Sunday, May 6, 2018

The Osterman Omega

Filed under: General,Geometry — Tags: , — m759 @ 5:01 PM

From "The Osterman Weekend" (1983) —

Counting symmetries of the R. T. Curtis Omega:

An Illustration from Shakespeare's birthday

Counting symmetries with the orbit-stabilizer theorem

Sunday, March 5, 2017

The Omega Matrix

Filed under: General,Geometry — Tags: — m759 @ 5:00 PM

Richard Evan Schwartz on
the mathematics of the 4×4 square

See also Priority in this journal.

Monday, June 15, 2015

Omega Matrix

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

See that phrase in this journal.

See also last night's post.

The Greek letter Ω is customarily used to
denote a set that is acted upon by a group.
If the group is the affine group of 322,560
transformations of the four-dimensional
affine space over the two-element Galois
field, the appropriate &Omegais the 4×4 grid above.

See the Cullinane diamond theorem.

Monday, January 12, 2015

Points Omega*

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

The previous post displayed a set of
24 unit-square "points" within a rectangular array.
These are the points of the 
Miracle Octad Generator  of R. T. Curtis.

The array was labeled  Ω
because that is the usual designation for
a set acted upon by a group:

* The title is an allusion to Point Omega , a novel by
   Don DeLillo published on Groundhog Day 2010.
   See "Point Omega" in this journal.

Friday, August 15, 2014

The Omega Matrix

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


The webpage Rosenhain and Göpel Tetrads in PG(3,2)
has been updated to include more material on symplectic structure.

Thursday, August 7, 2014

The Omega Mystery

Filed under: General — Tags: — m759 @ 11:00 PM

See a post,  The Omega Matrix, from the date of her death.

Related material:

"When Death tells a story, you really have to listen."
— Cover of The Book Thief

A scene from the film of the above book —

“Looking carefully at Golay’s code is like staring into the sun.”

— Richard Evan Schwartz

Some context — "Mathematics, Magic, and Mystery" —
See posts tagged April Awareness 2014.

Tuesday, August 5, 2014

The Omega Story

Filed under: General — Tags: — m759 @ 1:00 AM

"We tell ourselves stories in order to live…. We interpret what we see, select the most workable of multiple choices. We live entirely, especially if we are writers, by the imposition of a narrative line upon disparate images, by the 'ideas' with which we have learned to freeze the shifting phantasmagoria which is our actual experience."

Joan Didion

See also a post from May 4, 2011 (the date, according to a Google
search, of untitled notes regarding a matrix called Omega).

Monday, August 4, 2014

The Omega Portal

Filed under: General — m759 @ 11:00 AM

Version from “The Avengers” (2012) —

Version from Josefine Lyche (2009) —

Image- Josefine Lyche work (with 1986 figures by Cullinane) in a 2009 exhibition in Oslo

See also this journal on the date that the above Avengers  video was uploaded.

Sunday, August 3, 2014

The Omega Matrix

Filed under: General,Geometry — Tags: , , — m759 @ 10:31 PM

Shown below is the matrix Omega from notes of Richard Evan Schwartz.
See also earlier versions (1976-1979) by Steven H. Cullinane.

IMAGE- The matrix Omega from notes of Richard Evan Schwartz. See also earlier versions (1977-1979) by Steven H. Cullinane.

Backstory:  The Schwartz Notes (June 1, 2011), and Schwartz on
the American Mathematical Society's current home page:

(Click to enlarge.)

Tuesday, April 19, 2011

Romancing the Omega

Filed under: General,Geometry — m759 @ 12:25 PM

Today's news from Oslo suggests a review—

Image- Josefine Lyche work (with 1986 figures by Cullinane) in a 2009 exhibition in Oslo

Click for further details.

The circular sculpture in the foreground
is called by the artist "The Omega Point."
This has been described as
"a portal that leads in or out of time and space."

Some related philosophical remarks—

Oslo Connection and some notes on Galois connections.

