Log24

Saturday, May 21, 2011

Unity and Multiplicity (continued*)

Filed under: General — m759 @ 10:48 AM

Heisenberg on Heraclitus

From Physics and Philosophy , by Werner Heisenberg, 1958, reprinted by Penguin Classics, 2003—

Page 28—

… In the philosophy of Heraclitus of Ephesus the concept of Becoming occupies the foremost
place. He regarded that which moves, the fire, as the basic element. The difficulty, to reconcile
the idea of one fundamental principle with the infinite variety of phenomena, is solved for him by
recognizing that the strife of the opposites is really a kind of harmony. For Heraclitus the world is
at once one and many, it is just 'the opposite tension' of the opposites that constitutes the unity
of the One. He says: 'We must know that war is common to all and strife is justice, and that all
things come into being and pass away through strife.'

Looking back to the development of Greek philosophy up to this point one realizes that it has
been borne from the beginning to this

Page 29—

stage by the tension between the One and the Many. For our senses the world consists of an
infinite variety of things and events, colors and sounds. But in order to understand it we have to
introduce some kind of order, and order means to recognize what is equal, it means some sort
of unity. From this springs the belief that there is one fundamental principle, and at the same
time the difficulty to derive from it the infinite variety of things. That there should be a material
cause for all things was a natural starting point since the world consists of matter. But when one
carried the idea of fundamental unity to the extreme one came to that infinite and eternal
undifferentiated Being which, whether material or not, cannot in itself explain the infinite variety
of things. This leads to the antithesis of Being and Becoming and finally to the solution of
Heraclitus, that the change itself is the fundamental principle; the 'imperishable change, that
renovates the world,' as the poets have called it. But the change in itself is not a material cause
and therefore is represented in the philosophy of Heraclitus by the fire as the basic element,
which is both matter and a moving force.

We may remark at this point that modern physics is in some way extremely near to the
doctrines of Heraclitus. If we replace the word 'fire' by the word 'energy' we can almost repeat
his statements word for word from our modern point of view. Energy is in fact the substance
from which all elementary particles, all atoms and therefore all things are made, and energy is
that which moves. Energy is a substance, since its total amount does not change, and the
elementary particles can actually be made from this substance as is seen in many experiments on
the creation of elementary particles. Energy can be changed into motion, into heat, into light
and into tension. Energy may be called the fundamental cause for all change in the world. But this
comparison of Greek philosophy with the ideas of modern science will be discussed later.

* See earlier uses of the phrase in this journal. Further background— Hopkins and Heraclitus.

Wednesday, May 4, 2011

Unity and Multiplicity

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

Continued from Crimson Walpurgisnacht.

EpigraphsTwo quotations from  
Shakespeare's Birthday last year

Rebecca Goldstein
   on first encountering Plato
 

"I was reading Durant's section on Plato, struggling to understand his theory of the ideal Forms that lay in inviolable perfection out beyond the phantasmagoria. (That was the first, and I think the last, time that I encountered that word.)"

Screenwriter Joan Didion

"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."

From Thomas Mann, "Schopenhauer," 1938, in Essays of Three Decades , translated by H. T. Lowe-Porter, Alfred A. Knopf, 1947, pp. 372-410—

Page 372: THE PLEASURE we take in a metaphysical system, the gratification purveyed by the intellectual organization of the world into a closely reasoned, complete, and balanced structure of thought, is always of a pre-eminently aesthetic kind. It flows from the same source as the joy, the high and ever happy satisfaction we get from art, with its power to shape and order its material, to sort out life's manifold confusions so as to give us a clear and general view.

Truth and beauty must always be referred the one to the other. Each by itself, without the support given by the other, remains a very fluctuating value. Beauty that has not truth on its side and cannot have reference to it, does not live in it and through it, would be an empty chimera— and "What is truth?"

….

