though I will not give a proof here.

— "Conformal Field Theory in Four and Six Dimensions"

and a discussion by Peter J. Cameron

There are exactly two non-isomorphic ways
to partition the 4-subsets of a 9-set
into nine copies of AG( 3,2).
Both admit 2-transitive groups.

— "The Klein Quadric and Triality"

Exercise: Is Witten's triality related to Cameron's?
(For some historical background, see the triality  link from above
and Cameron's Klein Correspondence and Triality.)

Cameron applies his  triality to the pure geometry of a 9-set.
For a 9-set viewed in the context of physics, see A Beginning

From MIT Commencement Day, 2011—

A symbol related to Apollo, to nine, and to "nothing"

A minimalist favicon—

IMAGE- Generic 3x3 square as favicon

This miniature 3×3 square— http://log24.com/log/pix11A/110518-3x3favicon.ico — may, if one likes,
be viewed as the "nothing" present at the Creation. 
See Feb. 19, 2011, and Jim Holt on physics.

Happy April 1.

Saturday, February 18, 2012

Symmetry

Filed under: General,Geometry — m759 @ 7:35 PM

From the current Wikipedia article "Symmetry (physics)"—

"In physics, symmetry includes all features of a physical system that exhibit the property of symmetry—that is, under certain transformations, aspects of these systems are 'unchanged', according to a particular observation. A symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is 'preserved' under some change.

A family of particular transformations may be continuous  (such as rotation of a circle) or discrete  (e.g., reflection of a bilaterally symmetric figure, or rotation of a regular polygon). Continuous and discrete transformations give rise to corresponding types of symmetries. Continuous symmetries can be described by Lie groups while discrete symmetries are described by finite groups (see Symmetry group)."….

"A discrete symmetry is a symmetry that describes non-continuous changes in a system. For example, a square possesses discrete rotational symmetry, as only rotations by multiples of right angles will preserve the square's original appearance."

Note the confusion here between continuous (or discontinuous) transformations  and "continuous" (or "discontinuous," i.e. "discrete") groups .

This confusion may impede efforts to think clearly about some pure mathematics related to current physics— in particular, about the geometry of spaces made up of individual units ("points") that are not joined together in a continuous manifold.

For an attempt to forestall such confusion, see Noncontinuous Groups.

For related material, see Erlanger and Galois as well as the opening paragraphs of Diamond Theory

Symmetry is often described as invariance under a group of transformations. An unspoken assumption about symmetry in Euclidean 3-space is that the transformations involved are continuous.

Diamond theory rejects this assumption, and in so doing reveals that Euclidean symmetry may itself  be invariant under rather interesting groups of non-continuous (and a-symmetric) transformations. (These might be called noncontinuous  groups, as opposed to so-called discontinuous  (or discrete ) symmetry groups. See Weyl's Symmetry .)

For example, the affine group A on the 4-space over the 2-element field has a natural noncontinuous and asymmetric but symmetry-preserving action on the elements of a 4×4 array. (Details)

(Version first archived on March 27, 2002)

Update of Sunday, February 19, 2012—

The abuse of language by the anonymous authors
of the above Wikipedia article occurs also in more
reputable sources. For instance—

IMAGE- Brading and Castellani, 'Symmetries in Physics'- Four main sections include 'Continuous Symmetries' and 'Discrete Symmetries.'

Some transformations referred to by Brading and Castellani
and their editees as "discrete symmetries" are, in fact, as
linear transformations of continuous spaces, themselves
continuous  transformations.

This unfortunate abuse of language is at least made explicit
in a 2003 text, Mathematical Perspectives on Theoretical
Physics 
(Nirmala Prakash, Imperial College Press)—

"… associated[*] with any given symmetry there always exists
a continuous or a discrete group of transformations….
A symmetry whose associated group is continuous (discrete)
is called a continuous  (discrete ) symmetry ." — Pp. 235, 236

[* Associated how?]

Tuesday, January 10, 2012

Defining Form

Filed under: General,Geometry — Tags: , — m759 @ 9:00 AM

(Continued from Epiphany and from yesterday.)

Detail from the current American Mathematical Society homepage

http://www.log24.com/log/pix12/120110-AMS_page-Detail.jpg

Further detail, with a comparison to Dürer's magic square—

http://www.log24.com/log/pix12/120110-Donmoyer-Still-Life-Detail.jpg http://www.log24.com/log/pix12/120110-DurerSquare.jpg

The three interpenetrating planes in the foreground of Donmoyer's picture
provide a clue to the structure of the the magic square array behind them.

Group the 16 elements of Donmoyer's array into four 4-sets corresponding to the
four rows of Dürer's square, and apply the 4-color decomposition theorem.
Note the symmetry of the set of 3 line diagrams that result.

Now consider the 4-sets 1-4, 5-8, 9-12, and 13-16, and note that these
occupy the same positions in the Donmoyer square that 4-sets of
like elements occupy in the diamond-puzzle figure below—

http://www.log24.com/log/pix12/120110-DiamondPuzzleFigure.jpg

Thus the Donmoyer array also enjoys the structural  symmetry,
invariant under 322,560 transformations, of the diamond-puzzle figure.

Just as the decomposition theorem's interpenetrating lines  explain the structure
of a 4×4 square , the foreground's interpenetrating planes  explain the structure
of a 2x2x2 cube .

For an application to theology, recall that interpenetration  is a technical term
in that field, and see the following post from last year—

Saturday, June 25, 2011

 

Theology for Antichristmas

— m759 @ 12:00 PM

Hypostasis (philosophy)

"… the formula 'Three Hypostases  in one Ousia '
came to be everywhere accepted as an epitome
of the orthodox doctrine of the Holy Trinity.
This consensus, however, was not achieved
without some confusion…." —Wikipedia

http://www.log24.com/log/pix11A/110625-CubeHypostases.gif

Ousia

Click for further details:

http://www.log24.com/log/pix11A/110625-ProjectiveTrinitySm.jpg

 

Tuesday, May 10, 2011

Groups Acting

Filed under: General,Geometry — Tags: , — m759 @ 10:10 AM

The LA Times  on last weekend's film "Thor"—

"… the film… attempts to bridge director Kenneth Branagh's high-minded Shakespearean intentions with Marvel Entertainment's bottom-line-oriented need to crank out entertainment product."

Those averse to Nordic religion may contemplate a different approach to entertainment (such as Taymor's recent approach to Spider-Man).

A high-minded— if not Shakespearean— non-Nordic approach to groups acting—

"What was wrong? I had taken almost four semesters of algebra in college. I had read every page of Herstein, tried every exercise. Somehow, a message had been lost on me. Groups act . The elements of a group do not have to just sit there, abstract and implacable; they can do  things, they can 'produce changes.' In particular, groups arise naturally as the symmetries of a set with structure. And if a group is given abstractly, such as the fundamental group of a simplical complex or a presentation in terms of generators and relators, then it might be a good idea to find something for the group to act on, such as the universal covering space or a graph."

— Thomas W. Tucker, review of Lyndon's Groups and Geometry  in The American Mathematical Monthly , Vol. 94, No. 4 (April 1987), pp. 392-394

"Groups act "… For some examples, see

Related entertainment—

High-minded— Many Dimensions

Not so high-minded— The Cosmic Cube

http://www.log24.com/log/pix11A/110509-SpideySuperStories39Sm.jpg

One way of blending high and low—

The high-minded Charles Williams tells a story
in his novel Many Dimensions about a cosmically
significant cube inscribed with the Tetragrammaton—
the name, in Hebrew, of God.

The following figure can be interpreted as
the Hebrew letter Aleph inscribed in a 3×3 square—

http://www.log24.com/log/pix11A/110510-GaloisAleph.GIF

The above illustration is from undated software by Ed Pegg Jr.

For mathematical background, see a 1985 note, "Visualizing GL(2,p)."

For entertainment purposes, that note can be generalized from square to cube
(as Pegg does with his "GL(3,3)" software button).

For the Nordic-averse, some background on the Hebrew connection—

Saturday, January 22, 2011

High School Squares*

Filed under: General,Geometry — Tags: — m759 @ 1:20 AM

The following is from the weblog of a high school mathematics teacher—

http://www.log24.com/log/pix11/110121-LatinSquares4x4.jpg

This is related to the structure of the figure on the cover of the 1976 monograph Diamond Theory

http://www.log24.com/log/pix11/110122-DiamondTheoryCover.jpg

Each small square pattern on the cover is a Latin square,
with elements that are geometric figures rather than letters or numerals.
All order-four Latin squares are represented.

For a deeper look at the structure of such squares, let the high-school
chart above be labeled with the letters A through X, and apply the
four-color decomposition theorem.  The result is 24 structural diagrams—

    Click to enlarge

IMAGE- The Order-4 (4x4) Latin Squares

Some of the squares are structurally congruent under the group of 8 symmetries of the square.

This can be seen in the following regrouping—

   Click to enlarge

IMAGE- The Order-4 (4x4) Latin Squares, with Congruent Squares Adjacent

      (Image corrected on Jan. 25, 2011– "seven" replaced "eight.")

* Retitled "The Order-4 (i.e., 4×4) Latin Squares" in the copy at finitegeometry.org/sc.

Saturday, July 3, 2010

Beyond the Limits

Filed under: General,Geometry — Tags: — m759 @ 7:29 PM

"Human perception is a saga of created reality. But we were devising entities beyond the agreed-upon limits of recognition or interpretation…."

– Don DeLillo, Point Omega

Capitalized, the letter omega figures in the theology of two Jesuits, Teilhard de Chardin and Gerard Manley Hopkins. For the former, see a review of DeLillo. For the latter, see James Finn Cotter's Inscape  and "Hopkins and Augustine."

The lower-case omega is found in the standard symbolic representation of the Galois field GF(4)—

GF(4) = {0, 1, ω, ω2}

A representation of GF(4) that goes beyond the standard representation—

http://www.log24.com/log/pix10A/100703-Elements.gif

Here the four diagonally-divided two-color squares represent the four elements of GF(4).

The graphic properties of these design elements are closely related to the algebraic properties of GF(4).

This is demonstrated by a decomposition theorem used in the proof of the diamond theorem.

To what extent these theorems are part of "a saga of created reality" may be debated.

I prefer the Platonist's "discovered, not created" side of the debate.

Saturday, June 19, 2010

Imago Creationis

Filed under: General,Geometry — Tags: , , , , — m759 @ 6:00 PM

Image-- The Four-Diamond Tesseract

In the above view, four of the tesseract's 16
vertices are overlaid by other vertices.
For views that are more complete and
moveable, see Smith's tesseract page.

Four-Part Tesseract Divisions

http://www.log24.com/log/pix10A/100619-TesseractAnd4x4.gif

The above figure shows how four-part partitions
of the 16 vertices  of a tesseract in an infinite
Euclidean  space are related to four-part partitions
of the 16 points  in a finite Galois  space

Euclidean spaces versus Galois spaces
in a larger context—

 

 


Infinite versus Finite

The central aim of Western religion —

"Each of us has something to offer the Creator...
the bridging of
                 masculine and feminine,
                      life and death.
It's redemption.... nothing else matters."
-- Martha Cooley in The Archivist  (1998)

The central aim of Western philosophy —

              Dualities of Pythagoras
              as reconstructed by Aristotle:
                 Limited     Unlimited
                     Odd     Even
                    Male     Female
                   Light      Dark
                Straight    Curved
                  ... and so on ....

"Of these dualities, the first is the most important; all the others may be seen as different aspects of this fundamental dichotomy. To establish a rational and consistent relationship between the limited [man, etc.] and the unlimited [the cosmos, etc.] is… the central aim of all Western philosophy."
— Jamie James in The Music of the Spheres  (1993)

Another picture related to philosophy and religion—

Jung's Four-Diamond Figure from Aion

http://www.log24.com/log/pix10A/100615-JungImago.gif

This figure was devised by Jung
to represent the Self. Compare the
remarks of Paul Valéry on the Self—

Flight from Eden: The Origins of Modern Literary Criticism and Theory, by Steven Cassedy, U. of California Press, 1990, pages 156-157—

 

 

Valéry saw the mind as essentially a relational system whose operation he attempted to describe in the language of group mathematics. "Every act of understanding is based on a group," he says (C, 1:331). "My specialty— reducing everything to the study of a system closed on itself and finite" (C, 19: 645). The transformation model came into play, too. At each moment of mental life the mind is like a group, or relational system, but since mental life is continuous over time, one "group" undergoes a "transformation" and becomes a different group in the next moment. If the mind is constantly being transformed, how do we account for the continuity of the self? Simple; by invoking the notion of the invariant. And so we find passages like this one: "The S[elf] is invariant, origin, locus or field, it's a functional property of consciousness" (C, 15:170 [2:315]). Just as in transformational geometry, something remains fixed in all the projective transformations of the mind's momentary systems, and that something is the Self (le Moi, or just M, as Valéry notates it so that it will look like an algebraic variable). Transformation theory is all over the place. "Mathematical science…  reduced to algebra, that is, to the analysis of the transformations of a purely differential being made up of homogeneous elements, is the most faithful document of the properties of grouping, disjunction, and variation in the mind" (O, 1:36). "Psychology is a theory of transformations, we just need to isolate the invariants and the groups" (C, 1:915). "Man is a system that transforms itself" (C, 2:896).

Notes:

  Paul Valéry, Oeuvres  (Paris: Pléiade, 1957-60)

C   Valéry, Cahiers, 29 vols. (Paris: Centre National de le Recherche Scientifique, 1957-61)

Note also the remarks of George David Birkhoff at Rice University
in 1940 (pdf) on Galois's theory of groups and the related
"theory of ambiguity" in Galois's testamentary letter—

… metaphysical reasoning always relies on the Principle of Sufficient Reason, and… the true meaning of this Principle is to be found in the “Theory of Ambiguity” and in the associated mathematical “Theory of Groups.”

If I were a Leibnizian mystic, believing in his “preestablished harmony,” and the “best possible world” so satirized by Voltaire in “Candide,” I would say that the metaphysical importance of the Principle of Sufficient Reason and the cognate Theory of Groups arises from the fact that God thinks multi-dimensionally* whereas men can only think in linear syllogistic series, and the Theory of Groups is the appropriate instrument of thought to remedy our deficiency in this respect.

* That is, uses multi-dimensional symbols beyond our grasp.

Related material:

Imago Creationis

A medal designed by Leibniz to show how
binary arithmetic mirrors the creation by God
of something (1) from nothing (0).

http://www.log24.com/log/pix10A/100618-LeibnizMedaille.jpg

Another array of 16 strings of 0's and 1's, this time
regarded as coordinates rather than binary numbers—

Frame of Reference

http://www.log24.com/log/pix10A/100619-ReferenceFrame.gif

The Diamond Theorem

http://www.log24.com/log/pix10A/100619-Dtheorem.gif

Some context by a British mathematician —

http://www.log24.com/log/pix10A/100619-Cameron.gif

Imago

by Wallace Stevens

Who can pick up the weight of Britain, 
Who can move the German load 
Or say to the French here is France again? 
Imago. Imago. Imago. 

It is nothing, no great thing, nor man 
Of ten brilliancies of battered gold 
And fortunate stone. It moves its parade 
Of motions in the mind and heart, 

A gorgeous fortitude. Medium man 
In February hears the imagination's hymns 
And sees its images, its motions 
And multitudes of motions 

And feels the imagination's mercies, 
In a season more than sun and south wind, 
Something returning from a deeper quarter, 
A glacier running through delirium, 

Making this heavy rock a place, 
Which is not of our lives composed . . . 
Lightly and lightly, O my land, 
Move lightly through the air again.

Tuesday, June 15, 2010

Imago, Imago, Imago

Filed under: General,Geometry — Tags: , , , — m759 @ 11:07 AM

Recommended— an online book—

Flight from Eden: The Origins of Modern Literary Criticism and Theory,
by Steven Cassedy, U. of California Press, 1990.

See in particular

Valéry and the Discourse On His Method.

Pages 156-157—

Valéry saw the mind as essentially a relational system whose operation he attempted to describe in the language of group mathematics. “Every act of understanding is based on a group,” he says (C, 1:331). “My specialty—reducing everything to the study of a system closed on itself and finite” (C, 19: 645). The transformation model came into play, too. At each moment of mental life the mind is like a group, or relational system, but since mental life is continuous over time, one “group” undergoes a “transformation” and becomes a different group in the next moment. If the mind is constantly being transformed, how do we account for the continuity of the self? Simple; by invoking the notion of the invariant. And so we find passages like this one: “The S[elf] is invariant, origin, locus or field, it’s a functional property of consciousness” (C, 15:170 [2: 315]). Just as in transformational geometry, something remains fixed in all the projective transformations of the mind’s momentary systems, and that something is the Self (le Moi, or just M, as Valéry notates it so that it will look like an algebraic variable). Transformation theory is all over the place. “Mathematical science . . . reduced to algebra, that is, to the analysis of the transformations of a purely differential being made up of homogeneous elements, is the most faithful document of the properties of grouping, disjunction, and variation in the mind” (O, 1:36). “Psychology is a theory of transformations, we just need to isolate the invariants and the groups” (C, 1:915). “Man is a system that transforms itself” (C, 2:896).