Friday, February 11, 2011


Filed under: General — m759 @ 1:00 PM

And I'd  like to thank the heroine of Finale

Image-- Josefine Lyche as Diamond Girl, representing the soul's triumph over evil

*  The title refers to a 2009 sculpture by Lyche


   For a related shape, see today's noon post.

Thursday, July 1, 2010

Omega at Eight

Filed under: General,Geometry — Tags: — m759 @ 8:00 PM

The "compact key to universal wisdom" passage in the previous post seemed
too well written to be the work of an anonymous webforum author.

Here is a slightly expanded version—

Throughout history mystics and philosophers have sought
a compact key to universal wisdom, a finite formula or text
that would provide the answer to every question. The use of
the Bible, the Koran and the I Ching for divination and the
tradition of the secret books of Hermes Trismegistus and the
medieval Jewish Cabala exemplify this belief or hope.  Such
sources of universal wisdom are traditionally protected from
casual use by being difficult to find as well as difficult to un-
derstand and dangerous to use, tending to answer more quest-
ions and deeper ones than the searcher wishes to ask. The
esoteric book is, like God, simple yet undescribable. It is om-
niscient, and it transforms all who know it. The use of clas-
sical texts to foretell mundane events is considered supersti-
tious nowadays, yet in another sense science is in quest of its
own Cabala, a concise set of natural laws that would explain
all phenomena. In mathematics, where no set of axioms can
hope to prove all true statements, the goal might be a concise
axiomatization of all "interesting" true statements.
      Ω is in many senses a Cabalistic number. It can be known
of through human reason, but not known. To know it in detail
one must accept its uncomputable sequence of digits on faith,
like words of a sacred text.   

This is Martin Gardner's* and Charles H. Bennett's
revised version of a passage from Bennett's  paper
"On Random and Hard-to-Describe Numbers," 1979.

The original passage from Bennett's paper—

Throughout history mystics and philosophers have sought a compact key to
universal wisdom, a finite formula or text which, when known and understood,
would provide the answer to every question. The Bible, the Koran, the mythical
secret books of Hermes Trismegistus, and the medieval Jewish Cabala have
been so regarded. Sources of universal wisdom are traditionally protected from
casual use by being hard to find, hard to understand when found, and dangerous
to use, tending to answer more and deeper questions than the user wishes to
ask. Like God the esoteric book is simple yet undescribable, omniscient, and
transforms all who know It. The use of classical texts to fortell [sic] mundane events
is considered superstitious nowadays, yet, in another sense, science is in quest of
its own Cabala, a concise set of natural laws which would explain all phenomena.
In mathematics, where no set of axioms can hope to prove all true statements,
the goal might be a concise axiomatization of all "interesting" true statements.
      Ω is in many senses a Cabalistic number. It can be known of, but not known,
through human reason. To know it in detail, one would have to accept its un-
computable digit sequence on faith, like words of a sacred text.

The Bennett paper deals with Gregory Chaitin's concept of an "Omega Number."

I prefer the Omega of Josefine Lyche

Image-- Uncertified copy of 1986 figures by Cullinane in a 2009 art exhibit in Oslo

Click for further details.

See also All Hallows' Eve, 2002.

* Martin Gardner's Mathematical Games  column
"The Random Number Omega Bids Fair to Hold the Mysteries of the Universe,"
Scientific American, November 1979, 241(5), pp. 20–34.
The column is reprinted as "Chaitin's Omega," Ch. 21, pp. 307-319 in the
collection of Gardner's columns titled Fractal Music, Hypercards and More,
W.H. Freeman & Co., 1991

Friday, April 26, 2013


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

The Oslo Version and The Lyche Omega

Those who prefer more traditional art 
may consult The Portal Project.

Monday, June 1, 2020

The Gefter Boundary

Filed under: General — Tags: , — m759 @ 5:09 PM

“The message was clear: having a finite frame of reference
creates the illusion of a world, but even the reference frame itself
is an illusion. Observers create reality, but observers aren’t real.
There is nothing ontologically distinct about an observer, because
you can always find a frame in which that observer disappears:
the frame of the frame itself, the boundary of the boundary.”