Page 376: … the life of Plato was a very great event in the history of the human spirit; and first of all it was a scientific and a moral event. Everyone feels that something profoundly moral attaches to this elevation of the ideal as the only actual, above the ephemeralness and multiplicity of the phenomenal, this devaluation  of the senses to the advantage of the spirit, of the temporal to the advantage of the eternal— quite in the spirit of the Christianity that came after it. For in a way the transitory phenomenon, and the sensual attaching to it, are put thereby into a state of sin: he alone finds truth and salvation who turns his face to the eternal. From this point of view Plato's philosophy exhibits the connection between science and ascetic morality.

But it exhibits another relationship: that with the world of art. According to such a philosophy time itself is merely the partial and piecemeal view which an individual holds of ideas— the latter, being outside time, are thus eternal. "Time"— so runs a beautiful phrase of Plato— "is the moving image of eternity." And so this pre-Christian, already Christian doctrine, with all its ascetic wisdom, possesses on the other hand extraordinary charm of a sensuous and creative kind; for a conception of the world as a colourful and moving phantasmagoria of pictures, which are transparencies for the ideal and the spiritual, eminently savours of the world of art, and through it the artist, as it were, first comes into his own.

From last night's online NY Times  obituaries index—

http://www.log24.com/log/pix11A/110503-NYTobits.jpg

"How much story do you want?" — George Balanchine

Tuesday, April 26, 2011

Unity and Multiplicity

Filed under: General,Geometry — m759 @ 5:48 PM

Today's earlier post mentions one approach to the concepts of unity and multiplicity. Here is another.

http://www.log24.com/log/pix11A/110427-Cube27.jpg
Unity:
The 3×3×3 Galois Cube

Ed Pegg Jr.'s program at Wolfram demonstrating concepts of a 1985 note by Cullinane

Multiplicity:

One of a group, GL(3,3), of 11,232
natural transformations of the 3×3×3 Cube

See also the earlier 1985 3×3 version by Cullinane.

Tuesday, August 20, 2019

Ice Giants and Fire Gods: Mind the Gap

Filed under: General — Tags: — m759 @ 5:52 PM

From my reading Monday morning —

From the online New York Times  this afternoon —

Related literature —

For the Church of Synchronology

The Gigantomachia page above is dated September 20, 2003.
See as well my own webpage from that date: "The Form, the Pattern."

Monday, August 19, 2019

Gods and Giants

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

For "the Pergamum altar," see Pergamon in this journal.

See also . . .

Tuesday, June 25, 2019

Medium and Message

Filed under: General — Tags: — m759 @ 10:31 AM

Wednesday, May 1, 2019

The Medium and the Message

Filed under: General — Tags: — m759 @ 6:45 PM

In memory of Quentin Fiore — from a Log24 search for McLuhan,
an item related to today's previous post . . .

Related material from Log24 on the above-reported date of death —

See also, from a search for Analogy in this journal . . .

 .

Sunday, August 26, 2018

Mathematics and Narrative: Queensland

Filed under: General — Tags: — m759 @ 3:56 AM

Mathematics —

http://www.log24.com/log/pix18/180826-Cameron-on-U_of_Queensland-slide-500w.jpg

Narrative —

http://www.log24.com/log/pix18/180826-Huntsman-Winters-War-Billboard-Art-500w.jpg

See also other recent posts tagged Queensland.

Wednesday, May 16, 2018

Church and Temple

Filed under: General — Tags: , , , — m759 @ 10:48 AM

Friday, June 23, 2017

Annals of Art and Design

Filed under: General — Tags: — m759 @ 10:12 PM

The life of Mr. Breder is not unrelated to that of Carl Andre.

See also, in this  journal, Bulk Apperception.

Wednesday, May 10, 2017

In the Park with Yin and Yang

Filed under: General,Geometry — Tags: — m759 @ 10:35 PM

In memory of an art dealer who 
reportedly died on Sunday, May 7—

Decorations for a Cartoon Graveyard

Tuesday, May 9, 2017

Text and Context

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

Some context for the previous post, which was about
a new Art Space  Pinterest board

Tuesday, July 12, 2016

Group Elements and Skew Lines

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

The following passage by Igor Dolgachev (Good Friday, 2003
seems somewhat relevant (via its connection to Kummer's 166 )
to previous remarks here on Dirac matrices and geometry