Notes:

  Paul Valéry, Oeuvres (Paris: Pléiade, 1957-60)

C   Valéry, Cahiers, 29 vols. (Paris: Centre National de le Recherche Scientifique, 1957-61)

Compare Jung’s image in Aion  of the Self as a four-diamond figure:

http://www.log24.com/log/pix10A/100615-JungImago.gif

and Cullinane’s purely geometric four-diamond figure:

http://www.log24.com/log/pix10A/100615-FourD.gif

For a natural group of 322,560 transformations acting on the latter figure, see the diamond theorem.

What remains fixed (globally, not pointwise) under these transformations is the system  of points and hyperplanes from the diamond theorem. This system was depicted by artist Josefine Lyche in her installation “Theme and Variations” in Oslo in 2009.  Lyche titled this part of her installation “The Smallest Perfect Universe,” a phrase used earlier by Burkard Polster to describe the projective 3-space PG(3,2) that contains these points (at right below) and hyperplanes (at left below).

Image-- Josefine Lyche's combination of Polster's phrase with<br /> Cullinane's images in her gallery show, Oslo, 2009-- 'The Smallest<br /> Perfect Universe -- Points and Hyperplanes'

Although the system of points (at right above) and hyperplanes (at left above) exemplifies Valéry’s notion of invariant, it seems unlikely to be the sort of thing he had in mind as an image of the Self.

Tuesday, March 17, 2009

Tuesday March 17, 2009

Filed under: General,Geometry — m759 @ 11:07 AM
Deep Structures

The traditional 'Square of Opposition'

The Square of Oppositon
at Stanford Encylopedia of Philosophy


The Square of Opposition diagram in its earliest known form

The Square of Opposition
in its original form

"The diagram above is from a ninth century manuscript of Apuleius' commentary on Aristotle's Perihermaneias, probably one of the oldest surviving pictures of the square."

Edward Buckner at The Logic Museum

From the webpage "Semiotics for Beginners: Paradigmatic Analysis," by Daniel Chandler:
 

The Semiotic Square of Greimas

The Semiotic Square

"The structuralist semiotician Algirdas Greimas introduced the semiotic square (which he adapted from the 'logical square' of scholastic philosophy) as a means of analysing paired concepts more fully (Greimas 1987,* xiv, 49). The semiotic square is intended to map the logical conjunctions and disjunctions relating key semantic features in a text. Fredric Jameson notes that 'the entire mechanism… is capable of generating at least ten conceivable positions out of a rudimentary binary opposition' (in Greimas 1987,* xiv). Whilst this suggests that the possibilities for signification in a semiotic system are richer than the either/or of binary logic, but that [sic] they are nevertheless subject to 'semiotic constraints' – 'deep structures' providing basic axes of signification."

* Greimas, Algirdas (1987): On Meaning: Selected Writings in Semiotic Theory (trans. Paul J Perron & Frank H Collins). London: Frances Pinter

Another version of the semiotic square:
 

Rosalind Krauss's version of the semiotic square, which she calls the Klein group

Krauss says that her figure "is, of course, a Klein Group."

Here is a more explicit figure representing the Klein group:

The Klein Four-Group, illustration by Steven H. Cullinane

There is also the logical
    diamond of opposition

The Diamond of Opposition (figure from Wikipedia)

A semiotic (as opposed to logical)
diamond has been used to illustrate
remarks by Fredric Jameson,
 a Marxist literary theorist:

"Introduction to Algirdas Greimas, Module on the Semiotic Square," by Dino Felluga at Purdue University–

The semiotic square has proven to be an influential concept not only in narrative theory but in the ideological criticism of Fredric Jameson, who uses the square as "a virtual map of conceptual closure, or better still, of the closure of ideology itself" ("Foreword"* xv). (For more on Jameson, see the [Purdue University] Jameson module on ideology.)

Greimas' schema is useful since it illustrates the full complexity of any given semantic term (seme). Greimas points out that any given seme entails its opposite or "contrary." "Life" (s1) for example is understood in relation to its contrary, "death" (s2). Rather than rest at this simple binary opposition (S), however, Greimas points out that the opposition, "life" and "death," suggests what Greimas terms a contradictory pair (-S), i.e., "not-life" (-s1) and "not-death" (-s2). We would therefore be left with the following semiotic square (Fig. 1):

A semiotic 'diamond of opposition'

As Jameson explains in the Foreword to Greimas' On Meaning, "-s1 and -s2"—which in this example are taken up by "not-death" and "not-life"—"are the simple negatives of the two dominant terms, but include far more than either: thus 'nonwhite' includes more than 'black,' 'nonmale' more than 'female'" (xiv); in our example, not-life would include more than merely death and not-death more than life.

* Jameson, Fredric. "Foreword." On Meaning: Selected Writings in Semiotic Theory. By Algirdas Greimas. Trans. Paul J. Perron and Frank H. Collins. Minneapolis: U of Minnesota P, 1976.

"The Game in the Ship cannot be approached as a job, a vocation, a career, or a recreation. To the contrary, it is Life and Death itself at work there. In the Inner Game, we call the Game Dhum Welur, the Mind of God."

The Gameplayers of Zan, by M.A. Foster

"For every kind of vampire,
there is a kind of cross."
— Thomas Pynchon,
 Gravity's Rainbow

Crosses used by semioticians
to baffle their opponents
are illustrated above.

Some other kinds of crosses,
and another kind of opponent:

Monday, July 11, 2005

Logos
for St. Benedict's Day

Click on either of the logos below for religious meditations– on the left, a Jewish meditation from the Conference of Catholic Bishops; on the right, an Aryan meditation from Stormfront.org.

Logo of Conference of Catholic Bishops     Logo of Stormfront website

Both logos represent different embodiments of the "story theory" of truth, as opposed to the "diamond theory" of truth.  Both logos claim, in their own ways, to represent the eternal Logos of the Christian religion.  I personally prefer the "diamond theory" of truth, represented by the logo below.

Illustration of the 2x2 case of the diamond theorem

See also the previous entry
(below) and the entries
  of 7/11, 2003.
 

Sunday, July 10, 2005

Mathematics
and Narrative

 
Click on the title
for a narrative about

Nikolaos K. Artemiadis

Nikolaos K. Artemiadis,
 (co-) author of

Artemiadis's 'History of Mathematics,' published by the American Mathematical Society
 

From Artemiadis's website:
1986: Elected Regular Member
of the Academy of Athens
1999: Vice President
of the Academy of Athens
2000: President
of the Academy of Athens
Seal of the American Mathematical Society with picture of Plato's Academy

"First of all, I'd like to
   thank the Academy…"

— Remark attributed to Plato

Monday, March 16, 2009

Monday March 16, 2009

Filed under: General — m759 @ 8:00 PM
Damnation Morning
continued

Annals of Prose Style

  Film Review

“No offense to either of them, but ‘Georgia Rule’ suggests an Ingmar Bergman script as directed by Jerry Lewis. The subject matter is grim, the relationships are gnarled, the worldview is bleak, and, at any given moment, you suspect someone’s going to be hit with a pie.” –John Anderson at Variety.com, May 8, 2007

Sounds perfect to me.


Through a Glass Darkly

“Preserving a strict unity of time and place, this stark tale of a young woman’s decline into insanity is set in a summer home on a holiday island. It is the first part of the trilogy

Bergman's trilogy including 'Through a Glass Darkly'

that comprises Winter Light and The Silence, films which are generally seen as addressing Bergman’s increasing disillusionment with the emotional coldness of his inherited Lutheran religion. In particular here, Bergman focuses on the absence of familial love which might perhaps have pulled Karin (Andersson) back from the brink; while Karin’s mental disintegration manifests itself in the belief that God is a spider. As she slips inexorably into madness, she is observed with terrifying objectivity by her emotionally paralyzed father (Björnstrand) and seemingly helpless husband (von Sydow).”

— Nigel Floyd, Time Out, quoted at Bergmanorama

Related material:

1. The “spider” symbol of Fritz Leiber’s short story “Damnation Morning“–

Fritz Leiber's 'spider' figure

2. The Illuminati Diamond of Hollywood’s “Angels & Demons” (to open May 15), and

3. The following diagram by one “John Opsopaus“–

Elemental square by John Opsopaus from 'The Rotation of the Elements'

Sunday, March 15, 2009

Sunday March 15, 2009

Filed under: General,Geometry — Tags: — m759 @ 11:00 AM

Angels, Demons,
"Symbology"

"On Monday morning, 9 March, after visiting the Mayor of Rome and the Municipal Council on the Capitoline Hill, the Holy Father spoke to the Romans who gathered in the square outside the Senatorial Palace…

'… a verse by Ovid, the great Latin poet, springs to mind. In one of his elegies he encouraged the Romans of his time with these words:

"Perfer et obdura: multo graviora tulisti."

 "Hold out and persist:
  you have got through
  far more difficult situations."

 (Tristia, Liber  V, Elegia  XI, verse 7).'"

This journal
on 9 March:

Diamond Theory version of 'The Square Inch Space' with yin-yang symbol for comparison

Note the color-interchange
symmetry
of each symbol
under 180-degree rotation.

Related material:
The Illuminati Diamond:

IMAGE- Illuminati Diamond, pp. 359-360 in 'Angels & Demons,' Simon & Schuster Pocket Books 2005, 448 pages, ISBN 0743412397

Dan Brown's novel Angels & Demons introduced in the year 2000 the fictional academic discipline of "symbology" and a fictional Harvard professor of that discipline, Robert Langdon (named after ambigram* artist John Langdon).

Fictional Harvard professor of symbology Robert Langdon, as portrayed by Tom Hanks

Tom Hanks as Robert Langdon


A possible source for Brown's term "symbology" is a 1995 web page, "The Rotation of the Elements," by one "John Opsopaus." (Cf. Art History Club.)

"The four qualities are the key to understanding the rotation of the elements and many other applications of the symbology of the four elements." –John Opsopaus

* "…ambigrams were common in symbology…." —Angels & Demons
 

Monday, March 9, 2009

Monday March 9, 2009

Filed under: General,Geometry — Tags: — m759 @ 12:00 PM

Humorism

'The Manchurian Candidate' campaign button

"Always with a
little humor."
Dr. Yen Lo  

Diamond diagram of the four humors, the four qualities, the four elements, the four seasons, and four colors

From Temperament: A Brief Survey

For other interpretations
of the above shape, see
The Illuminati Diamond.

from Jung's Aion:

"From the circle and quaternity motif is derived the symbol of the geometrically formed crystal and the wonder-working stone. From here analogy formation leads on to the city, castle, church, house, room, and vessel. Another variant is the wheel. The former motif emphasizes the ego’s containment in the greater dimension of the self; the latter emphasizes the rotation which also appears as a ritual circumambulation. Psychologically, it denotes concentration on and preoccupation with a centre…." –Jung, Collected Works, Vol. 9, Part II, paragraph 352

As for rotation, see the ambigrams in Dan Brown's Angels & Demons (to appear as a film May 15) and the following figures:

Diamond Theory version of 'The Square Inch Space' with yin-yang symbol for comparison
 
Click on image
for a related puzzle.
For a solution, see
 The Diamond Theorem.

A related note on
"Angels & Demons"
director Ron Howard:

Director Ron Howard with illustration of the fictional discipline 'symbology'
 
Click image for details.

Saturday, March 7, 2009

Saturday March 7, 2009

Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

One or Two Ideas
 
Today's birthday: Piet Mondrian
 
From James Joyce's A Portrait of the Artist as a Young Man:

he hearth and began to stroke his chin.

–When may we expect to have something from you on the esthetic question? he asked.

–From me! said Stephen in astonishment. I stumble on an idea once a fortnight if I am lucky.

–These questions are very profound, Mr Dedalus, said the dean. It is like looking down from the cliffs of Moher into the depths. Many go down into the depths and never come up. Only the trained diver can go down into those depths and explore them and come to the surface again.

–If you mean speculation, sir, said Stephen, I also am sure that there is no such thing as free thinking inasmuch as all thinking must be bound by its own laws.

–Ha!

–For my purpose I can work on at present by the light of one or two ideas of Aristotle and Aquinas.

–I see. I quite see your point.

Besides being Mondrian's birthday, today is also the dies natalis (in the birth-into-heaven sense) of St. Thomas Aquinas and, for those who believe worthy pre-Christians also enter heaven, possibly of Aristotle.

Pope Benedict XVI explained the dies natalis concept on Dec. 26, 2006:

"For believers the day of death, and even more the day of martyrdom, is not the end of all; rather, it is the 'transit' towards immortal life. It is the day of definitive birth, in Latin, dies natalis."

The Pope's remarks on that date
were in St. Peter's Square.
 
From this journal on that date,
a different square —
 
The Seventh Symbol:
 

Box symbol

Pictorial version
of Hexagram 20,
Contemplation (View)

The square may be regarded as
symbolizing art itself.
(See Nov.30 – Dec.1, 2008.)

In honor of
Aristotle and Aquinas,
here is a new web site,
illuminati-diamond.com,
with versions of the diamond shape
made famous by Mondrian

Cover of  Mondrian: The Diamond Compositions

— a shape symbolizing
possibility within modal logic
 as well as the potentiality of
 Aristotle's prima materia.

Friday, March 6, 2009

Friday March 6, 2009

Filed under: General,Geometry — Tags: — m759 @ 7:30 PM
The Illuminati Stone

TV listing for this evening —
Family Channel, 7:30 PM:

"Harry Potter and
  the Sorcerer's Stone"

In other entertainment news —
Scheduled to open May 15:

IMAGE- Illuminati Diamond, pp. 359-360 in 'Angels & Demons,' Simon & Schuster Pocket Books 2005, 448 pages, ISBN 0743412397

"Only gradually did I discover
what the mandala really is:
'Formation, Transformation,
Eternal Mind's eternal recreation'"
(Faust, Part Two)

Carl Gustav Jung  

Related material:

"For just about half a century, E.J. Holmyard's concisely-titled Alchemy has served as a literate, well-informed, and charming introduction to the history and literature of Western alchemy." —Ian Myles Slater

From 'Alchemy,' by Holmyard, the diamond of Aristotle's 4 elements and 4 qualities

For more about this
"prime matter" (prima materia)
see The Diamond Archetype

The Diamond Cross

and Holy the Firm.

 

Background:

Holmyard —

'Alchemy,' by Holmyard, back cover of Dover edition

— and Aristotle's
On Generation and Corruption.

Monday, March 2, 2009

Monday March 2, 2009

Filed under: General,Geometry — Tags: — m759 @ 11:30 AM
Joyce's Nightmare
continues

Today in History – March 2

Today is Monday, March 2, the 61st day of 2009. There are 304 days left in the year.

Today's Highlight in History:

On March 2, 1939, Roman Catholic Cardinal Eugenio Pacelli was elected Pope on his 63rd birthday; he took the name Pius XII.

IMAGE- Illuminati Diamond, pp. 359-360 in 'Angels & Demons,' Simon & Schuster Pocket Books 2005, 448 pages, ISBN 0743412397

 

Log24 on June 9, 2008

From Gravity's Rainbow (Penguin Classics, 1995), page 563:

"He brings out the mandala he found.
'What's it mean?'
[….]

Slothrop gives him the mandala. He hopes it will work like the mantra that Enzian told him once, mba-kayere (I am passed over), mba-kayere… a spell […]. A mezuzah. Safe passage through a bad night…."

 

In lieu of Slothrop's mandala, here is another…

Christ and the four elements, 1495
 

Christ and the Four Elements

This 1495 image is found in
The Janus Faces of Genius:
The Role of Alchemy
in Newton's Thought,
by B. J. T. Dobbs,
Cambridge University Press,
2002, p. 85


Related mandalas:Diamond arrangement of the four elements
and

Logo by Steven H. Cullinane for website on finite geometry

For further details,
click on any of the
three mandalas above.

 

Angels and Demons cross within a diamond (page 306), and Finite Geometry logo

Happy birthday to
Tom Wolfe, author of
The Painted Word.
 

Monday, December 29, 2008

Monday December 29, 2008

Filed under: General — Tags: — m759 @ 12:21 PM
The Gift
 

Plato's Diamond

Robert Stone,
A Flag for Sunrise:

"'That old Jew gave me this here.' Egan looked at the diamond. 'I ain't giving this to you, understand? The old man gave it to me for my boy. It's worth a whole lot of money– you can tell that just by looking– but it means something, I think. It's got a meaning, like.'