— Amanda Gefter in 2014, quoted here on Mayday 2020.

Image- Josefine Lyche work (with 1986 figures by Cullinane) in a 2009 exhibition in Oslo

See as well the previous post.

A Graveyard Smash: Galois Geometry Meets Nordic Aliens

Filed under: General — Tags: — m759 @ 4:02 PM

See also Vril Chick.

Wednesday, May 20, 2020

Cue the Violins

Filed under: General — Tags: , — m759 @ 11:36 PM

In memory of a music editor.

Blangsted reportedly died on May 1.
See also that date in this journal, among
other posts tagged The Next Level.

Mathematics as a Black Art

Filed under: General — Tags: , , — m759 @ 11:34 AM

Saturday, June 1, 2019

Crystals for Dabblers

Filed under: General — m759 @ 11:47 AM

The title was suggested by the "Crystal Cult" installations
of Oslo artist Josefine Lyche and by a post of May 30 —

Thursday, May 30, 2019


Filed under: General — m759 @ 8:02 PM Edit This

Jeff Nichols, director of Midnight Special  (2016) —

"When asked about the film's similarities to
the 2015 Disney movie Tomorrowland , which
also posits a futuristic world that exists in an
alternative dimension, Nichols sighed.
'I was a little bummed, I guess,' he said of
when he first learned about the project. . . . 
'Our die was cast. Sometimes this kind of 
collective unconscious that we're all dabbling in,
sometimes you're not the first one out of the gate.' "

See also Jung's four-diamond figure and the previous post.

Writers of fiction are, of course, also dabblers in the collective unconscious.
For instance . . .

A 1971 British paperback edition of The Dreaming Jewels,  
a story by Theodore Sturgeon (first version published in 1950):

The above book cover, together with the Death Valley location
Zabriskie Point, suggests . . .

Those less enchanted by the collective unconscious may prefer a
different weblog's remarks on the same date as the above Borax post . . .

Sunday, March 10, 2019

Vocabulary for SXSW:

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

Foursquare, Inscape, Subway 

Foursquare —

Inscape —

Subway —

Art installation, "Crystal Cult" by Josefine Lyche, at an Oslo subway station —

See also today's previous post.


Filed under: General — Tags: — m759 @ 12:08 PM

Related material —

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

Nietzsche on Heraclitus— 'play in necessity' and 'law in becoming'— illustrated.

Thursday, February 7, 2019

Geometry of the 4×4 Square: The Kummer Configuration

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

From the series of posts tagged Kummerhenge

A Wikipedia article relating the above 4×4 square to the work of Kummer —

A somewhat more interesting aspect of the geometry of the 4×4 square
is its relationship to the 4×6 grid underlying the Miracle Octad Generator
(MOG) of R. T. Curtis.  Hudson's 1905 classic Kummer's Quartic Surface
deals with the Kummer properties above and also foreshadows, without
explicitly describing, the finite-geometry properties of the 4×4 square as
a finite affine 4-space — properties that are of use in studying the Mathieu
group M24  with the aid of the MOG.

Thursday, July 12, 2018

Kummerhenge Illustrated

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


“… the utterly real thing in writing is the only thing that counts…."

— Maxwell Perkins to Ernest Hemingway, Aug. 30, 1935

"Omega is as real  as we need it to be."

— Burt Lancaster in "The Osterman Weekend"

Monday, May 7, 2018

Fish Babel

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

Stanley Fish in the online New York Times  today —

". . . Because it is an article of their faith that politics are bad
and the unmediated encounter with data is good,
internet prophets will fail to see the political implications
of what they are trying to do, for in their eyes political implications
are what they are doing away with.

Indeed, their deepest claim — so deep that they are largely
unaware of it — is that politics can be eliminated. They don’t
regard politics as an unavoidable feature of mortal life but as
an unhappy consequence of the secular equivalent of the
Tower of Babel: too many languages, too many points of view.
Politics (faction and difference) will just wither away when
the defect that generates it (distorted communication) has
been eliminated by unmodified data circulated freely among
free and equal consumers; everyone will be on the same page,
reading from the same script and apprehending the same
universal meanings. Back to Eden!"