Note related remarks from E. M. Bruins in 1959 —

First page of 'Configurations in Quantum Mechanics,' by E.M. Bruins, 1959

Friday, June 3, 2016

Bruins and van Dam

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

A review of some recent posts on Dirac and geometry,
each of which mentions the late physicist Hendrik van Dam:

The first of these posts mentions the work of E. M. Bruins.
Some earlier posts that cite Bruins:

Wednesday, May 25, 2016

Kummer and Dirac

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

From "Projective Geometry and PT-Symmetric Dirac Hamiltonian,"
Y. Jack Ng  and H. van Dam, 
Physics Letters B , Volume 673, Issue 3,
23 March 2009, Pages 237–239

(http://arxiv.org/abs/0901.2579v2, last revised Feb. 20, 2009)

" Studies of spin-½ theories in the framework of projective geometry
have been undertaken before. See, e.g., Ref. [4]. 1 "

1 These papers are rather mathematical and technical.
The authors of the first two papers discuss the Dirac equation
in terms of the Plucker-Klein correspondence between lines of
a three-dimensional projective space and points of a quadric
in a five-dimensional projective space. The last paper shows
that the Dirac equation bears a certain relation to Kummer’s
surface, viz., the structure of the Dirac ring of matrices is 
related to that of Kummer’s 166 configuration . . . ."

[4]

O. Veblen
Proc. Natl. Acad. Sci. USA , 19 (1933), p. 503
Full Text via CrossRef

E.M. Bruins
Proc. Nederl. Akad. Wetensch. , 52 (1949), p. 1135

F.C. Taylor Jr., Master thesis, University of North Carolina
at Chapel Hill (1968), unpublished


A remark of my own on the structure of Kummer’s 166 configuration . . . .

See that structure in this  journal, for instance —

See as well yesterday morning's post.

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, December 14, 2015

Dirac and Geometry

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

(Continued)

See a post by Peter Woit from Sept. 24, 2005 — Dirac's Hidden Geometry.

The connection, if any, with recent Log24 posts on Dirac and Geometry
is not immediately apparent.  Some related remarks from a novel —

From Broken Symmetries by Paul Preuss
(first published by Simon and Schuster in 1983) —

"He pondered the source of her fascination with the occult, which sooner or later seemed to entangle a lot of thoughtful people who were not already mired in establishmentarian science or religion. It was  the religious impulse, at base. Even reason itself could function as a religion, he supposed— but only for those of severely limited imagination. 

He’d toyed with 'psi' himself, written a couple of papers now much quoted by crackpots, to his chagrin. The reason he and so many other theoretical physicists were suckers for the stuff was easy to understand— for two-thirds of a century an enigma had rested at the heart of theoretical physics, a contradiction, a hard kernel of paradox. Quantum theory was inextricable from the uncertainty relations. 

The classical fox knows many things, but the quantum-mechanical hedgehog knows only one big thing— at a time. 'Complementarity,' Bohr had called it, a rubbery notion the great professor had stretched to include numerous pairs of opposites. Peter Slater was willing to call it absurdity, and unlike some of his older colleagues who, following in Einstein’s footsteps, demanded causal explanations for everything (at least in principle), Peter had never thirsted after 'hidden variables' to explain what could not be pictured. Mathematical relationships were enough to satisfy him, mere formal relationships which existed at all times, everywhere, at once. It was a thin nectar, but he was convinced it was the nectar of the gods. 

The psychic investigators, on the other hand, demanded to know how  the mind and the psychical world were related. Through ectoplasm, perhaps? Some fifth force of nature? Extra dimensions of spacetime? All these naive explanations were on a par with the assumption that psi is propagated by a species of nonlocal hidden variables, the favored explanation of sophisticates; ignotum per ignotius

'In this connection one should particularly remember that the human language permits the construction of sentences which do not involve any consequences and which therefore have no content at all…' The words were Heisenberg’s, lecturing in 1929 on the irreducible ambiguity of the uncertainty relations. They reminded Peter of Evan Harris Walker’s ingenious theory of the psi force, a theory that assigned psi both positive and negative values in such a way that the mere presence of a skeptic in the near vicinity of a sensitive psychic investigation could force null results. Neat, Dr. Walker, thought Peter Slater— neat, and totally without content. 