'Let's see,' Egan said, 'what would it mean?' He took hold of Pablo's hand cupping the stone and held his own hand under it. '"The jewel is in the lotus," perhaps that's what it means. The eternal in the temporal. The Boddhisattva declining nirvana out of compassion. Contemplating the ignorance of you and me, eh? That's a metaphor of our Buddhist friends.'

Pablo's eyes glazed over. 'Holy shit,' he said. 'Santa Maria.' He stared at the diamond in his palm with passion."

For further details, click on the diamond.

 

Related narratives:

Today's online Times on
the Saturday, Dec. 27,
death of an artist:

Robert Graham obituary, NY Times, 12/29/08

"Dale Wasserman… the playwright responsible for two Broadway hits of the 1960s, 'One Flew Over the Cuckoo’s Nest' and 'Man of La Mancha,' died on Sunday [December 21, 2008] at his home in Paradise Valley, Ariz., near Phoenix….

Mr. Wasserman wrote more than 75 scripts for television, the stage and the movies, including screenplays for 'The Vikings' (1958), a seafaring epic with Tony Curtis and Kirk Douglas, and 'A Walk With Love and Death' (1969), a John Huston film set in 14th-century Europe….

He feuded with… John Huston, who gave the lead female role in 'Walk' to his teenage daughter, Anjelica, against Mr. Wasserman's wishes. And he never attended ceremonies to receive the awards he won."

Accepting for Mr. Wasserman:
Mr. Graham's widow,
Anjelica Huston

Anjelica Huston and Jack Nicholson

"Well…"

Wednesday, June 25, 2008

Wednesday June 25, 2008

Filed under: General,Geometry — Tags: — m759 @ 7:20 PM
The Cycle of
the Elements

John Baez, Week 266
(June 20, 2008):

“The Renaissance thinkers liked to
organize the four elements using
a chain of analogies running
from light to heavy:

fire : air :: air : water :: water : earth

They also organized them
in a diamond, like this:”

Diamond of the four ancient elements, figure by John Baez

This figure of Baez
is related to a saying
attributed to Heraclitus:

Diamond  showing transformation of the four ancient elements

For related thoughts by Jung,
see Aion, which contains the
following diagram:

Jung's four-diamond figure showing transformations of the self as Imago Dei

“The formula reproduces exactly the essential features of the symbolic process of transformation. It shows the rotation of the mandala, the antithetical play of complementary (or compensatory) processes, then the apocatastasis, i.e., the restoration of an original state of wholeness, which the alchemists expressed through the symbol of the uroboros, and finally the formula repeats the ancient alchemical tetrameria, which is implicit in the fourfold structure of unity.”

— Carl Gustav Jung

That the words Maximus of Tyre (second century A.D.) attributed to Heraclitus imply a cycle of the elements (analogous to the rotation in Jung’s diagram) is not a new concept. For further details, see “The Rotation of the Elements,” a 1995 webpage by one  “John Opsopaus.”

Related material:

Log24 entries of June 9, 2008, and

Quintessence: A Glass Bead Game,”
by Charles Cameron.

Monday, June 9, 2008

Monday June 9, 2008

Filed under: General,Geometry — m759 @ 10:20 PM
Lying Rhymes

Readers of the previous entry
who wish to practice their pardes
may contemplate the following:

NY Lottery June 9, 2008: mid-day 007, evening 563

 
The evening 563 may, as in other recent entries, be interpreted as a page number in Gravity’s Rainbow (Penguin Classics, 1995). From that page:

“He brings out the mandala he found.
‘What’s it mean?’
[….]
Slothrop gives him the mandala. He hopes it will work like the mantra that Enzian told him once, mba-kayere (I am passed over), mba-kayere… a spell […]. A mezuzah. Safe passage through a bad night….”

In lieu of Slothrop’s mandala, here
is another, from the Dante link
in today’s previous entry:

Christ and the four elements, 1495

Christ and the Four Elements

This 1495 image is found in
The Janus Faces of Genius:
The Role of Alchemy

in Newton’s Thought,
by B. J. T. Dobbs,
Cambridge University Press,
2002, p. 85


Related mandalas:

Diamond arrangement of the four elements

and

Logo by Steven H. Cullinane for website on finite geometry

For further details,
click on any of the
three mandalas above.

“For every kind of vampire,
there is a kind of cross.”

— Thomas Pynchon, quoted
here on 9/13, 2007

(As for today’s New York Lottery midday number 007, see (for instance) Edward Rothstein in today’s New York Times on paradise, and also Tom Stoppard on heaven as “just a lying rhyme” for seven.)

Time of entry: 10:20:55 PM

Tuesday, April 29, 2008

Tuesday April 29, 2008

Filed under: General,Geometry — Tags: , , — m759 @ 11:09 AM
Sacerdotal Jargon
at Harvard:

Thomas Wolfe

Thomas Wolfe
(Harvard M.A., 1922)

versus

Rosalind Krauss

Rosalind Krauss
(Harvard M.A., 1964,
Ph.D., 1969)

on

The Kernel of Eternity

"No culture has a pact with eternity."
George Steiner, interview in  
The Guardian of April 19

"At that instant he saw,
in one blaze of light, an image
of unutterable conviction….
the core of life, the essential
pattern whence all other things
proceed, the kernel of eternity."

— Thomas Wolfe, Of Time
and the River, quoted in
Log24 on June 9, 2005

 

From today's online Harvard Crimson:

"… under the leadership of Faust,
Harvard students should look forward
to an ever-growing opportunity for
international experience
and artistic endeavor."

 

Wolfgang Pauli as Mephistopheles

Pauli as Mephistopheles
in a 1932 parody of
Goethe's
Faust at Niels Bohr's
institute in Copenhagen

From a recent book
on Wolfgang Pauli,
The Innermost Kernel:

Pauli's Dream Square (square plus the two diagonals)

A belated happy birthday
to the late
Felix Christian Klein
  (born on April 25) —

The Klein Group: The four elements in four colors, with black points representing the identity

Another Harvard figure quoted here on Dec. 5, 2002:

"The theory of poetry, that is to say, the total of the theories of poetry, often seems to become in time a mystical theology or, more simply, a mystique. The reason for this must by now be clear. The reason is the same reason why the pictures in a museum of modern art often seem to become in time a mystical aesthetic, a prodigious search of appearance, as if to find a way of saying and of establishing that all things, whether below or above appearance, are one and that it is only through reality, in which they are reflected or, it may be, joined together, that we can reach them. Under such stress, reality changes from substance to subtlety, a subtlety in which it was natural for Cézanne to say: 'I see planes bestriding each other and sometimes straight lines seem to me to fall' or 'Planes in color…. The colored area where shimmer the souls of the planes, in the blaze of the kindled prism, the meeting of planes in the sunlight.' The conversion of our Lumpenwelt went far beyond this. It was from the point of view of another subtlety that Klee could write: 'But he is one chosen that today comes near to the secret places where original law fosters all evolution. And what artist would not establish himself there where the organic center of all movement in time and space– which he calls the mind or heart of creation– determines every function.' Conceding that this sounds a bit like sacerdotal jargon, that is not too much to allow to those that have helped to create a new reality, a modern reality, since what has been created is nothing less."

— Wallace Stevens, Harvard College Class of 1901, "The Relations between Poetry and Painting" in The Necessary Angel (Knopf, 1951)

From a review of Rosalind Krauss's The Optical Unconscious  (MIT Press hardcover, 1993):

Krauss is concerned to present Modernism less in terms of its history than its structure, which she seeks to represent by means of a kind of diagram: "It is more interesting to think of modernism as a graph or table than a history." The "table" is a square with diagonally connected corners, of the kind most likely to be familiar to readers as the Square of Opposition, found in elementary logic texts since the mid-19th century. The square, as Krauss sees it, defines a kind of idealized space "within which to work out unbearable contradictions produced within the real field of history." This she calls, using the inevitable gallicism, "the site of Jameson's Political Unconscious" and then, in art, the optical unconscious, which consists of what Utopian Modernism had to kick downstairs, to repress, to "evacuate… from its field."

— Arthur C. Danto in ArtForum, Summer 1993

Rosalind Kraus in The Optical Unconscious (MIT Press paperback, 1994):

For a presentation of the Klein Group, see Marc Barbut, "On the Meaning of the Word 'Structure' in Mathematics," in Introduction to Structuralism, ed. Michael Lane (New York: Basic Books, 1970). Claude Lévi-Strauss uses the Klein group in his analysis of the relation between Kwakiutl and Salish masks in The Way of the Masks, trans. Sylvia Modelski (Seattle: University of Washington Press, 1982), p. 125; and in relation to the Oedipus myth in "The Structural Analysis of Myth," Structural Anthropology, trans. Claire Jackobson [sic] and Brooke Grundfest Schoepf (New York: Basic Books, 1963). In a transformation of the Klein Group, A. J. Greimas has developed the semiotic square, which he describes as giving "a slightly different formulation to the same structure," in "The Interaction of Semiotic Constraints," On Meaning (Minneapolis: University of Minnesota Press, 1987), p. 50. Jameson uses the semiotic square in The Political Unconscious (see pp. 167, 254, 256, 277) [Fredric Jameson, The Political Unconscious: Narrative as a Socially Symbolic Act (Ithaca: Cornell University Press, 1981)], as does Louis Marin in "Disneyland: A Degenerate Utopia," Glyph, no. 1 (1977), p. 64.

For related non-sacerdotal jargon, see…
 

Wikipedia on the Klein group (denoted V, for Vierergruppe):

In this representation, V is a normal subgroup of the alternating group A4 (and also the symmetric group S4) on 4 letters. In fact, it is the kernel of a surjective map from S4 to S3. According to Galois theory, the existence of the Klein four-group (and in particular, this representation of it) explains the existence of the formula for calculating the roots of quartic equations in terms of radicals.

For radicals of another sort, see A Logocentric Meditation, A Mass for Lucero, and [update of 7 PM] Steven Erlanger in today's New York Times— "France Still Divided Over Lessons of 1968 Unrest."

For material related to Klee's phrase mentioned above by Stevens, "the organic center of all movement in time and space," see the following Google search:

April 29, 2008, Google search on 'penrose space time'

Click on the above
 image for details.

See also yesterday's
Religious Art.

Saturday, April 19, 2008

Saturday April 19, 2008

Filed under: General,Geometry — m759 @ 5:01 AM
A Midrash for Benedict

On April 16, the Pope’s birthday, the evening lottery number in Pennsylvania was 441. The Log24 entries of April 17 and April 18 supplied commentaries based on 441’s incarnation as a page number in an edition of Heidegger’s writings.  Here is a related commentary on a different incarnation of 441.  (For a context that includes both today’s commentary and those of April 17 and 18, see Gian-Carlo Rota– a Heidegger scholar as well as a mathematician– on mathematical Lichtung.)

From R. D. Carmichael, Introduction to the Theory of Groups of Finite Order (Boston, Ginn and Co., 1937)– an exercise from the final page, 441, of the final chapter, “Tactical Configurations”–

“23. Let G be a multiply transitive group of degree n whose degree of transitivity is k; and let G have the property that a set S of m elements exists in G such that when k of the elements S are changed by a permutation of G into k of these elements, then all these m elements are permuted among themselves; moreover, let G have the property P, namely, that the identity is the only element in G which leaves fixed the nm elements not in S.  Then show that G permutes the m elements S into

n(n -1) … (nk + 1)
____________________

m(m – 1) … (mk + 1)

sets of m elements each, these sets forming a configuration having the property that any (whatever) set of k elements appears in one and just one of these sets of m elements each. Discuss necessary conditions on m, n, k in order that the foregoing conditions may be realized. Exhibit groups illustrating the theorem.”

This exercise concerns an important mathematical structure said to have been discovered independently by the American Carmichael and by the German Ernst Witt.

For some perhaps more comprehensible material from the preceding page in Carmichael– 440– see Diamond Theory in 1937.

Sunday, August 12, 2007

Sunday August 12, 2007

Filed under: General,Geometry — Tags: — m759 @ 9:00 AM

The Geometry of Qubits

In the context of quantum information theory, the following structure seems to be of interest–

"… the full two-by-two matrix ring with entries in GF(2), M2(GF(2))– the unique simple non-commutative ring of order 16 featuring six units (invertible elements) and ten zero-divisors."

— "Geometry of Two-Qubits," by Metod Saniga (pdf, 17 pp.), Jan. 25, 2007

A 16-element affine space and a corresponding 16-element matrix ring

This ring is another way of looking at the 16 elements of the affine space A4(GF(2)) over the 2-element field.  (Arrange the four coordinates of each element– 1's and 0's– into a square instead of a straight line, and regard the resulting squares as matrices.)  (For more on A4(GF(2)), see Finite Relativity and related notes at Finite Geometry of the Square and Cube.)  Using the above ring, Saniga constructs a system of 35 objects (not unlike the 35 lines of the finite geometry PG(3,2)) that he calls a "projective line" over the ring.  This system of 35 objects has a subconfiguration isomorphic to the (2,2) generalized quadrangle W2 (which occurs naturally as a subconfiguration of PG(3,2)– see Inscapes.)

Saniga concludes:
 

"We have demonstrated that the basic properties of a system of two interacting spin-1/2 particles are uniquely embodied in the (sub)geometry of a particular projective line, found to be equivalent to the generalized quadrangle of order two. As such systems are the simplest ones exhibiting phenomena like quantum entanglement and quantum non-locality and play, therefore, a crucial role in numerous applications like quantum cryptography, quantum coding, quantum cloning/teleportation and/or quantum computing to mention the most salient ones, our discovery thus

  • not only offers a principally new geometrically-underlined insight into their intrinsic nature,
  • but also gives their applications a wholly new perspective
  • and opens up rather unexpected vistas for an algebraic geometrical modelling of their higher-dimensional counterparts."
It would seem that my own
study of pure mathematics
for instance, of the following
"diamond ring"–
 
The image “http://www.log24.com/theory/images/FourD.gif” cannot be displayed, because it contains errors.
 
is not without relevance to
the physics of quantum theory.

Friday, May 25, 2007

Friday May 25, 2007

Filed under: General,Geometry — Tags: , , , — m759 @ 7:11 AM
Dance and the Soul

From Log24 on
this date last year:

"May there be an ennui
of the first idea?
What else,
prodigious scholar,
should there be?"

— Wallace Stevens,
"Notes Toward a
Supreme Fiction"

The Associated Press,
May 25, 2007–

Thought for Today:
"I hate quotations.
 Tell me what you know."
— Ralph Waldo Emerson

[Journals, on May 3, 1849]

The First Idea:

The Line, by S. H. Cullinane

Four Elements:
 

Four Elements (Diamond)

Square Dance:

Square Dance (Diamond Theorem)

This "telling of what
I know" will of course
mean little to those
who, like Emerson,
have refused to learn
through quotations.

For those less obdurate
than Emerson —Harold Bloom
on Wallace Stevens

and Paul Valery's
   "Dance and the Soul"–

"Stevens may be playful, yet seriously so, in describing desire, at winter's end, observing not only the emergence of the blue woman of early spring, but seeing also the myosotis, whose other name is 'forget-me-not.' Desire, hearing the calendar hymn, repudiates the negativity of the mind of winter, unable to bear what Valery's Eryximachus had called 'this cold, exact, reasonable, and moderate consideration of human life as it is.' The final form of this realization in Stevens comes in 1950, in The Course of a Particular, in the great monosyllabic line 'One feels the life of that which gives life as it is.' But even Stevens cannot bear that feeling for long. As Eryximachus goes on to say in Dance and the Soul:

A cold and perfect clarity is a poison impossible to combat. The real, in its pure state, stops the heart instantaneously….[…] To a handful of ashes is the past reduced, and the future to a tiny icicle. The soul appears to itself as an empty and measurable form. –Here, then, things as they are come together, limit one another, and are thus chained together in the most rigorous and mortal* fashion…. O Socrates, the universe cannot for one instant endure to be only what it is.

Valery's formula for reimagining the First Idea is, 'The idea introduces into what is, the leaven of what is not.' This 'murderous lucidity' can be cured only by what Valery's Socrates calls 'the intoxication due to act,' particularly Nietzschean or Dionysiac dance, for this will rescue us from the state of the Snow Man, 'the motionless and lucid observer.'" —Wallace Stevens: The Poems of Our Climate

* "la sorte… la plus mortelle":
    mortal in the sense
   "deadly, lethal"

Other quotations

(from March 28,
the birthday of
Reba McEntire):

Logical Songs

Reba McEntire, Saturday Evening Post, Mar/Apr 1995

Logical Song I
(Supertramp)

"When I was young, it seemed that
Life was so wonderful, a miracle,
Oh it was beautiful, magical
And all the birds in the trees,
Well they'd be singing so happily,
Joyfully, playfully watching me"

Logical Song II
(Sinatra)

"You make me feel so young,
You make me feel like
Spring has sprung
And every time I see you grin
I'm such a happy in-
dividual….