The final page, 759, of the Harry Potter saga —

"Talk about magical thinking!" — Fish, ibidem .

See also the above Harry Potter page 
in this  journal Sunday morning.

Thursday, April 20, 2017

Stone Logic

Filed under: General,Geometry — m759 @ 9:48 PM

See also "Romancing the Omega" —

Image- Josefine Lyche work (with 1986 figures by Cullinane) in a 2009 exhibition in Oslo

Related mathematics — Guitart in this journal —

From 'Moving Logic, from Boole to Galois,' by René Guitart, 2005

See also Weyl + Palermo in this journal —


Tuesday, March 7, 2017

Signature Backdrop

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

"The Bitter End’s signature stage backdrop —
a bare 150-year-old brick wall — helped distinguish it from
other popular bohemian hangouts like the Village Gate  
and the Village Vanguard. It appeared on the cover of
Peter, Paul and Mary’s first album."

The New York Times  this evening on a Sunday death


“Looking carefully at Golay’s code is like staring into the sun.”

— Richard Evan Schwartz

See also Schwartz in "The Omega Matrix," a post of 5 PM ET Sunday:

Tuesday, May 24, 2016

Rosenhain and Göpel Revisited

The authors Taormina and Wendland in the previous post
discussed some mathematics they apparently did not know was
related to a classic 1905 book by R. W. H. T. Hudson, Kummer's
Quartic Surface

"This famous book is a prototype for the possibility
of explaining and exploring a many-faceted topic of
research, without focussing on general definitions,
formal techniques, or even fancy machinery. In this
regard, the book still stands as a highly recommendable,
unparalleled introduction to Kummer surfaces, as a
permanent source of inspiration and, last but not least, 
as an everlasting symbol of mathematical culture."

— Werner Kleinert, Mathematical Reviews ,
     as quoted at Amazon.com

Some 4×4 diagrams from that book are highly relevant to the
discussion by Taormina and Wendland of the 4×4 squares within
the 1974 Miracle Octad Generator of R. T. Curtis that were later,
in 1987, described by Curtis as pictures of the vector 4-space over
the two-element Galois field GF(2).

Hudson did not think of his 4×4 diagrams as illustrating a vector space,
but he did use them to picture certain subsets of the 16 cells in each
diagram that he called Rosenhain and Göpel tetrads .

Some related work of my own (click images for related posts)—

Rosenhain tetrads as 20 of the 35 projective lines in PG(3,2)

IMAGE- Desargues's theorem in light of Galois geometry

Göpel tetrads as 15 of the 35 projective lines in PG(3,2)

Anticommuting Dirac matrices as spreads of projective lines

Related terminology describing the Göpel tetrads above

Ron Shaw on symplectic geometry and a linear complex in PG(3,2)

Monday, June 15, 2015

Slow Art

Filed under: General — Tags: — m759 @ 2:03 PM

Slowness is sometimes in the eye of the beholder.

See this journal on Slow Art Day 2015.

Related material: Epistemic States in this journal.

Wednesday, May 13, 2015


Filed under: General — Tags: — m759 @ 9:48 PM

See the previous post, "Space," as well as

SymOmega in this journal and a suggested motto
for The University of Western Australia.


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

Notes on space for day 13 of May, 2015 —

The 13 symmetry axes of the cube may be viewed as
the 13 points of the Galois projective space PG(2,3).
This space (a plane) may also be viewed as the nine points
of the Galois affine space AG(2,3) plus the four points on
an added "line at infinity."

Related poetic material:

The ninefold square and Apollo, as well as 


Tuesday, May 12, 2015

Writing Well*

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

See Stevens + New Haven.

* The above figure may be viewed as
   the Chinese "Holy Field" or as the
   Chinese character for "Well"
   inscribed in a square.