One had to be willing to tolerate ambiguity; one had to be willing to be crazy. Heisenberg himself was only human— he’d persuasively woven ambiguity into the fabric of the universe itself, but in that same set of 1929 lectures he’d rejected Dirac’s then-new wave equations with the remark, 'Here spontaneous transitions may occur to the states of negative energy; as these have never been observed, the theory is certainly wrong.' It was a reasonable conclusion, and that was its fault, for Dirac’s equations suggested the existence of antimatter: the first antiparticles, whose existence might never have been suspected without Dirac’s crazy results, were found less than three years later. 

Those so-called crazy psychics were too sane, that was their problem— they were too stubborn to admit that the universe was already more bizarre than anything they could imagine in their wildest dreams of wizardry."

Particularly relevant

"Mathematical relationships were enough to satisfy him,
mere formal relationships which existed at all times,
everywhere, at once."

Some related pure  mathematics

Anticommuting Dirac matrices as spreads of projective lines

Friday, November 27, 2015

Einstein and Geometry

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

(A Prequel to Dirac and Geometry)

"So Einstein went back to the blackboard.
And on Nov. 25, 1915, he set down
the equation that rules the universe.
As compact and mysterious as a Viking rune,
it describes space-time as a kind of sagging mattress…."

— Dennis Overbye in The New York Times  online,
     November 24, 2015

Some pure  mathematics I prefer to the sagging Viking mattress —

Readings closely related to the above passage —

Thomas Hawkins, "From General Relativity to Group Representations:
the Background to Weyl's Papers of 1925-26
," in Matériaux pour
l'histoire des mathématiques au XXe siècle:
Actes du colloque
à la mémoire de Jean Dieudonné
, Nice, 1996  (Soc. Math.
de France, Paris, 1998), pp. 69-100.

The 19th-century algebraic theory of invariants is discussed
as what Weitzenböck called a guide "through the thicket
of formulas of general relativity."

Wallace Givens, "Tensor Coordinates of Linear Spaces," in
Annals of Mathematics  Second Series, Vol. 38, No. 2, April 1937, 
pp. 355-385.

Tensors (also used by Einstein in 1915) are related to 
the theory of line complexes in three-dimensional
projective space and to the matrices used by Dirac
in his 1928 work on quantum mechanics.

For those who prefer metaphors to mathematics —

"We acknowledge a theorem's beauty
when we see how the theorem 'fits' in its place,
how it sheds light around itself, like a Lichtung ,
a clearing in the woods." 
— Gian-Carlo Rota, Indiscrete Thoughts ,
Birkhäuser Boston, 1997, page 132

Rota fails to cite the source of his metaphor.
It is Heidegger's 1964 essay, "The End of Philosophy
and the Task of Thinking" —

"The forest clearing [ Lichtung ] is experienced
in contrast to dense forest, called Dickung  
in our older language." 
— Heidegger's Basic Writings 
edited by David Farrell Krell, 
Harper Collins paperback, 1993, page 441

Monday, November 23, 2015

Dirac and Line Geometry

Filed under: General,Geometry — Tags: , , — m759 @ 2:29 AM

Some background for my post of Nov. 20,
"Anticommuting Dirac Matrices as Skew Lines" —

First page of 'Configurations in Quantum Mechanics,' by E.M. Bruins, 1959

His earlier paper that Bruins refers to, "Line Geometry
and Quantum Mechanics," is available in a free PDF.

For a biography of Bruins translated by Google, click here.

For some additional historical background going back to
Eddington, see Gary W. Gibbons, "The Kummer
Configuration and the Geometry of Majorana Spinors,"
pages 39-52 in Oziewicz et al., eds., Spinors, Twistors,
Clifford Algebras, and Quantum Deformations:
Proceedings of the Second Max Born Symposium held
near Wrocław, Poland, September 1992
 . (Springer, 2012,
originally published by Kluwer in 1993.)