You and I are
Just like a couple of tots
Running across the meadow
Picking up lots
Of forget-me-nots"

Tuesday, January 9, 2007

Tuesday January 9, 2007

Filed under: General — Tags: — m759 @ 12:00 PM
For Balanchine's Birthday

(continued from
January 9, 2003)

George Balanchine

Encyclopædia Britannica Article

born January 22
[January 9, Old Style], 1904,
St. Petersburg, Russia
died April 30, 1983, New York,
New York, U.S.

Photograph:George Balanchine.
George Balanchine.
©1983 Martha Swope

original name 
Georgy Melitonovich Balanchivadze

most influential choreographer of classical ballet in the United States in the 20th century.  His works, characterized by a cool neoclassicism, include The Nutcracker (1954) and Don Quixote (1965), both pieces choreographed for the New York City Ballet, of which he was a founder (1948), the artistic director, and the…


Balanchine,  George… (75 of 1212 words)

"What on earth is
a concrete universal?"
— Robert M. Pirsig

Review:

From Wikipedia's
"Upper Ontology"
and
Epiphany 2007:

"There is no neutral ground
that can serve as
a means of translating between
specialized (lower) ontologies."

There is, however,
"the field of reason"–

the 3×3 grid:

The image “http://www.log24.com/theory/images/grid3x3.gif” cannot be displayed, because it contains errors.

Click on grid
for details.

As Rosalind Krauss
has noted, some artists
regard the grid as

"a staircase to
  the Universal."

Other artists regard
Epiphany itself as an
approach to
the Universal:

"Epiphany signals the traversal
of the finite by the infinite,
of the particular by the universal,
of the mundane by the mystical,
of time by eternity.
"

Richard Kearney, 2005,
in The New Arcadia Review

The image “http://www.log24.com/log/pix07/070109-Kearney2.jpg” cannot be displayed, because it contains errors.

Kearney (right) with
Martin Scorsese (left)
and Gregory Peck
in 1997.

"… one of the things that worried me about traditional metaphysics, at least as I imbibed it in a very Scholastic manner at University College Dublin in the seventies, is that philosophy was realism and realism was truth. What disturbed me about that was that everything was already acquired; truth was always a systematic given and it was there to be learned from Creation onwards; it was spoken by Jesus Christ and then published by St. Thomas Aquinas: the system as perfect synthesis. Hence, my philosophy grew out of a hunger for the 'possible' and it was definitely a reaction to my own philosophical formation. Yet that wasn't my only reaction. I was also reacting to what I considered to be the deep pessimism, and even at times 'nihilism' of the postmodern turn."

— Richard Kearney, interview (pdf) in The Leuven Philosophy Newsletter, Vol. 14, 2005-2006

For more on "the possible," see Kearney's The God Who May Be, Diamonds Are Forever, and the conclusion of Mathematics and Narrative:

 

"We symbolize
logical necessity
with the box (box.gif (75 bytes))
and logical possibility
with the diamond (diamond.gif (82 bytes))."

 

Keith Allen Korcz 

The image “http://www.log24.com/log/pix05B/050802-Stone.gif” cannot be displayed, because it contains errors.

"The possibilia that exist,
and out of which
the Universe arose,
are located in
     a necessary being…."

Michael Sudduth,
Notes on
God, Chance, and Necessity
by Keith Ward,
Regius Professor of Divinity,
Christ Church College, Oxford
(the home of Lewis Carroll)

Friday, December 29, 2006

Friday December 29, 2006

Filed under: General — Tags: — m759 @ 11:01 AM
Tools
of Christ Church

"For every kind of vampire,
there is a kind of cross."
— Thomas Pynchon

Cover of Thomas, by Shelley Mydans: Sword and its shadow, a cross

Click on picture for details.

Today is the feast
of St. Thomas Becket.

In his honor, a meditation
on tools and causation:

"Lewis Wolpert, an eminent developmental biologist at University College London, has just published Six Impossible Things Before Breakfast, a pleasant, though rambling, look at the biological basis of belief. While the book focuses on our ability to form causal beliefs about everyday matters (the wind moved the trees, for example), it spends considerable time on the origins of religious and moral beliefs. Wolpert defends the unusual idea that causal thinking is an adaptation required for tool-making. Religious beliefs can thus be seen as an odd extension of causal thinking about technology to more mysterious matters. Only a species that can reason causally could assert that 'this storm was sent by God because we sinned.' While Wolpert's attitude toward religion is tolerant, he's an atheist who seems to find religion more puzzling than absorbing."

Review by H. Allen Orr in
The New York Review of Books,
Vol. 54, No. 1, January 11, 2007    


"An odd extension"–

Wolpert's title is, of course,
from Lewis Carroll.

Related material:

"It's a poor sort of memory
that only works backwards."
Through the Looking-Glass

An event at the Kennedy Center
broadcast on
December 26, 2006
(St. Steven's Day):

"Conductor John Williams, a 2004 Honoree, says, 'Steven, sharing our 34-year collaboration has been a great privilege for me. It's been an inspiration to watch you dream your dreams, nurture them and make them grow. And, in the process, entertain and edify billions of people around the world. Tonight we'd like to salute you, musically, with a piece that expresses that spirit beautifully … It was written by Leonard Bernstein, a 1980 Kennedy Center Honoree who was, incidentally, the first composer to be performed in this hall.' Backed by The United States Army Chorus and The Choral Arts Society, soprano Harolyn Blackwell and tenor Gregory Turay sing the closing number for Spielberg's tribute and the gala itself. It's the finale to the opera 'Candide,' 'Make Our Garden Grow,' and Williams conducts."

CBS press release

See also the following,
from the conclusion to

"Mathematics and Narrative"

(Log24, Aug. 22, 2005):

Diamond on cover of Narrative Form, by Suzanne Keen

"At times, bullshit can
only be countered
   with superior bullshit."
Norman Mailer

Many Worlds and Possible Worlds in Literature and Art, in Wikipedia:

    "The concept of possible worlds dates back to at least Leibniz who in his Théodicée tries to justify the apparent imperfections of the world by claiming that it is optimal among all possible worlds.  Voltaire satirized this view in his picaresque novel Candide….
    Borges' seminal short story El jardín de senderos que se bifurcan ("The Garden of Forking Paths") is an early example of many worlds in fiction."

"Il faut cultiver notre jardin."
— Voltaire

"We symbolize
logical necessity
with the box (box.gif (75 bytes))
and logical possibility
with the diamond (diamond.gif (82 bytes))."

Keith Allen Korcz 

Diamond in a square

"The possibilia that exist,
and out of which
the Universe arose,
are located in
     a necessary being…."

Michael Sudduth,
Notes on
God, Chance, and Necessity
by Keith Ward,
 Regius Professor of Divinity,
  Christ Church College, Oxford
(the home of Lewis Carroll)

For further details,
click on the
Christ Church diamond.

Tuesday, October 10, 2006

Tuesday October 10, 2006

Filed under: General — Tags: , — m759 @ 8:00 PM
Mate in
Two Seconds

From Oct. 14 last year:

The image “http://www.log24.com/log/pix05B/051014-Tick.gif” cannot be displayed, because it contains errors.

From Oct. 13 last year
(Yom Kippur):

A Poem for Pinter
Oct. 13, 2005

The Guardian on Harold Pinter, winner of this year's Nobel Prize for Literature:

"Earlier this year, he announced his decision to retire from playwriting in favour of poetry,"

Michael Muskal in today's Los Angeles Times:

"Pinter, 75, is known for his sparse and thin style as well as his etched characters whose crystal patter cuts through the mood like diamond drill bits."

Robert Stone, A Flag for Sunrise (See Jan. 25):

"'That old Jew gave me this here.'  Egan looked at the diamond….  'It's worth a whole lot of money– you can tell that just by looking– but it means something, I think.  It's got a meaning, like.'

'Let's see,' Egan said, 'what would it mean?'  He took hold of Pablo's hand cupping the stone and held his own hand under it.  '"The jewel is in the lotus," perhaps that's what it means.  The eternal in the temporal….'"

Notes on Modal Logic:

"Modal logic was originally developed to investigate logic under the modes of necessary and possible truth.  The words 'necessary' and 'possible' are called modal connectives, or modalities.  A modality is a word that when applied to a statement indicates when, where, how, or under what circumstances the statement may be true.  In terms of notation, it is common to use a box [] for the modality 'necessary' and a diamond <> for the modality 'possible.'"

A Poem for Pinter

The image “http://www.log24.com/log/pix05B/051013-Waka.gif” cannot be displayed, because it contains errors.

Commentary:

"Waka" also means Japanese poem or Maori canoe.  (For instance, this Japanese poem and this Maori canoe.)

For a meditation on "bang splat," see Sept. 25-29.

For the meaning of "tick tick," see Emily Dickinson on "degreeless noon."

"Hash," of course, signifies "checkmate."  (See previous three entries.)

For language more suited to
the year's most holy day, see
this year's Yom Kippur entry,
from October 2.

That was also the day of the
Amish school killings in
Pennsylvania and the day that
mathematician Paul Halmos died.

For more on the former, see
Death in Two Seconds.

For more on the latter, see
The Halmos Tombstone.

4x9 black monolith

Tuesday, July 11, 2006

Tuesday July 11, 2006

Filed under: General,Geometry — Tags: — m759 @ 9:11 PM

Not Crazy Enough?

Some children of the sixties may feel that today's previous two entries, on Syd Barrett, the Crazy Diamond, are not crazy enough.  Let them consult the times of those entries– 2:11 and 8:15– and interpret those times, crazily, as dates: 2/11 and 8/15.

This brings us to Stephen King territory– apparently the natural habitat of Syd Barrett.

See Log24 on a 2/11, Along Came a Dreamcatcher, and Log24 on an 8/15, The Line.

From 8/15, a remark of Plato:

"There appears to be a sort of war of Giants and Gods going on…"

(Compare with the remarks by Abraham Cowley for Tom Stoppard's recent birthday.)

From 2/11, two links: Halloween Meditations  and We Are the Key.

From Dreamcatcher (the film and the book):

The image “http://www.log24.com/log/pix06/060211-Freeman2.jpg” cannot be displayed, because it contains errors.

The image “http://www.log24.com/log/pix06/060324-Dreamcatcher.gif” cannot be displayed, because it contains errors.

For Syd Barrett as Duddits,

The image “http://www.log24.com/log/pix06A/060711-Duddits.jpg” cannot be displayed, because it contains errors.

see Terry Kirby on Syd Barrett
(edited– as in Stephen King
and the New Testament
for narrative effect):

"He appeared as the Floyd performed the song 'Shine On You Crazy Diamond.' It contains the words: 'Remember when you were young, you shone like the sun. Shine on you crazy diamond. Now there's a look in your eyes, like black holes in the sky.'

At first, they didn't recognise the man, whose head and eyebrows were shaved….

But this was the 'crazy diamond' himself: Syd Barrett, the subject of the song….

When Roger Waters saw his old friend, he broke down….

Rick Wright, the keyboards player, later told an interviewer:

… 'Roger [Waters] was in tears, I think I was; we were both in tears. It was very shocking… seven years of no contact and then to walk in while we're actually doing that particular track. I don't know – coincidence, karma, fate, who knows? But it was very, very, very powerful.'"

Remarks suitable for Duddits's opponent, Mister Gray, may be found in the 1994 Ph.D. thesis of Noel Gray.

"I refer here to Plato's utilisation in the Meno of graphic austerity as the tool to bring to the surface, literally and figuratively, the inherent presence of geometry in the mind of the slave."

Plato's Diamond

Shine on, gentle Duddits.

Tuesday July 11, 2006

Filed under: General — Tags: — m759 @ 8:15 PM
"In Tom Stoppard's new play 'Rock 'n' Roll,' showing in the West End, he [Syd Barrett] is portrayed in the opening scene, and his life and music are a recurring theme."

— Terry Kirby, Syd Barrett: The Crazy Diamond, in The Independent of July 12

Keynote

"Each scene is punctuated with a rock track from such acts as the Velvet Underground, the Doors, the Rolling Stones, Bob Dylan and Pink Floyd. Songs by Floyd's lost founder, Syd Barrett, are the keynote for Stoppard's theme that rock music sounded the death knell for repression but also heralded a freedom filled with its own perils."

— Ray Bennett, today's review of a new play, "Rock 'n' Roll," by Tom Stoppard

Related material:

Dance of the Numbers,
for Tom Stoppard
on his birthday,
July 3, 2006,

and
Knock, Knock, Knockin',
from yesterday.

'Cause I'm a poet
Don't you know it

— Syd Barrett,
Bob Dylan Blues

Tuesday July 11, 2006

Filed under: General — Tags: — m759 @ 2:11 PM

Pink Floyd co-founder
Syd Barrett dies

"Pink Floyd's 1975 track 'Shine On You Crazy Diamond,' from the album 'Wish You Were Here,' is widely believed to be a tribute to Barrett."– Reuters

Monday, July 10, 2006

Monday July 10, 2006

Filed under: General — Tags: — m759 @ 2:48 AM
Knock, Knock, Knockin'

An obituary in this morning's New York Times suggests a flashback. The Times says that Paul Nelson, 69, a music critic once famously ripped off by the young Bobby Zimmerman, was found dead in his Manhattan apartment last Wednesday. Here is a Log24 entry for that date. (The obituary, by Jon Pareles, notes that Nelson "prized hard-boiled detective novels and film noir.")

Wednesday, July 5, 2006  7:35 PM

Dance of the Numbers
continued–

A music review:

"… in the mode of
 a film noir murder mystery"

"For Bach, as Sellars explains,
 death is not an exit but an entrance."

Seven is Heaven,
Eight is a Gate,
Nine is a Vine.

Monday, January 23, 2006

Monday January 23, 2006

Filed under: General — Tags: — m759 @ 12:00 PM

The Case

An entry suggested by today's New York Times story by Tom Zeller Jr., A Million Little Skeptics:

From The Hustler, by Walter Tevis:

The only light in the room was from the lamp over the couch where she was reading.
    He looked at her face.  She was very drunk.  Her eyes were swollen, pink at the corners.  "What's the book?" he said, trying to make his voice conversational. But it sounded loud in the room, and hard.
    She blinked up at him, smiled sleepily, and said nothing.
    "What's the book?"  His voice had an edge now.
    "Oh," she said.  "It's Kierkegaard.  Soren Kierkegaard."  She pushed her legs out straight on the couch, stretching her feet.  Her skirt fell back a few inches from her knees.  He looked away.
    "What's that?" he said.
    "Well, I don't exactly know, myself."  Her voice was soft and thick.
    He turned his face away from her again, not knowing what he was angry with.  "What does that mean, you don't know, yourself?"
    She blinked at him.  "It means, Eddie, that I don't exactly know what the book is about.  Somebody told me to read it, once, and that's what I'm doing.  Reading it."
    He looked at her, tried to grin at her– the old, meaningless, automatic grin, the grin that made everybody like him– but he could not.  "That's great," he said, and it came out with more irritation than he had intended.
    She closed the book, tucked it beside her on the couch.  "I guess this isn't your night, Eddie.  Why don't we have a drink?"
    "No."  He did not like that, did not want her being nice to him, forgiving.  Nor did he want a drink.
    Her smile, her drunk, amused smile, did not change.  "Then let's talk about something else," she said.  "What about that case you have?  What's in it?"  Her voice was not prying, only friendly.  "Pencils?"
    "That's it," he said.  "Pencils."
    She raised her eyebrows slightly.  Her voice seemed thick.  "What's in it, Eddie?"
    "Figure it out yourself."  He tossed the case on the couch.

 

Related material:

Soren Kierkegaard on necessity and possibility
in The Sickness Unto Death, Chapter 3,

The Diamond of Possibility,

The image “http://www.log24.com/theory/images/Modal-diamondinbox.gif” cannot be displayed, because it contains errors.

the Baseball Almanac,

The image “http://www.log24.com/log/pix06/060123-BaseballLogo75.gif” cannot be displayed, because it contains errors.

and this morning's entry, "Natural Hustler."

Thursday, January 19, 2006

Thursday January 19, 2006

Filed under: General — Tags: — m759 @ 9:00 AM

Logos
 

The image “http://www.log24.com/log/pix06/060119-BibleLogoSmall.gif” cannot be displayed, because it contains errors.

The image “http://www.log24.com/log/pix06/060119-AlvinPlantinga2.jpg” cannot be displayed, because it contains errors.

Alvin Plantinga

The image “http://www.log24.com/theory/images/Modal-diamondinbox.gif” cannot be displayed, because it contains errors.