Zinsser Obituary

Filed under: General — Tags: — m759 @ 10:48 PM

"William Zinsser, a writer, editor and teacher
whose book ‘On Writing Well’ sold more than
1.5 million copies by employing his own literary
craftsmanship to urge clarity, simplicity, brevity
and humanity, died on Tuesday [May 12, 2015]
at his home in Manhattan. He was 92." 

— Douglas Martin in the online New York Times

Monday, February 2, 2015

Spielraum as Ω

Filed under: General,Geometry — Tags: , — m759 @ 6:29 PM

From "Origins of the Logical Theory of Probability: von Kries, Wittgenstein, Waismann," by Michael Heidelberger —

"Von Kries calls a range of objective possibilities of a hypothesis or event (under given laws) its Spielraum   (literally: play space), which can mean ‘room to move’, ‘leeway’, ‘latitude of choice’, ‘degree of freedom’ or ‘free play’ and ‘clearance’ – or even ‘scope’. John Maynard Keynes translated it as ‘field’, but the term ‘range’ has generally been adopted in English. Von Kries now holds that if numerical probability were to make any sense at all it must be through this concept of the Spielraum  . Von Kries’s theory is therefore called a ‘Spielraum  theory’ or ‘range theory of probability’."

— International Studies in the Philosophy of Science , Volume 15, Issue 2, 2001, pp. 177-188

See also the tag Points Omega
(Scroll down to January 11-12, 2015.)

Related material:

"Now, for example, in how far are
the six sides of a symmetric die
'equally possible' upon throwing?"

— From "The Natural-Range Conception
     of Probability," by Dr. Jacob Rosenthal,
     page 73 in Time, Chance, and
     Reduction: Philosophical Aspects of
Mechanics , ed. by 
     Gerhard Ernst and Andreas Hüttemann, 
     Cambridge U. Press, 2010, pp. 71-90

Sunday, January 11, 2015

Real Beyond Artifice

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

A professor at Harvard has written about
"the urge to seize and display something
real beyond artifice."

He reportedly died on January 3, 2015.

An image from this journal on that date:

Another Gitterkrieg  image:

 The 24-set   Ω  of  R. T. Curtis

Click on the images for related material.

Thursday, September 11, 2014


Filed under: General — Tags: — m759 @ 10:28 PM

Part I

Image- Josefine Lyche's work (with 1986 figures by Cullinane) in a 2009 exhibition in Oslo

Part II

Part III

Wednesday, August 13, 2014

Stranger than Dreams*

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

Illustration from a discussion of a symplectic structure 
in a 4×4 array quoted here on January 17, 2014 —

See symplectic structure in this journal.

* The final words of Point Omega , a 2010 novel by Don DeLillo.
See also Omega Matrix in this journal.

Monday, August 4, 2014

A Wrinkle in Space

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

"There is  such a thing as a tesseract." — Madeleine L'Engle

An approach via the Omega Matrix:


See, too, Rosenhain and Göpel as The Shadow Guests .

Friday, April 26, 2013

High White

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


For Times Square Church
Click image for a video.

Sunday, September 23, 2012

Plan 9 (continued)–

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

In Like Flynn

From the Wall Street Journal  site Friday evening—

ESSAY September 21, 2012, 9:10 p.m. ET

Are We Really Getting Smarter?

Americans' IQ scores have risen steadily over the past century.
James R. Flynn examines why.

IMAGE- Raven's Progressive Matrices problem with ninth configuration a four-diamonds grid

No, thank you. I prefer the ninth configuration as is—

IMAGE- Four-diamonds grid, the ninth configuration in a Raven's Progressive Matrices problem

Why? See Josefine Lyche's art installation "Grids, you say?"

Her reference there to "High White Noon" is perhaps
related to the use of that phrase in this journal.

The phrase is from a 2010 novel by Don DeLillo.
See "Point Omega," as well as Lyche's "Omega Point,"
in this journal.