For more-recent remarks on quantum geometry, see a
paper by Saniga cited in today's update to my Nov. 20 post

Friday, November 13, 2015

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

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



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)—

Sunday, October 25, 2015

Buyers and Sellers of Children

Filed under: General — Tags: — m759 @ 6:45 AM

(Continued.)

Featured on this morning's online front page of
The New York Times

Some further details —

An example of New York Times  culture is shown above —

"… Mondrian paintings at the Museum of Modern Art
blend symmetry with a tensile volatility."

(To be fair, this contemptible bullshit is from a picture caption,
not from the art review being summarized.)

Related cultural observations —

Math for Child Buyers  and  Fiction for Child Sellers.

Friday, May 22, 2015

Mathmagic Land

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

Donald Duck with Pythagorean pentagram on hand

Donald in Mathmagic Land

Manly P. Hall

Saturday, February 7, 2015

Words and Pictures, continued

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

Word and Object

Filed under: General — Tags: , — m759 @ 7:00 PM

From actor James Spader, whose birthday is today —

"… my father taught English. My mother taught art…."

— Spader in a 2014 interview

See as well the 2013 film "Words and Pictures"
and Log24 posts on a 2007 film, "The Last Mimzy."

Above: A scene from Spader's TV series "The Blacklist"
that was aired on Thursday, February 5, 2015.

Monday, November 3, 2014

Gods and Giants

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

A weblog reports Chris Rock's remarks
on Saturday Night Live this past weekend:

"It’s America, we commercialize everything.
Look at what we did to Christmas.
Christmas.  Christmas is Jesus’ birthday.
It’s Jesus’ birthday.  Now, I don’t know Jesus
but from what I’ve read, Jesus is the least
materialistic person to ever roam the earth.
No bling on Jesus.
Jesus kept a low profile and we turned his
birthday into the most materialistic day of the
year.  Matter of fact, we have the Jesus birthday
season.  It’s a whole season of materialism.
Then, at the end of the Jesus birthday season
we have the nerve to have an economist come
on TV and tell you how horrible the Jesus birthday
season was this year.  Oh, we had a horrible Jesus’
birthday this year.  Hopefully, business will pick up
by his Crucifixion.”

Related music and image:

"Show us the way to the next little girl …"

Natalie Wood in "Miracle on 34th Street" (1947)

Related non-materialistic meditations:
The Rhetoric of Abstract Concepts and Gods and Giants.

Friday, October 10, 2014

Both Hands and an Ass Map

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

(Continued from Grids and Space and posts tagged Riddle for Caltech)

IMAGE- Scene from 'Deathtrap,' with subtitle

Tuesday, September 9, 2014

Smoke and Mirrors

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

This post is continued from a March 12, 2013, post titled
"Smoke and Mirrors" on art in Tromsø, Norway, and from
a June 22, 2014, post on the nineteenth-century 
mathematicians Rosenhain and Göpel.

The latter day was the day of death for 
mathematician Loren D. Olson, Harvard '64.

For some background on that June 22 post, see the tag 
Rosenhain and Göpel in this journal.

Some background on Olson, who taught at the
University of Tromsø, from the American Mathematical
Society yesterday:

Olson died not long after attending the 50th reunion of the
Harvard Class of 1964.

For another connection between that class (also my own) 
and Tromsø, see posts tagged "Elegantly Packaged."
This phrase was taken from today's (print) 
New York Times  review of a new play titled "Smoke."
The phrase refers here  to the following "package" for 
some mathematical objects that were named after 
Rosenhain and Göpel — a 4×4 array —

For the way these objects were packaged within the array
in 1905 by British mathematician R. W. H. T. Hudson, see
a page at finitegometry.org/sc. For the connection to the art 
in Tromsø mentioned above, see the diamond theorem.

Thursday, July 24, 2014

Death in Mathmagic Land

Filed under: General — Tags: , , — m759 @ 11:29 AM

(Continued from May 14, 2014.)

See also consciousness growth and the previous post.

Sunday, June 22, 2014

Göpel and Rosenhain

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

(Continued)

Some bizarre remarks on “purity” in the previous post
suggest a review of some pure  mathematics.