Philosophy
logo

The image “http://www.log24.com/log/pix06/060119-HarryPlantinga2.jpg” cannot be displayed, because it contains errors.

Harry Plantinga

The image “http://www.log24.com/log/pix06/060119-CCEL.gif” cannot be displayed, because it contains errors.

CCEL
logo

The image “http://www.log24.com/log/pix06/060119-LEngle2.jpg” cannot be displayed, because it contains errors.

Madeleine
L'Engle

The image “http://www.log24.com/log/pix06/060119-Tesseract2.jpg” cannot be displayed, because it contains errors.

Tesseract
logo

Sunday, January 15, 2006

Sunday January 15, 2006

Filed under: General,Geometry — Tags: , — m759 @ 7:59 AM

Inscape

My entry for New Year's Day links to a paper by Robert T. Curtis*
from The Arabian Journal for Science and Engineering
(King Fahd University, Dhahran, Saudi Arabia),
Volume 27, Number 1A, January 2002.

From that paper:

"Combinatorially, an outer automorphism [of S6] can exist because the number of unordered pairs of 6 letters is equal to the number of ways in which 6 letters can be partitioned into three pairs. Which is to say that the two conjugacy classes of odd permutations of order 2 in S6 contain the same number of elements, namely 15. Sylvester… refers to the unordered pairs as duads and the partitions as synthemes. Certain collections of five synthemes… he refers to as synthematic totals or simply totals; each total is stabilized within S6 by a subgroup acting triply transitively on the 6 letters as PGL2(5) acts on the projective line. If we draw a bipartite graph on (15+15) vertices by joining each syntheme to the three duads it contains, we obtain the famous 8-cage (a graph of valence 3 with minimal cycles of length 8)…."

Here is a way of picturing the 8-cage and a related configuration of points and lines:

The image “http://www.log24.com/theory/images/Cremona-Richmond.gif” cannot be displayed, because it contains errors.

Diamond Theory shows that this structure
can also be modeled by an "inscape"
made up of subsets of a
4×4 square array:

The image “http://www.log24.com/theory/images/Inscape.gif” cannot be displayed, because it contains errors.

The illustration below shows how the
points and lines of the inscape may
be identified with those of the
Cremona-Richmond configuration.

The image “http://www.log24.com/theory/images/Inscape2.gif” cannot be displayed, because it contains errors.

* "A fresh approach to the exceptional automorphism and covers of the symmetric groups"

Thursday, October 13, 2005

Thursday October 13, 2005

Filed under: General — Tags: , , — m759 @ 12:00 PM

A Poem for Pinter

The Guardian on Harold Pinter, winner of this year's Nobel Prize for Literature:

"Earlier this year, he announced his decision to retire from playwriting in favour of poetry,"

Michael Muskal in today's Los Angeles Times:

"Pinter, 75, is known for his sparse and thin style as well as his etched characters whose crystal patter cuts through the mood like diamond drill bits."

Robert Stone, A Flag for Sunrise  (See Jan. 25):

"'That old Jew gave me this here.'  Egan looked at the diamond….  'It's worth a whole lot of money– you can tell that just by looking– but it means something, I think.  It's got a meaning, like.'

'Let's see,' Egan said, 'what would it mean?'  He took hold of Pablo's hand cupping the stone and held his own hand under it.  '"The jewel is in the lotus," perhaps that's what it means.  The eternal in the temporal….'"

Notes on Modal Logic:

"Modal logic was originally developed to investigate logic under the modes of necessary  and possible  truth.  The words 'necessary' and 'possible' are called modal connectives , or modalities .  A modality is a word that when applied to a statement indicates when, where, how, or under what circumstances the statement may be true.  In terms of notation, it is common to use a box [] for the modality 'necessary' and a diamond <> for the modality 'possible.'"

A Poem for Pinter

The image “http://www.log24.com/log/pix05B/051013-Waka.gif” cannot be displayed, because it contains errors.

Commentary:

"Waka" also means Japanese poem or Maori canoe.

(For instance, this Japanese poem and this Maori canoe.)

For a meditation on "bang splat," see Sept. 25-29.

For the meaning of "tick tick," see Emily Dickinson on "degreeless noon."

"Hash," of course, signifies "checkmate."  (See previous three entries.)

Monday, August 22, 2005

Monday August 22, 2005

Filed under: General — Tags: — m759 @ 4:07 PM
The Hole

Part I: Mathematics and Narrative

The image “http://www.log24.com/log/pix05B/050822-Narr.jpg” cannot be displayed, because it contains errors.

Apostolos Doxiadis on last month's conference on "mathematics and narrative"–

Doxiadis is describing how talks by two noted mathematicians were related to

    "… a sense of a 'general theory bubbling up' at the meeting… a general theory of the deeper relationship of mathematics to narrative…. "

Doxiadis says both talks had "a big hole in the middle."  

    "Both began by saying something like: 'I believe there is an important connection between story and mathematical thinking. So, my talk has two parts.  [In one part] I’ll tell you a few things about proofs.  [And in the other part] I’ll tell you about stories.' …. And in both talks it was in fact implied by a variation of the post hoc propter hoc, the principle of consecutiveness implying causality, that the two parts of the lectures were intimately related, the one somehow led directly to the other."
  "And the hole?"
  "This was exactly at the point of the link… [connecting math and narrative]… There is this very well-known Sidney Harris cartoon… where two huge arrays of formulas on a blackboard are connected by the sentence ‘THEN A MIRACLE OCCURS.’ And one of the two mathematicians standing before it points at this and tells the other: ‘I think you should be more explicit here at step two.’ Both… talks were one half fascinating expositions of lay narratology– in fact, I was exhilarated to hear the two most purely narratological talks at the meeting coming from number theorists!– and one half a discussion of a purely mathematical kind, the two parts separated by a conjunction roughly synonymous to ‘this is very similar to this.’  But the similarity was not clearly explained: the hole, you see, the ‘miracle.’  Of course, both [speakers]… are brilliant men, and honest too, and so they were very clear about the location of the hole, they did not try to fool us by saying that there was no hole where there was one."
 

Part II: Possible Worlds

"At times, bullshit can only be countered with superior bullshit."
Norman Mailer

Many Worlds and Possible Worlds in Literature and Art, in Wikipedia:

    "The concept of possible worlds dates back to a least Leibniz who in his Théodicée tries to justify the apparent imperfections of the world by claiming that it is optimal among all possible worlds.  Voltaire satirized this view in his picaresque novel Candide….
    Borges' seminal short story El jardín de senderos que se bifurcan ("The Garden of Forking Paths") is an early example of many worlds in fiction."

 

Background:

Modal Logic in Wikipedia

Possible Worlds in Wikipedia

Possible-Worlds Theory, by Marie-Laure Ryan
(entry for The Routledge Encyclopedia of Narrative Theory)

The God-Shaped Hole
 

Part III: Modal Theology

  "'What is this Stone?' Chloe asked….
  '…It is told that, when the Merciful One made the worlds, first of all He created that Stone and gave it to the Divine One whom the Jews call Shekinah, and as she gazed upon it the universes arose and had being.'"

  — Many Dimensions, by Charles Williams, 1931 (Eerdmans paperback, April 1979, pp. 43-44)


"The lapis was thought of as a unity and therefore often stands for the prima materia in general."

  — Aion, by C. G. Jung, 1951 (Princeton paperback, 1979, p. 236)

"Its discoverer was of the opinion that he had produced the equivalent of the primordial protomatter which exploded into the Universe."

 
  — The Stars My Destination, by Alfred Bester, 1956 (Vintage hardcover, July 1996, p. 216)
 
"We symbolize
logical necessity
with the box (box.gif (75 bytes))
and logical possibility
with the diamond (diamond.gif (82 bytes))."

Keith Allen Korcz 

The image “http://www.log24.com/log/pix05B/050802-Stone.gif” cannot be displayed, because it contains errors.

"The possibilia that exist,
and out of which
the Universe arose,
are located in
     a necessary being…."

Michael Sudduth,
Notes on
God, Chance, and Necessity
by Keith Ward,
Regius Professor of Divinity
at Christ Church College, Oxford
(the home of Lewis Carroll)

Tuesday, March 22, 2005

Tuesday March 22, 2005

Filed under: General,Geometry — Tags: , , — m759 @ 4:01 PM

Make a Différance

From Frida Saal's
Lacan The image “http://www.log24.com/log/pix05/050322-Diamond.gif” cannot be displayed, because it contains errors. Derrida:

"Our proposal includes the lozenge (diamond) in between the names, because in the relationship / non-relationship that is established among them, a tension is created that implies simultaneously a union and a disjunction, in the perspective of a theoretical encounter that is at the same time necessary and impossible. That is the meaning of the lozenge that joins and separates the two proper names. For that reason their respective works become totally non-superposable and at the same time they were built with an awareness, or at least a partial awareness, of each other. What prevails between both of them is the différance, the Derridean signifier that will become one of the main issues in this presentation."

 


From a Contemporary Literary Theory website:

"Différance is that which all signs have, what constitutes them as signs, as signs are not that to which they refer: i) they differ, and hence open a space from that which they represent, and ii) they defer, and hence open up a temporal chain, or, participate in temporality. As well, following de Sassure's famous argument, signs 'mean' by differing from other signs. The coined word 'différance' refers to at once the differing and the deferring of signs. Taken to the ontological level†, the differing and deferring of signs from what they mean, means that every sign repeats the creation of space and time; and ultimately, that différance is the ultimate phenomenon in the universe, an operation that is not an operation, both active and passive, that which enables and results from Being itself."

From a text purchased on
Make a Difference Day, Oct. 23, 1999:

The image “http://www.log24.com/log/pix05/050322-Fig39.gif” cannot be displayed, because it contains errors.22. Without using the Pythagorean Theorem prove that the hypotenuse of  an isosceles right triangle will have the length The image “http://www.log24.com/log/pix05/050322-Sqtr2.gif” cannot be displayed, because it contains errors.  if the equal legs have the length 1.  Suggestion: Consider the similar triangles in Fig. 39.
23.  The ancient Greeks regarded the Pythagorean Theorem as involving areas, and they proved it by means of areas.  We cannot do so now because we have not yet considered the idea of area.  Assuming for the moment, however, the idea of the area of a square, use this idea instead of similar triangles and proportion in Ex. 22 above to show that x = The image “http://www.log24.com/log/pix05/050322-Sqtr2.gif” cannot be displayed, because it contains errors. .

 

— Page 98 of Basic Geometry, by George David Birkhoff, Professor of Mathematics at Harvard University, and Ralph Beatley, Associate Professor of Education at Harvard University (Scott, Foresman 1941)



Though it may be true, as the president of Harvard recently surmised, that women are inherently inferior to men at abstract thought — in particular, pure mathematics*  — they may in other respects be quite superior to men:

The image “http://www.log24.com/log/pix05/050322-Reba2.jpg” cannot be displayed, because it contains errors.

The above is from October 1999.
See also Naturalized Epistemology,
from Women's History Month, 2001.

* See the remarks of Frida Saal above and of Barbara Johnson on mathematics (The Shining of May 29, cited in Readings for St. Patrick's Day).


† For the diamond symbol at "the ontological level," see Modal Theology, Feb. 21, 2005.  See also Socrates on the immortality of the soul in Plato's Meno, source of the above Basic Geometry diamond.

Thursday, March 3, 2005

Thursday March 3, 2005

Filed under: General — Tags: — m759 @ 3:26 PM
Necessity, Possibility, Symmetry

The image “http://www.log24.com/log/pix05/050303-Symmetry.gif” cannot be displayed, because it contains errors.


Matrix group actions,
March 26, 1985

"We symbolize logical necessity
with the box (box.gif (75 bytes))
and logical possibility
with the diamond (diamond.gif (82 bytes))."

Keith Allen Korcz,
(Log24.net, 1/25/05)

And what do we           
   symbolize by  The image “http://www.log24.com/theory/images/Modal-diamondbox.gif” cannot be displayed, because it contains errors. ?

"The possibilia that exist,
and out of which
the Universe arose,
are located in
     a necessary being…."

Michael Sudduth,
Notes on
God, Chance, and Necessity
by Keith Ward,
Regius Professor of Divinity
at Christ Church College, Oxford
(the home of Lewis Carroll)

Sunday, February 20, 2005

Sunday February 20, 2005

Filed under: General,Geometry — Tags: , — m759 @ 2:20 PM

Relativity Blues

Today, February 20, is the 19th anniversary of my note The Relativity Problem in Finite Geometry.  Here is some related material.

In 1931, the Christian writer Charles Williams grappled with the theology of time, space, free will, and the many-worlds interpretation of quantum mechanics (anticipating by many years the discussion of this topic by physicists beginning in the 1950's).

(Some pure mathematics — untainted by physics or theology — that is nevertheless related, if only by poetic analogy, to Williams's 1931 novel, Many Dimensions, is discussed in the above-mentioned note and in a generalization, Solomon's Cube.)

On the back cover of Williams's 1931 novel, the current publisher, William B. Eerdmans Publishing Company of Grand Rapids, Michigan, makes the following statement:

"Replete with rich religious imagery, Many Dimensions explores the relation between predestination and free will as it depicts different human responses to redemptive transcendence."

One possible response to such statements was recently provided in some detail by a Princeton philosophy professor.  See On Bullshit, by Harry G. Frankfurt, Princeton University Press, 2005.

A more thoughtful response would take into account the following:

1. The arguments presented in favor of philosopher John Calvin, who discussed predestination, in The Death of Adam: Essays on Modern Thought, by Marilynne Robinson

2. The physics underlying Einstein's remarks on free will, God, and dice
 
3. The physics underlying Rebecca Goldstein's novel Properties of Light and Paul Preuss's novels  Secret Passages and Broken Symmetries

4. The physics underlying the recent so-called "free will theorem" of John Conway and Simon Kochen of Princeton University

5. The recent novel Gilead, by Marilynne Robinson, which deals not with philosophy, but with lives influenced by philosophy — indirectly, by the philosophy of the aforementioned John Calvin.

From a review of Gilead by Jane Vandenburgh:  

"In The Death of Adam, Robinson shows Jean Cauvin to be the foremost prophet of humanism whose Protestant teachings against the hierarchies of the Roman church set in motion the intellectual movements that promoted widespread literacy among the middle and lower classes, led to both the American and French revolutions, and not only freed African slaves in the United States but brought about suffrage for women. It's odd then that through our culture's reverse historicism, the term 'Calvinism' has come to mean 'moralistic repression.'"

For more on what the Calvinist publishing firm Eerdmans calls "redemptive transcendence," see various July 2003 Log24.net entries.  If these entries include a fair amount of what Princeton philosophers call bullshit, let the Princeton philosophers meditate on the summary of Harvard philosophy quoted here on November 5 of last year, as well as the remarks of November 5, 2003,  and those of November 5, 2002.

From Many Dimensions (Eerdmans paperback, 1963, page 53):

"Lord Arglay had a suspicion that the Stone would be purely logical.  Yes, he thought, but what, in that sense, were the rules of its pure logic?"

A recent answer:

Modal Theology

"We symbolize logical necessity
with the box (box.gif (75 bytes))
and logical possibility
with the diamond (diamond.gif (82 bytes))."

Keith Allen Korcz,
(Log24.net, 1/25/05)

And what do we           
   symbolize by  The image “http://www.log24.com/theory/images/Modal-diamondbox.gif” cannot be displayed, because it contains errors. ?

"The possibilia that exist,
and out of which
the Universe arose,
are located in
     a necessary being…."

Michael Sudduth,
Notes on
God, Chance, and Necessity
by Keith Ward,
Regius Professor of Divinity
at Christ Church College, Oxford
(the home of Lewis Carroll)

Thursday, February 17, 2005

Thursday February 17, 2005

Filed under: General,Geometry — Tags: — m759 @ 1:00 PM
Modal Theology

"We symbolize logical necessity
with the box (box.gif (75 bytes))
and logical possibility
with the diamond (diamond.gif (82 bytes))."

Keith Allen Korcz,
(Log24.net, 1/25/05)

And what do we           
   symbolize by  The image “http://www.log24.com/theory/images/Modal-diamondbox.gif” cannot be displayed, because it contains errors. ?

On the Lapis Philosophorum,
the Philosophers' Stone –

"'What is this Stone?' Chloe asked….
'…It is told that, when the Merciful One
made the worlds, first of all He created
that Stone and gave it to the Divine One
whom the Jews call Shekinah,
and as she gazed upon it
the universes arose and had being.'"
Many Dimensions,
by Charles Williams, 1931
(Eerdmans paperback,
April 1979, pp. 43-44)

"The lapis was thought of as a unity
and therefore often stands for
the prima materia in general."
Aion, by C. G. Jung, 1951
(Princeton paperback,
1979, p. 236)

"Its discoverer was of the opinion that
he had produced the equivalent of
the primordial protomatter
which exploded into the Universe."
The Stars My Destination,
by Alfred Bester, 1956
(Vintage hardcover,
July 1996, p. 216)

"The possibilia that exist,
and out of which
the Universe arose,
are located in
     a necessary being…."