The Wall Street Journal  author above, James R. Flynn (born in 1934)
"is famous for his discovery of the Flynn effect, the continued
year-after-year increase of IQ scores in all parts of the world."

His son Eugene Victor Flynn is a mathematician, co-author
of the following chapter on the Kummer surface— 

For use of the Kummer surface in Buddhist metaphysics, see last night's
post "Occupy Space (continued)" and the letters of Nanavira Thera from the 
late 1950s at nanavira.blogspot.com.

These letters, together with Lyche's use of the phrase "high white noon,"
suggest a further quotation

You know that it would be untrue
You know that I would be a liar
If I was to say to you
Girl, we couldn't get much higher

See also the Kummer surface at the web page Configurations and Squares.

Monday, May 21, 2012

Child’s Play (continued*)

Filed under: General — Tags: — m759 @ 7:59 PM

You and I …

we are just like a couple of tots…



Born 1973 in Bergen. Lives and works in Oslo.


2000 – 2004 National Academy of Fine Arts, Oslo
1998 – 2000 Strykejernet Art School, Oslo, NO
1995 – 1998 Philosophy, University of Bergen

University of Bergen—

 It might therefore seem that the idea of digital and analogical systems as rival fundaments to human experience is a new suggestion and, like digital technology, very modern. In fact, however, the idea is as old as philosophy itself (and may be much older). In his Sophist, Plato sets out the following ‘battle’ over the question of ‘true reality’:

What we shall see is something like a battle of gods and giants going on between them over their quarrel about reality [γιγαντομαχία περì της ουσίας] ….One party is trying to drag everything down to earth out of heaven and the unseen, literally grasping rocks and trees in their hands, for they lay hold upon every stock and stone and strenuously affirm that real existence belongs only to that which can be handled and offers resistance to the touch. They define reality as the same thing as body, and as soon as one of the opposite party asserts that anything without a body is real, they are utterly contemptuous and will not listen to another word. (…) Their adversaries are very wary in defending their position somewhere in the heights of the unseen, maintaining with all their force that true reality [την αληθινήν ουσίαν] consists in certain intelligible and bodiless forms. In the clash of argument they shatter and pulverize those bodies which their opponents wield, and what those others allege to be true reality they call, not real being, but a sort of moving process of becoming. On this issue an interminable battle is always going on between the two camps [εν μέσω δε περι ταυτα απλετος αμφοτέρων μάχη τις (…) αει συνέστηκεν]. (…) It seems that only one course is open to the philosopher who values knowledge and truth above all else. He must refuse to accept from the champions of the forms the doctrine that all reality is changeless [and exclusively immaterial], and he must turn a deaf ear to the other party who represent reality as everywhere changing [and as only material]. Like a child begging for 'both', he must declare that reality or the sum of things is both at once [το όν τε και το παν συναμφότερα] (Sophist 246a-249d).

The gods and the giants in Plato’s battle present two varieties of the analog position. Each believes that ‘true reality’ is singular, that "real existence belongs only to" one side or other of competing possibilities. For them, difference and complexity are secondary and, as secondary, deficient in respect to truth, reality and being (την αληθινήν ουσίαν, το όν τε και το παν). Difference and complexity are therefore matters of "interminable battle" whose intended end for each is, and must be (given their shared analogical logic), only to eradicate the other. The philosophical child, by contrast, holds to ‘both’ and therefore represents the digital position where the differentiated two yet belong originally together. Here difference, complexity and systematicity are primary and exemplary.

It is an unfailing mark of the greatest thinkers of the tradition, like Plato, that they recognize the digital possibility and therefore recognize the principal difference of it from analog possibilities.

— Cameron McEwen, "The Digital Wittgenstein,"
    The Wittgenstein Archives at the University of Bergen

* See that phrase in this journal.

Saturday, February 11, 2012


Filed under: General — Tags: , — m759 @ 3:17 PM

See Notes for a Haiku.

Related material—

A novel published on Groundhog Day, 2010—

IMAGE- 'Point Omega' by DeLillo

— as well as Conceptual Art, Josefine Lyche's
"Grids, You Say?" and The Speed of Thought.