Michael Sudduth,
Notes on
God, Chance, and Necessity
by Keith Ward,
Regius Professor of Divinity
at Christ Church College, Oxford
(the home of Lewis Carroll)

See also
The Diamond Archetype.

For more on modal theology, see

Kurt Gödel's Ontological Argument
and

 The Ontological Argument
 from Anselm to Gödel.

Tuesday, January 25, 2005

Tuesday January 25, 2005

Filed under: General — Tags: — m759 @ 12:00 PM

Diamonds Are Forever

 
The image “http://www.log24.com/theory/images/Modal-diamondinbox.gif” cannot be displayed, because it contains errors.

 

Robert Stone,
A Flag for Sunrise:

" 'That old Jew gave me this here.'  Egan looked at the diamond.  'I ain't giving this to you, understand?  The old man gave it to me for my boy.  It's worth a whole lot of money– you can tell that just by looking– but it means something, I think.  It's got a meaning, like.'

'Let's see,' Egan said, 'what would it mean?'  He took hold of Pablo's hand cupping the stone and held his own hand under it.  '"The jewel is in the lotus," perhaps that's what it means.  The eternal in the temporal.  The Boddhisattva declining nirvana out of compassion.   Contemplating the ignorance of you and me, eh?  That's a metaphor of our Buddhist friends.'

Pablo's eyes glazed over.  'Holy shit,' he said.  'Santa Maria.'  He stared at the diamond in his palm with passion.

'Hey,' he said to the priest, 'diamonds are forever!  You heard of that, right?  That means something, don't it?'

'I have heard it,' Egan said.  'Perhaps it has a religious meaning.' "
 


"We symbolize logical necessity
with the box (box.gif (75 bytes))
and logical possibility
with the diamond (diamond.gif (82 bytes))."

Keith Allen Korcz

From

DIALECTIC AND EXISTENCE
IN KIERKEGAARD AND KANT

Nythamar Fernandes de Oliveira

Pontifical Catholic University
at Porto Alegre, Brazil

"Such is the paradoxical 'encounter' of the eternal with the temporal. Just like the Moment of the Incarnation, when the Eternal entered the temporal, Kierkegaard refers to the category of the Instant (Danish Ojeblikket, 'a glance of the eye, eyeblink,' German Augenblick) as the dialectical kernel of our existential consciousness:

If the instant is posited, so is the eternal –but also the future, which comes again like the past … The concept around which everything turns in Christianity, the concept which makes all things new, is the fullness of time, is the instant as eternity, and yet this eternity is at once the future and the past.

Although I cannot examine here the Kierkegaardian conception of time, the dialectical articulation of time and existence, as can be seen, underlies his entire philosophy of existence, just as the opposition between 'eternity' and 'temporality': the instant, as 'an atom of eternity,' serves to restructure the whole synthesis of selfhood into a spiritual one, in man’s 'ascent' toward its Other and the Unknown. In the last analysis, the Eternal transcends every synthesis between eternity and time, infinity and finiteness, preserving not only the Absolute Paradox in itself but above all the wholly otherness of God. It is only because of the Eternal, therefore, that humans can still hope to attain their ultimate vocation of becoming a Chistian. As Kierkegaard writes in Works of Love (1847),

The possibility of the good is more than possibility, for it is the eternal. This is the basis of the fact that one who hopes can never be deceived, for to hope is to expect the possibility of the good; but the possibility of the good is eternal. …But if there is less love in him, there is also less of the eternal in him; but if there is less of the eternal in him, there is also less possibility, less awareness of possibility (for possibility appears through the temporal movement of the eternal within the eternal in a human being)."

Monday, January 24, 2005

Monday January 24, 2005

Filed under: General — Tags: — m759 @ 10:23 PM

The Diamond
of Possibility

by  Keith Allen Korcz

"We symbolize logical necessity with the box (box.gif (75 bytes)) and logical possibility with the diamond (diamond.gif (82 bytes)).  I first discuss combining negations with the box and diamond, noting that logical possibility and logical necessity are inter-definable with the help of negation: box.gif (75 bytes)p = ~diamond.gif (82 bytes)~p and diamond.gif (82 bytes)p = ~box.gif (75 bytes)~p. Thus, possibility and necessity are two sides of the same coin."

And what do we           
 symbolize by  The image “http://www.log24.com/theory/images/Modal-diamondbox.gif” cannot be displayed, because it contains errors. ?

Monday January 24, 2005

Filed under: General,Geometry — Tags: — m759 @ 2:45 PM

Old School Tie

From a review of A Beautiful Mind:

"We are introduced to John Nash, fuddling flat-footed about the Princeton courtyard, uninterested in his classmates' yammering about their various accolades. One chap has a rather unfortunate sense of style, but rather than tritely insult him, Nash holds a patterned glass to the sun, [director Ron] Howard shows us refracted patterns of light that take shape in a punch bowl, which Nash then displaces onto the neckwear, replying, 'There must be a formula for how ugly your tie is.' "

The image “http://www.log24.com/log/pix05/050124-Tie.gif” cannot be displayed, because it contains errors.

"Three readings of diamond and box
have been extremely influential."

Draft of
Computing with Modal Logics
(pdf), by Carlos Areces
and Maarten de Rijke

"Algebra in general is particularly suited for structuring and abstracting. Here, structure is imposed via symmetries and dualities, for instance in terms of Galois connections….

diamonds and boxes are upper and lower adjoints of Galois connections…."

— "Modal Kleene Algebra
and Applications: A Survey"
(pdf), by Jules Desharnais,
Bernhard Möller, and
Georg Struth, March 2004
 
See also
Galois Correspondence

The image “http://www.log24.com/log/pix05/050124-galois12s.jpg” cannot be displayed, because it contains errors.

Evariste Galois

and Log24.net, May 20, 2004:

"Perhaps every science must
start with metaphor
and end with algebra;
and perhaps without metaphor
there would never have been
any algebra."

— attributed, in varying forms
(1, 2, 3), to Max Black,
Models and Metaphors, 1962

For metaphor and
algebra combined, see

"Symmetry invariance
in a diamond ring,"

A.M.S. abstract 79T-A37,
Notices of the Amer. Math. Soc.,
February 1979, pages A-193, 194 —
the original version of the 4×4 case
of the diamond theorem.

Sunday, September 5, 2004

Sunday September 5, 2004

Filed under: General — m759 @ 4:00 PM

Symmetry and Change
in the Dreamtime

Notes from the Journal
of Steven H. Cullinane

Summary:

Aug 31 2004 
07:31:01 PM
Early Evening,
Shining Star 
Sep 01 2004
09:00:35 AM
Words
and Images
Sep 01 2004
12:07:28 PM
Whale Rider
Sep 02 2004
11:11:42 AM
Heaven
and Earth

Sep 02 2004
07:00:23 PM
Whale Road

Sep 03 2004
12:00:54 AM

Cinderella’s
Slipper
 
Sep 03 2004
10:01:56 AM
Another
September Morn

 

Sep 03 2004
12:00:25 PM

Noon

Sep 03 2004
01:13:49 PM

De Nada

Sep 03 2004
03:17:13 
PM

Ite, Missa Est 


Tuesday, August 31, 2004

Symmetry and Change, Part 1…

Early Evening,
Shining Star

7:31:01 PM ET

Hexagram 01
The Creative:

 

The Image

Heaven

Heaven

The movement of heaven
is full of power.

Click on picture
for details.

The Clare Lawler Prize
for Literature goes to…

Under the Volcano,
Chapter VI:

“What have I got out of my life? Contacts with famous men… The occasion Einstein asked me the time, for instance. That summer evening…. smiles when I say I don’t know. And yet asked me. Yes: the great Jew, who has upset the whole world’s notions of time and space, once leaned down… to ask me… ragged freshman… at the first approach of the evening star, the time. And smiled again when I pointed out the clock neither of us had noticed.”

For the thoughts on time
of another famous man,
from Mexico, see the
Nobel Prize acceptance speech
of Octavio Paz,
In Search of the Present.”


Wednesday, September 1, 2004

Symmetry and Change, Part 2…

Words and Images

9:00:35 AM ET

Hexagram 35
Progress:

The Image

Fire

Earth

The sun rises over the earth.

From Aug. 18, 2004:

“Oh, my Lolita. I have only words
to play with!” (Nabokov, Lolita)

“This is the best toy train set
a boy ever had!”
(Orson Welles, after first touring
RKO Studios, quoted in Halliwell)

“As the quotes above by Nabokov and Welles suggest, we need to be able to account for the specific functions available to narrative in each medium, for the specific elements that empirical creators will ‘play with’ in crafting their narratives.”

Donald F. Larsson

For
James Whale
and
William French Anderson —

Words
In the Spirit of
Dave Barry’s Book of Bad Songs:

Stay for just a while…
Stay, and let me look at you.
It’s been so long, I hardly knew you.
Standing in the door…
Stay with me a while.
I only want to talk to you.
We’ve traveled halfway ’round the world
To find ourselves again.

September morn…
We danced until the night
      became a brand new day,
Two lovers playing scenes
      from some romantic play.
September morning still can
      make me feel this way.

Look at what you’ve done…
Why, you’ve become a grown-up girl…

— Neil Diamond

Images
In the Spirit of
September Morn:

The Last Day of Summer:
Photographs by Jock Sturges

In 1990, the FBI entered Sturges’s studio and seized his work, claiming violation of child pornography laws.”

Related material:

Bill’s Diamond Theory

and

Log24 entries of
Aug. 15, 2004
.

Those interested in the political implications of Diamond’s songs may enjoy Neil Performs at Kerry Fundraiser.

I personally enjoyed this site’s description of Billy Crystal’s remarks, which included “a joke about former President Clinton’s forthcoming children’s book — ‘It’s called The Little Engine That Could Because It Could.'”

“Puff, puff, woo, woo, off we go!” 

 


 

Wednesday, September 1, 2004

Symmetry and Change, Part 3…

Whale Rider

12:07:28 PM

Hexagram 28
Preponderance of
the Great:

The Image

Lake

Wind

The lake rises
above
the trees.

 

Cullinane College News:

“Congratulations to Clare Lawler, who participated very successfully in the recently held Secondary Schools Judo Championships in Wellington.”

For an explanation of this entry’s title, see the previous two entries and

Oxford Word
(Log24, July 10, 2004) 


Thursday, September 2, 2004

Symmetry and Change, Part 4…

Heaven and Earth

11:11:42 AM ET

Hexagram 42
Increase:

The Image

Wind

Thunder

Wind and thunder:
the image of Increase.

“This time resembles that of
the marriage of heaven and earth”


Kylie


Finney

Well if you want to ride
you gotta ride it like you find it.
Get your ticket at the station
of the Rock Island Line.
Lonnie Donegan (d. Nov. 3)
and others
The Rock Island Line’s namesake depot 
in Rock Island, Illinois

“What it all boiled down to really was everybody giving everybody else a hard time for no good reason whatever… You just couldn’t march to your own music. Nowadays, you couldn’t even hear it… It was lost, the music which each person had inside himself, and which put him in step with things as they should be.”

The Grifters, Ch. 10, 1963, by
James Myers Thompson

“The Old Man’s still an artist
with a Thompson.”
— Terry in “Miller’s Crossing

For some of “the music which
each person had inside,”
click on the picture
with the Thompson.

It may be that Kylie is,
in her own way, an artist…
with a 357:

(Hits counter at
The Quality of Diamond
as of 11:05 AM Sept. 2, 2004)

For more on
“the marriage of heaven and earth,”
see
Plato, Pegasus, and the Evening Star


Thursday, September 2, 2004

Symmetry and Change, Part 5…

Whale Road

7:00:23 PM

Hexagram 23
Splitting Apart:

The Image

Mountain

Earth

The mountain rests
on the earth
.

“… the plot is different but the monsters, names, and manner of speaking will ring a bell.”

— Frank Pinto, Jr., review of Seamus Heaney’s translation of Beowulf 

Other recommended reading, found during a search for the implications of today’s previous entry, “Hexagram 42”:

Water Wings.

This excellent meditation
on symmetry and change
comes from a site whose
home page
has the following image:


Friday, September 3, 2004

 Symmetry and Change, Part 6…

Cinderella’s Slipper

12:00:54 AM ET

Hexagram 54
The Marrying Maiden:

 

The Image

Thunder


Lake
See
The hundredletter
thunderwords of
Finnegans Wake


“… a Thoreau-like retreat
by a nearby lake….
Both men have
a ‘touch of the poet’….
The symmetry is perfect.”

Friday, September 3, 2004  

Symmetry and Change, Part 7…

Another September Morn

10:01:56 AM ET

Hexagram 56:
The Wanderer

 

The Image

Fire


Mountain

Fire on the mountain,
Run boys run…
Devil’s in the House of
The Rising Sun!
 


Friday, September 3, 2004

Symmetry and Change, Part 8…

Noon

12:00:25 PM ET

Hexagram 25
Innocence:

The Image

Heaven


Thunder

Under heaven
thunder rolls.
 


Friday, September 3, 2004

Symmetry and Change, Part 9…

De Nada

Helen Lane

1:13:49 PM ET

Hexagram 49
Revolution:

The Image

Lake


Fire
 Fire in the lake:
the image of Revolution
.

“I sit now in a little room off the bar at four-thirty in the morning drinking ochas and then mescal and writing this on some Bella Vista notepaper I filched the other night…. But this is worst of all, to feel your soul dying. I wonder if it is because to-night my soul has really died that I feel at the moment something like peace. Or is it because right through hell there is a path, as Blake well knew, and though I may not take it, sometimes lately in dreams I have been able to see it? …And this is how I sometimes think of myself, as a great explorer who has discovered some extraordinary land from which he can never return to give his knowledge to the world: but the name of this land is hell. It is not Mexico of course but in the heart.”

— Malcolm Lowry, Under the Volcano 


Friday, September 3, 2004

Symmetry and Change, conclusion…

Ite, Missa Est

3:17:13 PM ET

Hexagram 13
Fellowship With Men:

The Image

Heaven


Fire

Heaven together with fire.

“A pretty girl —
is like a melody —- !”

 For details, see
A Mass for Lucero


Wednesday, September 1, 2004

Wednesday September 1, 2004

Filed under: General — m759 @ 9:00 AM

Symmetry and Change, Part 2…

Words and Images

9:00:35 AM ET

Hexagram 35
Progress:

The Image

Fire

Earth

The sun rises over the earth.

From Aug. 18, 2004:

“Oh, my Lolita. I have only words
to play with!” (Nabokov, Lolita)

“This is the best toy train set
a boy ever had!”
(Orson Welles, after first touring
RKO Studios, quoted in Halliwell)

“As the quotes above by Nabokov and Welles suggest, we need to be able to account for the specific functions available to narrative in each medium, for the specific elements that empirical creators will ‘play with’ in crafting their narratives.”

Donald F. Larsson

For
James Whale
and
William French Anderson —

Words
In the Spirit of
Dave Barry’s Book of Bad Songs:

Stay for just a while…
Stay, and let me look at you.
It’s been so long, I hardly knew you.
Standing in the door…
Stay with me a while.
I only want to talk to you.
We’ve traveled halfway ’round the world
To find ourselves again.

September morn…
We danced until the night
      became a brand new day,
Two lovers playing scenes
      from some romantic play.
September morning still can
      make me feel this way.

Look at what you’ve done…
Why, you’ve become a grown-up girl…

— Neil Diamond

Images
In the Spirit of
September Morn:

The Last Day of Summer:
Photographs by Jock Sturges

In 1990, the FBI entered Sturges’s studio and seized his work, claiming violation of child pornography laws.”

Related material:

Bill’s Diamond Theory

and

Log24 entries of
Aug. 15, 2004
.

Those interested in the political implications of Diamond’s songs may enjoy Neil Performs at Kerry Fundraiser.

I personally enjoyed this site’s description of Billy Crystal’s remarks, which included “a joke about former President Clinton’s forthcoming children’s book — ‘It’s called The Little Engine That Could Because It Could.'”

“Puff, puff, woo, woo, off we go!”

Monday, August 18, 2003

Monday August 18, 2003

Filed under: General,Geometry — Tags: — m759 @ 3:09 PM

Entries since Xanga’s
August 10 Failure:


Sunday, August 17, 2003  2:00 PM

A Thorny Crown of…

West Wing's Toby Ziegler

From the first episode of
the television series
The West Wing“:

 

Original airdate: Sept. 22, 1999
Written by Aaron Sorkin

MARY MARSH
That New York sense of humor. It always–

CALDWELL
Mary, there’s absolutely no need…

MARY MARSH
Please, Reverend, they think they’re so much smarter. They think it’s smart talk. But nobody else does.