Thursday, September 8, 2011


Filed under: General — m759 @ 8:00 PM

IMAGE- American Mathematical Society obituaries for Patrick C. Fischer and Frank Birtel published Sept. 8, 2011

For Fischer's fellow database enthusiast Codd, see Cross-Referenced (April 24, 2003).

For Birtel's fellow pseudoscience enthusiast Frank Tipler, see the artist's comment linked to in Romancing the Omega (April 19, 2011)—

"Omega Point" is a term used by mathematical physicist Frank Tipler for what he maintains is the ultimate fate of the universe required by the laws of physics."

Josefine Lyche, 2009

Thursday, August 4, 2011

Midnight in Oslo

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

For Norway's Niels Henrik Abel (1802-1829)
on his birthday, August Fifth

(6 PM Aug. 4, Eastern Time, is 12 AM Aug. 5 in Oslo.)


Plato's Diamond

The above version by Peter Pesic is from Chapter I of his book Abel's Proof , titled "The Scandal of the Irrational." Plato's diamond also occurs in a much later mathematical story that might be called "The Scandal of the Noncontinuous." The story—


"These passages suggest that the Form is a character or set of characters common to a number of things, i.e. the feature in reality which corresponds to a general word. But Plato also uses language which suggests not only that the forms exist separately (χωριστά ) from all the particulars, but also that each form is a peculiarly accurate or good particular of its own kind, i.e. the standard particular of the kind in question or the model (παράδειγμα ) [i.e. paradigm ] to which other particulars approximate….

… Both in the Republic  and in the Sophist  there is a strong suggestion that correct thinking is following out the connexions between Forms. The model is mathematical thinking, e.g. the proof given in the Meno  that the square on the diagonal is double the original square in area."

– William and Martha Kneale, The Development of Logic , Oxford University Press paperback, 1985

Plato's paradigm in the Meno


Changed paradigm in the diamond theorem (2×2 case) —


Aspects of the paradigm change—

Monochrome figures to
   colored figures

Areas to

Continuous transformations to
   non-continuous transformations

Euclidean geometry to
   finite geometry

Euclidean quantities to
   finite fields

The 24 patterns resulting from the paradigm change—


Each pattern has some ordinary or color-interchange symmetry.

This is the 2×2 case of a more general result. The patterns become more interesting in the 4×4 case. For their relationship to finite geometry and finite fields, see the diamond theorem.

Related material: Plato's Diamond by Oslo artist Josefine Lyche.

Plato’s Ghost  evokes Yeats’s lament that any claim to worldly perfection inevitably is proven wrong by the philosopher’s ghost….”

— Princeton University Press on Plato’s Ghost: The Modernist Transformation of Mathematics  (by Jeremy Gray, September 2008)

"Remember me to her."

— Closing words of the Algis Budrys novel Rogue Moon .

Background— Some posts in this journal related to Abel or to random thoughts from his birthday.

Wednesday, June 1, 2011

The Schwartz Notes

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

A Google search today for material on the Web that puts the diamond theorem
in context yielded a satisfyingly complete list. (See the first 21 results.)
(Customization based on signed-out search activity was disabled.)

The same search limited to results from only the past month yielded,
in addition, the following—


This turns out to be a document by one Richard Evan Schwartz,
Chancellor’s Professor of Mathematics at Brown University.

Pages 12-14 of the document, which is untitled, undated, and
unsigned, discuss the finite-geometry background of the R.T.
Curtis Miracle Octad Generator (MOG) . As today’s earlier search indicates,
this is closely related to the diamond theorem. The section relating
the geometry to the MOG is titled “The MOG and Projective Space.”
It does not mention my own work.

See Schwartz’s page 12, page 13, and page 14.

Compare to the web pages from today’s earlier search.

There are no references at the end of the Schwartz document,
but there is this at the beginning—

These are some notes on error correcting codes. Two good sources for
this material are
From Error Correcting Codes through Sphere Packings to Simple Groups ,
by Thomas Thompson.
Sphere Packings, Lattices, and Simple Groups  by J. H. Conway and N.
Planet Math (on the internet) also some information.