JOSH
I’m actually from Connecticut, but that’s neither here nor there. The point is that I hope…

TOBY
She meant Jewish.

[A stunned silence. Everyone stares at Toby.]

TOBY (CONT.)
When she said “New York sense of humor,” she was talking about you and me.

JOSH
You know what, Toby, let’s just not even go there.

 

Going There, Part I

 

Crown of Ideas

Kirk Varnedoe, 57, art historian and former curator of the Museum of Modern Art, died Thursday, August 14, 2003.

From his New York Times obituary:

” ‘He loved life in its most tangible forms, and so for him art was as physical and pleasurable as being knocked down by a wave,’ said Adam Gopnik, the writer and a former student of his who collaborated on Mr. Varnedoe’s first big show at the Modern, ‘High & Low.’ ‘Art was always material first — it was never, ever bound by a thorny crown of ideas.’ ”

For a mini-exhibit of ideas in honor of Varnedoe, see

Fahne Hoch.

Verlyn Klinkenborg on Varnedoe:

“I was always struck by the tangibility of the words he used….  It was as if he were laying words down on the table one by one as he used them, like brushes in an artist’s studio. That was why students crowded into his classes and why the National Gallery of Art had overflow audiences for his Mellon Lectures earlier this year. Something synaptic happened when you listened to Kirk Varnedoe, and, remarkably, something synaptic happened when he listened to you. You never knew what you might discover together.”

Perhaps even a “thorny crown of ideas“?

“Crown of Thorns”
Cathedral, Brasilia

Varnedoe’s death coincided with
the Great Blackout of 2003.

“To what extent does this idea of a civic life produced by sense of adversity correspond to actual life in Brasília? I wonder if it is something which the city actually cultivates. Consider, for example the cathedral, on the monumental axis, a circular, concrete framed building whose sixteen ribs are both structural and symbolic, making a structure that reads unambiguously as a crown of thorns; other symbolic elements include the subterranean entrance, the visitor passing through a subterranean passage before emerging in the light of the body of the cathedral. And it is light, shockingly so….”

Modernist Civic Space: The Case of Brasilia, by Richard J. Williams, Department of History of Art, University of Edinburgh, Scotland

 

Going There, Part II

Simple, Bold, Clear

Art historian Kirk Varnedoe was, of course, not the only one to die on the day of the Great Blackout.

Claude Martel, 34, a senior art director of The New York Times Magazine, also died on Thursday, August 14, 2003.

Janet Froelich, the magazine’s art director, describes below a sample of work that she and Martel did together:

“A new world of ideas”

Froelich notes that “the elements are simple, bold, and clear.”

For another example of elements with these qualities, see my journal entry

Fahne Hoch.

The flag design in that entry
might appeal to Aaron Sorkin’s
Christian antisemite:

 

Fahne,
S. H. Cullinane,
Aug. 15, 2003

Dr. Mengele,
according to
Hollywood

 

Note that the elements of the flag design have the qualities described so aptly by Froelich– simplicity, boldness, clarity:

They share these qualities with the Elements of Euclid, a treatise on geometrical ideas.

For the manner in which such concepts might serve as, in Gopnik’s memorable phrase, a “thorny crown of ideas,” see

“Geometry for Jews” in

ART WARS: Geometry as Conceptual Art.

See also the discussion of ideas in my journal entry on theology and art titled

Understanding: On Death and Truth

and the discussion of the wordidea” (as well as the word, and the concept, “Aryan”) in the following classic (introduced by poet W. H. Auden):

 

 

Saturday, August 16, 2003  6:00 AM

Varnedoe’s Crown

Kirk Varnedoe, 57, art historian and former curator of the Museum of Modern Art, died Thursday, August 14, 2003.

From his New York Times obituary:

” ‘He loved life in its most tangible forms, and so for him art was as physical and pleasurable as being knocked down by a wave,’ said Adam Gopnik, the writer and a former student of his who collaborated on Mr. Varnedoe’s first big show at the Modern, ‘High & Low.’ ‘Art was always material first — it was never, ever bound by a thorny crown of ideas.’ “

For a mini-exhibit of ideas in honor of Varnedoe, see

Fahne Hoch. 

Verlyn Klinkenborg on Varnedoe:

“I was always struck by the tangibility of the words he used….  It was as if he were laying words down on the table one by one as he used them, like brushes in an artist’s studio. That was why students crowded into his classes and why the National Gallery of Art had overflow audiences for his Mellon Lectures earlier this year. Something synaptic happened when you listened to Kirk Varnedoe, and, remarkably, something synaptic happened when he listened to you. You never knew what you might discover together.”

Perhaps even a “thorny crown of ideas”?

“Crown of Thorns”
Cathedral, Brasilia

Varnedoe’s death coincided with
the Great Blackout of 2003.

“To what extent does this idea of a civic life produced by sense of adversity correspond to actual life in Brasília? I wonder if it is something which the city actually cultivates. Consider, for example the cathedral, on the monumental axis, a circular, concrete framed building whose sixteen ribs are both structural and symbolic, making a structure that reads unambiguously as a crown of thorns; other symbolic elements include the subterranean entrance, the visitor passing through a subterranean passage before emerging in the light of the body of the cathedral. And it is light, shockingly so….”

Modernist Civic Space: The Case of Brasilia, by Richard J. Williams, Department of History of Art, University of Edinburgh, Scotland


Friday, August 15, 2003  3:30 PM

ART WARS:

The Boys from Brazil

It turns out that the elementary half-square designs used in Diamond Theory

 

also appear in the work of artist Nicole Sigaud.

Sigaud’s website The ANACOM Project  has a page that leads to the artist Athos Bulcão, famous for his work in Brasilia.

From the document

Conceptual Art in an
Authoritarian Political Context:
Brasilia, Brazil
,

by Angélica Madeira:

“Athos created unique visual plans, tiles of high poetic significance, icons inseparable from the city.”

As Sigaud notes, two-color diagonally-divided squares play a large part in the art of Bulcão.

The title of Madeira’s article, and the remarks of Anna Chave on the relationship of conceptual/minimalist art to fascist rhetoric (see my May 9, 2003, entries), suggest possible illustrations for a more politicized version of Diamond Theory:

 

Fahne,
S. H. Cullinane,
Aug. 15, 2003

Dr. Mengele,
according to
Hollywood

 

Is it safe?

These illustrations were suggested in part by the fact that today is the anniversary of the death of Macbeth, King of Scotland, and in part by the following illustrations from my journal entries of July 13, 2003 comparing a MOMA curator to Lady Macbeth:

 

Die Fahne Hoch,
Frank Stella,
1959


Dorothy Miller,
MOMA curator,
died at 99 on
July 11, 2003
.

 


Thursday, August 14, 2003  3:45 AM

Famous Last Words

The ending of an Aug. 14 Salon.com article on Mel Gibson’s new film, “The Passion”:

” ‘The Passion’ will most likely offer up the familiar puerile, stereotypical view of the evil Jew calling for Jesus’ blood and the clueless Pilate begging him to reconsider. It is a view guaranteed to stir anew the passions of the rabid Christian, and one that will send the Jews scurrying back to the dark corners of history.”

— Christopher Orlet

“Scurrying”?!  The ghost of Joseph Goebbels, who famously portrayed Jews as sewer rats doing just that, must be laughing — perhaps along with the ghost of Lady Diana Mosley (née Mitford), who died Monday.

This goes well with a story that Orlet tells at his website:

“… to me, the most genuine last words are those that arise naturally from the moment, such as

 

Joseph Goebbels

 

Voltaire’s response to a request that he foreswear Satan: ‘This is no time to make new enemies.’ ”

For a view of Satan as an old, familiar, acquaintance, see the link to Prince Ombra in my entry last October 29 for Goebbels’s birthday.


Wednesday, August 13, 2003  3:00 PM

Best Picture

For some reflections inspired in part by

click here.


Tuesday, August 12, 2003  4:44 PM

Atonement:

A sequel to my entry “Catholic Tastes” of July 27, 2003.

Some remarks of Wallace Stevens that seem appropriate on this date:

“It may be that one life is a punishment
For another, as the son’s life for the father’s.”

—  Esthétique du Mal, Wallace Stevens

Joseph Patrick Kennedy, Jr.

“Unless we believe in the hero, what is there
To believe? ….
Devise, devise, and make him of winter’s
Iciest core, a north star, central
In our oblivion, of summer’s
Imagination, the golden rescue:
The bread and wine of the mind….”

Examination of the Hero in a Time of War, Wallace Stevens

Etymology of “Atonement”:

Middle English atonen, to be reconciled, from at one, in agreement

At One

“… We found,
If we found the central evil, the central good….
… we and the diamond globe at last were one.”

Asides on the Oboe, Wallace Stevens


Tuesday, August 12, 2003  1:52 PM

Franken & ‘Stein,
Attorneys at Law

Tue August 12, 2003 04:10 AM ET
NEW YORK (Reuters) – Fox News Network is suing humor writer Al Franken for trademark infringement over the phrase ‘fair and balanced’ on the cover of his upcoming book, saying it has been ‘a signature slogan’ of the network since 1996.”

Franken:
Fair?

‘Stein:
Balanced?

For answers, click on the pictures
of Franken and ‘Stein.


Thursday, July 31, 2003

Thursday July 31, 2003

Filed under: General — Tags: — m759 @ 6:41 PM

Killer Radio

"See the girl with the diamond ring?
 She knows how to shake that thing."

— Jerry Lee "Killer" Lewis on
    KHYI 95.3 FM, Plano, Texas,
    at about 5:12 PM EDT 7/31/03,
    introduced by DJ Allen Peck Sr.

"And on this point I pass the same judgment as those who say that geometricians give them nothing new by these rules, because they possessed them in reality, but confounded with a multitude of others, either useless or false, from which they could not discriminate them, as those who, seeking a diamond of great price amidst a number of false ones, but from which they know not how to distinguish it, should boast, in holding them all together, of possessing the true one equally with him who without pausing at this mass of rubbish lays his hand upon the costly stone which they are seeking and for which they do not throw away the rest."

— Blaise Pascal, De l'Esprit Géométrique

"When the light came she was sitting on the bed beside an open suitcase, toying with her diamond rings.  She saw the light first in the depths of the largest stone."

— Paul Preuss, Broken Symmetries,
    scene at Diamond Head, Oahu, Hawaii

Now playing (6:41 PM EDT) on Killer Radio:

"Jack of Diamonds, that's
 a hard card to find."

"This Jack, joke, poor potsherd, patch, matchwood, immortal diamond…."

— Gerard Manley Hopkins, Society of Jesus

Perhaps Sam Phillips was twanged by a Hawaiian guitar. (See previous two entries.)

 

The Big Time

"The place outside the cosmos where I and my pals do our nursing job I simply call the Place.  A lot of my nursing consists of amusing and humanizing Soldiers fresh back from raids into time. In fact, my formal title is Entertainer…."

The Big Time,
    by Fritz Leiber

A Story That Works

  • "There is the dark, eternally silent, unknown universe;
  • there are the friend-enemy minds shouting and whispering their tales and always seeking the three miracles —
    • that minds should really touch, or
    • that the silent universe should speak, tell minds a story, or (perhaps the same thing)
    • that there should be a story that works, that is all hard facts, all reality, with no illusions and no fantasy;
  • and lastly, there is lonely, story-telling, wonder-questing, mortal me."

    Fritz Leiber in "The Button Molder"

 

 

See also "Top Ten Most Overheard Comments by new KHYI listeners" at Miss Lana's Anything Page, entry for

Pearl Harbor Day, December 7, 2002.
 

Wednesday, July 23, 2003

Wednesday July 23, 2003

Filed under: General,Geometry — Tags: — m759 @ 4:17 PM

Being Pascal Sauvage

Pascal

"Voilà ce que je sais par une longue expérience de toutes sortes de livres et de personnes. Et sur cela je fais le même jugement de ceux qui disent que les géomètres ne leur donnent rien de nouveau par ces règles, parce qu' ils les avaient en effet, mais confondues parmi une multitude d' autres inutiles ou fausses dont ils ne pouvaient pas les discerner, que de ceux qui cherchant un diamant de grand prix

Diamant

parmi un grand nombre de faux, mais qu' ils n' en sauraient pas distinguer, se vanteraient, en les tenant tous ensemble, de posséder le véritable aussi bien que celui qui, sans s' arrêter à ce vil amas, porte la main sur la pierre choisie que l' on recherche, et pour laquelle on ne jetait pas tout le reste."

— Blaise Pascal, De l'Esprit Géométrique

La Pensée Sauvage

"….the crowning image of the kaleido­scope, lavishly analogized to the mythwork in a three-hundred-word iconic apotheosis that served to put the wraps on the sustained personification of “la pensée sauvage” in the figure of the bricoleur, in an argument developed across two chapters and some twenty pages in his [Claude Lévi-Strauss's] most famous book…."

— Robert de Marrais in
Catastrophes, Kaleidoscopes,
String Quartets:
Deploying the Glass Bead Game


Pascal Sauvage

Chiasmus

For more on pensée sauvage, see

"Claude Lévi-Strauss,

Chiasmus

and the Ethnographic Journey."

Saturday, July 5, 2003

Saturday July 5, 2003

Filed under: General,Geometry — m759 @ 7:21 PM

Elementary,
My Dear Gropius

“What is space, how can it be understood and given a form?”
— Walter Gropius

Stoicheia:

Stoicheia,” Elements, is the title of
Euclid’s treatise on geometry.

Stoicheia is apparently also related to a Greek verb meaning “march” or “walk.”

According to a website on St. Paul’s phrase ta stoicheia tou kosmou,” which might be translated

The Elements of the Cosmos,

“… the verbal form of the root stoicheo was used to mean, ‘to be in a line,’ ‘to march in rank and file.’ … The general meaning of the noun form (stoicheion) was ‘what belongs to a series.’ “

As noted in my previous entry, St. Paul used a form of stoicheo to say “let us also walk (stoichomen) by the Spirit.” (Galatians 5:25) The lunatic ravings* of Saul of Tarsus aside, the concepts of walking, of a spirit, and of elements may be combined if we imagine the ghost of Gropius strolling with the ghosts of Plato, Aristotle, and Euclid, and posing his question about space.  Their reply might be along the following lines:

Combining stoicheia with a peripatetic peripateia (i.e., Aristotelian plot twist), we have the following diagram of Aristotle’s four stoicheia (elements),

which in turn is related, by the “Plato’s diamond” figure in the monograph Diamond Theory, to the Stoicheia, or Elements, of Euclid.

Quod erat demonstrandum.

* A phrase in memory of the Paulist Norman J. O’Connor, the “jazz priest” who died on St. Peter’s day, Sunday, June 29, 2003.  Paulists are not, of course, entirely mad; the classic The Other Side of Silence: A Guide to Christian Meditation, by the Episcopal priest Morton Kelsey, was published by the Paulist Press.

Its cover (above), a different version of the four-elements theme, emphasizes the important Jungian concept of quaternity.  Jung is perhaps the best guide to the bizarre world of Christian symbolism.  It is perhaps ironic, although just, that the Paulist Fathers should distribute a picture of “ta stoicheia tou kosmou,” the concept that St. Paul himself railed against.

The above book by Kelsey should not be confused with another The Other Side of Silence, a work on gay history, although confusion would be understandable in light of recent ecclesiastical revelations.

Let us pray that if there is a heaven, Father O’Connor encounters there his fellow music enthusiast Cole Porter rather than the obnoxious Saul of Tarsus.

Saturday July 5, 2003

Filed under: General,Geometry — m759 @ 4:17 AM

Elements

In memory of Walter Gropius, founder of the Bauhaus and head of the Harvard Graduate School of Design.  Gropius died on this date in 1969.  He said that

"The objective of all creative effort in the visual arts is to give form to space. … But what is space, how can it be understood and given a form?"

"Alle bildnerische Arbeit will Raum gestalten. … Was ist Raum, wie können wir ihn erfassen und gestalten?"


Gropius

— "The Theory and Organization
of the Bauhaus
" (1923)

I designed the following logo for my Diamond Theory site early this morning before reading in a calendar that today is the date of Gropius's death.  Hence the above quote.

"And still those voices are calling
from far away…"
— The Eagles
 

Stoicheia:

("Stoicheia," Elements, is the title of
Euclid's treatise on geometry.)

Friday, May 9, 2003

Friday May 9, 2003

Filed under: General,Geometry — m759 @ 7:20 PM

ART WARS:
The Religion of Cubism

In the dome of the Capitol at Washington, DC, a painting depicts The Apotheosis of Washington .  Personally, I prefer the following pair of pictures, which might be titled Apotheosis of the Cube.

logo

Die

A New York Times article says Tony Smith's instructions for fabricating Die  were as follows:

"a six-foot cube of quarter-inch hot-rolled steel with diagonal internal bracing."

The transparent cube in the upper picture above shows the internal diagonals.  The fact that there are four of these may be used to demonstrate the isomorphism of the group of rotations of the cube with the group of permutations on an arbitrary set of four elements.  For deeper results, see Diamond Theory.

For an explanation of why our current president might feel that the cube deserves an apotheosis, see the previous entry, "The Rhetoric of Power."

See, too, Nabokov's Transparent Things :

"Its ultimate vision was the incandescence of a book or a box grown completely transparent and hollow.  This is, I believe, it: not the crude anguish of physical death but the incomparable pangs of the mysterious mental maneuver needed to pass from one state of being to another.  Easy, you know, does it, son."

Friday, April 25, 2003

Friday April 25, 2003

Filed under: General,Geometry — Tags: , , — m759 @ 7:59 PM

Mark

Today is the feast of Saint Mark.  It seems an appropriate day to thank Dr. Gerald McDaniel for his online cultural calendar, which is invaluable for suggesting blog topics.

Yesterday's entry "Cross-Referenced" referred to a bizarre meditation of mine titled "The Matthias Defense," which combines some thoughts of Nabokov on lunacy with some of my own thoughts on the Judeo-Christian tradition (i.e., also on lunacy).  In this connection, the following is of interest:

From a site titled Meaning of the Twentieth Century —

"Freeman Dyson has expressed some thoughts on craziness. In a Scientific American article called 'Innovation in Physics,' he began by quoting Niels Bohr. Bohr had been in attendance at a lecture in which Wolfgang Pauli proposed a new theory of elementary particles. Pauli came under heavy criticism, which Bohr summed up for him: 'We are all agreed that your theory is crazy. The question which divides us is whether it is crazy enough to have a chance of being correct. My own feeling is that is not crazy enough.' To that Freeman added: 'When a great innovation appears, it will almost certainly be in a muddled, incomplete and confusing form. To the discoverer, himself, it will be only half understood; to everyone else, it will be a mystery. For any speculation which does not at first glance look crazy, there is no hope!' "

Kenneth Brower, The Starship and the Canoe, 1979, pp. 146, 147

It is my hope that the speculation, implied in The Matthias Defense, that the number 162 has astonishing mystical properties (as a page number, article number, etc.) is sufficiently crazy to satisfy Pauli and his friend Jung as well as the more conventional thinkers Bohr and Dyson.  It is no less crazy than Christianity, and has a certain mad simplicity that perhaps improves on some of that religion's lunatic doctrines. 

Some fruits of the "162 theory" —

Searching on Google for muses 162, we find the following Orphic Hymn to Apollo and a footnote of interest:

27 Tis thine all Nature's music to inspire,
28 With various-sounding, harmonising lyre;
29 Now the last string thou tun'ft to sweet accord,
30 Divinely warbling now the highest chord….

"Page 162 Verse 29…. Now the last string…. Gesner well observes, in his notes to this Hymn, that the comparison and conjunction of the musical and astronomical elements are most ancient; being derived from Orpheus and Pythagoras, to Plato. Now, according to the Orphic and Pythagoric doctrine, the lyre of Apollo is an image of the celestial harmony…."

For the "highest chord" in a metaphorical sense, see selection 162 of the 1919 edition of The Oxford Book of English Verse (whose editor apparently had a strong religious belief in the Muses (led by Apollo)).  This selection contains the phrase "an ever-fixèd mark" — appropriately enough for this saint's day.  The word "mark," in turn, suggests a Google search for the phrase "runes to grave" Hardy, after a poem quoted in G. H. Hardy's A Mathematician's Apology.

Such a search yields a website that quotes Housman as the source of the "runes" phrase, and a further search yields what is apparently the entire poem:

Smooth Between Sea and Land

by A. E. Housman

Smooth between sea and land
Is laid the yellow sand,
And here through summer days
The seed of Adam plays.

Here the child comes to found
His unremaining mound,
And the grown lad to score
Two names upon the shore.

Here, on the level sand,
Between the sea and land,
What shall I build or write
Against the fall of night?

Tell me of runes to grave
That hold the bursting wave,
Or bastions to design
For longer date than mine.

Shall it be Troy or Rome
I fence against the foam
Or my own name, to stay
When I depart for aye?

Nothing: too near at hand
Planing the figured sand,
Effacing clean and fast
Cities not built to last
And charms devised in vain,
Pours the confounding main.

(Said to be from More Poems (Knopf, 1936), p. 64)

Housman asks the reader to tell him of runes to grave or bastions to design.  Here, as examples, are one rune and one bastion.

 


The rune known as
"Dagaz"

Represents
the balance point or "still point."


The Nike Bastion

 Dagaz: (Pronounced thaw-gauze, but with the "th" voiced as in "the," not unvoiced as in "thick") (Day or dawn.)

From Rune Meanings:

 Dagaz means "breakthrough, awakening, awareness. Daylight clarity as opposed to nighttime uncertainty. A time to plan or embark upon an enterprise. The power of change directed by your own will, transformation. Hope/happiness, the ideal. Security and certainty. Growth and release. Balance point, the place where opposites meet."

Also known as "the rune of transformation."

For the Dagaz rune in another context, see Geometry of the I Ching.  The geometry discussed there does, in a sense, "hold the bursting wave," through its connection with Walsh functions, hence with harmonic analysis.

 Temple of Athena Nike on the Nike Bastion, the Acropolis, Athens.  Here is a relevant passage from Paul Valéry's Eupalinos ou L'Architecte about another temple of four columns:

Et puis… Écoute, Phèdre (me disait-il encore), ce petit temple que j'ai bâti pour Hermès, à quelques pas d'ici, si tu savais ce qu'il est pour moi ! — Où le passant ne voit qu'une élégante chapelle, — c'est peu de chose: quatre colonnes, un style très simple, — j'ai mis le souvenir d'un clair jour de ma vie. Ô douce métamorphose ! Ce temple délicat, nul ne le sait, est l'image mathématique d'une fille de Corinthe que j'ai heureusement aimée. Il en reproduit fidèlement les proportions particulières. Il vit pour moi !

Four columns, in a sense more suited to Hardy's interests, are also a recurrent theme in The Diamond 16 Puzzle and Diamond Theory.

Apart from the word "mark" in The Oxford Book of English Verse, as noted above, neither the rune nor the bastion discussed has any apparent connection with the number 162… but seek and ye shall find.
 

Thursday, March 13, 2003

Thursday March 13, 2003

Filed under: General — m759 @ 4:44 PM

ART WARS:

From The New Yorker, issue of March 17, 2003, Clive James on Aldous Huxley:

The Perennial Philosophy, his 1945 book compounding all the positive thoughts of West and East into a tutti-frutti of moral uplift, was the equivalent, in its day, of It Takes a Village: there was nothing in it to object to, and that, of course, was the objection.”

For a cultural artifact that is less questionably perennial, see Huxley’s story “Young Archimedes.”

Plato, Pythagoras, and
the diamond figure

Plato’s Diamond in the Meno
Plato as a precursor of Gerard Manley Hopkins’s “immortal diamond.” An illustration shows the ur-diamond figure.

Plato’s Diamond Revisited
Ivars Peterson’s Nov. 27, 2000 column “Square of the Hypotenuse” which discusses the diamond figure as used by Pythagoras (perhaps) and Plato. Other references to the use of Plato’s diamond in the proof of the Pythagorean theorem:

Huxley:

“… and he proceeded to prove the theorem of Pythagoras — not in Euclid’s way, but by the simpler and more satisfying method which was, in all probability, employed by Pythagoras himself….
‘You see,’ he said, ‘it seemed to me so beautiful….’
I nodded. ‘Yes, it’s very beautiful,’ I said — ‘it’s very beautiful indeed.'”
— Aldous Huxley, “Young Archimedes,” in Collected Short Stories, Harper, 1957, pp. 246 – 247

Heath:

Sir Thomas L. Heath, in his commentary on Euclid I.47, asks how Pythagoreans discovered the Pythagorean theorem and the irrationality of the diagonal of a unit square. His answer? Plato’s diamond.
(See Heath, Sir Thomas Little (1861-1940),
The thirteen books of Euclid’s Elements translated from the text of Heiberg with introduction and commentary. Three volumes. University Press, Cambridge, 1908. Second edition: University Press, Cambridge, 1925. Reprint: Dover Publications, New York, 1956.

Other sites on the alleged
diamond” proof of Pythagoras

Colorful diagrams at Cut-the-Knot

Illustrated legend of the diamond proof

Babylonian version of the diamond proof

For further details of Huxley’s story, see

The Practice of Mathematics,

Part I, by Robert P. Langlands, from a lecture series at the Institute for Advanced Study, Princeton.

From the New Yorker Contributors page for St. Patrick’s Day, 2003:

Clive James (Books, p. 143) has a new collection, As of This Writing: The Essential Essays, 1968-2002, which will be published in June.”

See also my entry “The Boys from Uruguay” and the later entry “Lichtung!” on the Deutsche Schule Montevideo in Uruguay.

Friday, November 29, 2002

Friday November 29, 2002

Filed under: General,Geometry — Tags: — m759 @ 1:06 PM

A Logocentric Archetype

Today we examine the relativist, nominalist, leftist, nihilist, despairing, depressing, absurd, and abominable work of Samuel Beckett, darling of the postmodernists.

One lens through which to view Beckett is an essay by Jennifer Martin, "Beckettian Drama as Protest: A Postmodern Examination of the 'Delogocentering' of Language." Martin begins her essay with two quotations: one from the contemptible French twerp Jacques Derrida, and one from Beckett's masterpiece of stupidity, Molloy. For a logocentric deconstruction of Derrida, see my note, "The Shining of May 29," which demonstrates how Derrida attempts to convert a rather important mathematical result to his brand of nauseating and pretentious nonsense, and of course gets it wrong. For a logocentric deconstruction of Molloy, consider the following passage:

"I took advantage of being at the seaside to lay in a store of sucking-stones. They were pebbles but I call them stones…. I distributed them equally among my four pockets, and sucked them turn and turn about. This raised a problem which I first solved in the following way. I had say sixteen stones, four in each of my four pockets these being the two pockets of my trousers and the two pockets of my greatcoat. Taking a stone from the right pocket of my greatcoat, and putting it in my mouth, I replaced it in the right pocket of my greatcoat by a stone from the right pocket of my trousers, which I replaced by a stone from the left pocket of my trousers, which I replaced by a stone from the left pocket of my greatcoat, which I replaced by the stone which was in my mouth, as soon as I had finished sucking it. Thus there were still four stones in each of my four pockets, but not quite the same stones….But this solution did not satisfy me fully. For it did not escape me that, by an extraordinary hazard, the four stones circulating thus might always be the same four."

Beckett is describing, in great detail, how a damned moron might approach the extraordinarily beautiful mathematical discipline known as group theory, founded by the French anticleric and leftist Evariste Galois. Disciples of Derrida may play at mimicking the politics of Galois, but will never come close to imitating his genius. For a worthwhile discussion of permutation groups acting on a set of 16 elements, see R. D. Carmichael's masterly work, Introduction to the Theory of Groups of Finite Order, Ginn, Boston, 1937, reprinted by Dover, New York, 1956.

There are at least two ways of approaching permutations on 16 elements in what Pascal calls "l'esprit géométrique." My website Diamond Theory discusses the action of the affine group in a four-dimensional finite geometry of 16 points. For a four-dimensional euclidean hypercube, or tesseract, with 16 vertices, see the highly logocentric movable illustration by Harry J. Smith. The concept of a tesseract was made famous, though seen through a glass darkly, by the Christian writer Madeleine L'Engle in her novel for children and young adults, A Wrinkle in Tme.

This tesseract may serve as an archetype for what Pascal, Simone Weil (see my earlier notes), Harry J. Smith, and Madeleine L'Engle might, borrowing their enemies' language, call their "logocentric" philosophy.

For a more literary antidote to postmodernist nihilism, see Archetypal Theory and Criticism, by Glen R. Gill.

For a discussion of the full range of meaning of the word "logos," which has rational as well as religious connotations, click here.

Wednesday, November 27, 2002

Wednesday November 27, 2002

Filed under: General,Geometry — Tags: — m759 @ 11:30 PM

Waiting for Logos

Searching for background on the phrase "logos and logic" in yesterday's "Notes toward a Supreme Fact," I found this passage:

"…a theory of psychology based on the idea of the soul as the dialectical, self-contradictory syzygy of a) soul as anima and b) soul as animus. Jungian and archetypal psychology appear to have taken heed more or less of only one half of the whole syzygy, predominantly serving an anima cut loose from her own Other, the animus as logos and logic (whose first and most extreme phenomenological image is the killer of the anima, Bluebeard). Thus psychology tends to defend the virginal innocence of the anima and her imagination…"

— Wolfgang Giegerich, "Once More the Reality/Irreality Issue: A Reply to Hillman's Reply," website 

The anima and other Jungian concepts are used to analyze Wallace Stevens in an excellent essay by Michael Bryson, "The Quest for the Fiction of an Absolute." Part of Bryson's motivation in this essay is the conflict between the trendy leftist nominalism of postmodern critics and the conservative realism of more traditional critics:

"David Jarraway, in his Stevens and the Question of Belief, writes about a Stevens figured as a proto-deconstructionist, insisting on 'Steven's insistence on dismantling the logocentric models of belief' (311) in 'An Ordinary Evening in New Haven.' In opposition to these readings comes a work like Janet McCann's Wallace Stevens Revisited: 'The Celestial Possible', in which the claim is made (speaking of the post-1940 period of Stevens' life) that 'God preoccupied him for the rest of his career.'"

Here "logocentric" is a buzz word for "Christian." Stevens, unlike the postmodernists, was not anti-Christian. He did, however, see that the old structures of belief could not be maintained indefinitely, and pondered what could be found to replace them. "Notes toward a Supreme Fiction" deals with this problem. In his essay on Stevens' "Notes," Bryson emphasizes the "negative capability" of Keats as a contemplative technique:

"The willingness to exist in a state of negative capability, to accept that sometimes what we are seeking is not that which reason can impose…."

For some related material, see Simone Weil's remarks on Electra waiting for her brother Orestes. Simone Weil's brother was one of the greatest mathematicians of the past century, André Weil.

"Electra did not seek Orestes, she waited for him…"

— Simone Weil

"…at the end, she pulls it all together brilliantly in the story of Electra and Orestes, where the importance of waiting on God rather than seeking is brought home forcefully."

— Tom Hinkle, review of Waiting for God

Compare her remarks on waiting for Orestes with the following passage from Waiting for God:

"We do not obtain the most precious gifts by going in search of them but by waiting for them. Man cannot discover them by his own powers, and if he sets out to seek for them he will find in their place counterfeits of which he will be unable to discern falsity.

The solution of a geometry problem does not in itself constitute a precious gift, but the same law applies to it because it is the image of something precious. Being a little fragment of particular truth, it is a pure image of the unique, eternal, and living Truth, the very Truth that once in a human voice declared: "I am the Truth."

Every school exercise, thought of in this way, is like a sacrament.

In every school exercise there is a special way of waiting upon truth, setting our hearts upon it, yet not allowing ourselves to go out in search of it. There is a way of giving our attention to the data of a problem in geometry without trying to find the solution…."

— Simone Weil, "Reflections on the Right Use of School Studies with a View to the Love of  God"

Weil concludes the preceding essay with the following passage:

"Academic work is one of those fields containing a pearl so precious that it is worth while to sell all of our possessions, keeping nothing for ourselves, in order to be able to acquire it."

This biblical metaphor is also echoed in the work of Pascal, who combined in one person the theological talent of Simone Weil and the mathematical talent of her brother. After discussing how proofs should be written, Pascal says

"The method of not erring is sought by all the world. The logicians profess to guide to it, the geometricians alone attain it, and apart from their science, and the imitations of it, there are no true demonstrations. The whole art is included in the simple precepts that we have given; they alone are sufficient, they alone afford proofs; all other rules are useless or injurious. This I know by long experience of all kinds of books and persons.

And on this point I pass the same judgment as those who say that geometricians give them nothing new by these rules, because they possessed them in reality, but confounded with a multitude of others, either useless or false, from which they could not discriminate them, as those who, seeking a diamond of great price amidst a number of false ones, but from which they know not how to distinguish it, should boast, in holding them all together, of possessing the true one equally with him who without pausing at this mass of rubbish lays his hand upon the costly stone which they are seeking and for which they do not throw away the rest."

— Blaise Pascal, The Art of Persuasion

 

For more diamond metaphors and Jungian analysis, see

The Diamond Archetype.

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