It seems clear that these inadequate remarks by Schwartz on his sources
can and should be expanded.

Sunday, February 20, 2011


Filed under: General,Geometry — m759 @ 10:30 AM

From Galleri MGM in Oslo —


Josefine Lyche
Theme and Variations
26. February – 28. March 2009
Opening reception 26. February 19.00 – 21.00

"Why do we remember the past, but not the future?"
Stephen Hawking, A Brief History of Time, Ch. 9, "The Arrow of Time"

"If you want a vision of the future, imagine a boot stamping on a human face – forever."
George Orwell

Galleri MGM is pleased to present the second solo exhibition by Josefine Lyche (b.1973).

Lyche presents a body of work consisting of sculptures and a wall painting, as well as a series of new paperwork, all using geometrical abstractions and light as a medium. Her work dissolves boundaries between fiction and documentation, depicting how fantasies and dreams collide with and yet help determine the shape of reality.

Theme and variations is a term most commonly used in the music genre as a musical form in which the fundamental musical idea, or theme, is repeated in altered form or accompanied in a different manner.

The exhibition explores geometric shapes and solids and revisits work of artists like Robert Morris and Ellsworth Kelly, giving it a disco treatment of glitter, neon and gloss. The mathematical, science-fiction and new age references incorporated in the works comments on the ambivalent foretelling of utopian hope and dystopian vision of a near, yet unknown future.

The transmission between past and future is shown in the sculpture "The Omega Point" a portal that leads in or out of time and space. …

A connection to today's earlier post, Sunday SchoolThe Oslo Version, from Friday, May 21, 2010.

Lyche's "Omega Point" portal, together with her last name, suggested three posts from the following Saturday morning— which later proved to be the date of Martin Gardner's death—

Art Space, Through the Lyche Gate and The Lyche Gate Asterisk.

For some further religious remarks, see November 9th, 2010— A Theory of Pure Design.

Thursday, July 15, 2010

Brightness at Noon, continued

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

"What exactly was Point Omega?"

This is Robert Wright in Nonzero: The Logic of Human Destiny.

Wright is discussing not the novel Point Omega  by Don DeLillo,
but rather a (related) concept of  the Jesuit philosopher Pierre Teilhard de Chardin.

My own idiosyncratic version of a personal "point omega"—

Image- Josefine Lyche work (with 1986 figures by Cullinane) in a 2009 exhibition in Oslo

Click for further details.

The circular sculpture in the foreground
is called by the artist "The Omega Point."
This has been described as
"a portal that leads in or out of time and space."

For some other sorts of points, see the drawings
on the wall and Geometry Simplified

Image-- The trivial two-point affine space and the trivial one-point projective space, visualized

The two points of the trivial affine space are represented by squares,
and the one point of the trivial projective space is represented by
a line segment separating the affine-space squares.

For related darkness  at noon, see Derrida on différance
as a version of Plato's khôra

(Click to enlarge.)

Image-- Fordham University Press on Derrida, differance, and khora

The above excerpts are from a work on and by Derrida
published in 1997 by Fordham University,
a Jesuit institutionDeconstruction in a Nutshell

Image-- A Catholic view of Derrida

For an alternative to the Villanova view of Derrida,
see Angels in the Architecture.

Wednesday, November 5, 2003

Wednesday November 5, 2003

Filed under: General — Tags: , — m759 @ 2:23 PM

Game Over

 "Everything that has a beginning
     has an end."

— The Matrix Revolutions

Matrix, by Knots, Inc., 1979.

"Easy to master — A lifetime to enjoy!"

The object for 2 players (8-adult)
is to be the first to form a line
consisting of 4 different
colored chips.

Imagist Poem

Digital 'tears in the rain'

Image suggesting the 'Go chip' in 'Wild Palms'

(Recall the Go-chip
in Wild Palms.)

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