Wednesday, May 14, 2014

Death in Mathmagic Land

Filed under: General — Tags: , — m759 @ 3:28 PM

"It was our old friend Pythagoras who discovered
that the pentagram was full of mathematics."

— Narrator, "Donald in Mathmagic Land," Disney, 1959

and it was Peter J. Cameron who discovered that
mathematics was full of pentagrams.

From Log24 on May 3:  Gray Space —

Robert J. Stewart (left) andpentagram photo posted May 2
by Oslo artist Josefine Lyche. See also Lyche in this journal.

From Log24 on May 13:  An Artist's Memorial —

The death mentioned in the above May 13 post occurred on
May 12, the date of a scheduled Black Mass at Harvard.

Related material:

Wednesday, March 5, 2014

Rosenhain and Göpel Again

Filed under: General — Tags: , — m759 @ 8:25 PM

IMAGE- Rosenhain, Göpel, and hyperelliptic curves

See also Rosenhain and Göpel in the Wikipedia
article Kummer surface, and in this journal.

Related material: user @hyperelliptic on Twitter.

Saturday, February 15, 2014

Rosenhain and Göpel

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

(Continued)

See The Oslo Version in this journal and the New Year’s Day 2014 post.
The pictures of the 56 spreads in that post (shown below) are based on
the 20 Rosenhain and 15 Göpel tetrads that make up the 35 lines of
PG(3,2), the finite projective 3-space over the 2-element Galois field.

IMAGE- The 56 spreads in PG(3,2)

Click for a larger image.

Saturday, September 21, 2013

Mathematics and Narrative (continued)

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

Mathematics:

A review of posts from earlier this month —

Wednesday, September 4, 2013

Moonshine

Filed under: Uncategorized — m759 @ 4:00 PM

Unexpected connections between areas of mathematics
previously thought to be unrelated are sometimes referred
to as "moonshine."  An example—  the apparent connections
between parts of complex analysis and groups related to the
large Mathieu group M24. Some recent work on such apparent
connections, by Anne Taormina and Katrin Wendland, among
others (for instance, Miranda C.N. Cheng and John F.R. Duncan),
involves structures related to Kummer surfaces .
In a classic book, Kummer's Quartic Surface  (1905),
R.W.H.T. Hudson pictured a set of 140 structures, the 80
Rosenhain tetrads and the 60 Göpel tetrads, as 4-element
subsets of a 16-element 4×4 array.  It turns out that these
140 structures are the planes of the finite affine geometry
AG(4,2) of four dimensions over the two-element Galois field.
(See Diamond Theory in 1937.)

Thursday, September 5, 2013

Moonshine II

Filed under: Uncategorized — Tags:  — m759 @ 10:31 AM

(Continued from yesterday)

The foreword by Wolf Barth in the 1990 Cambridge U. Press
reissue of Hudson's 1905 classic Kummer's Quartic Surface
covers some of the material in yesterday's post Moonshine.

The distinction that Barth described in 1990 was also described, and illustrated,
in my 1986 note "Picturing the smallest projective 3-space."  The affine 4-space
over the the finite Galois field GF(2) that Barth describes was earlier described—
within a 4×4 array like that pictured by Hudson in 1905— in a 1979 American
Mathematical Society abstract, "Symmetry invariance in a diamond ring."

"The distinction between Rosenhain and Goepel tetrads
is nothing but the distinction between isotropic and
non-isotropic planes in this affine space over the finite field."

The 1990 paragraph of Barth quoted above may be viewed as a summary
of these facts, and also of my March 17, 2013, note "Rosenhain and Göpel
Tetrads in PG(3,2)
."

Narrative:

Aooo.

Happy birthday to Stephen King.

Thursday, April 25, 2013

Rosenhain and Göpel Revisited

Filed under: General,Geometry — Tags: , — m759 @ 5:24 PM

Some historical background for today's note on the geometry
underlying the Curtis Miracle Octad Generator (MOG):

IMAGE- Bateman in 1906 on Rosenhain and Göpel tetrads

The above incidence diagram recalls those in today's previous post
on the MOG, which is used to construct the large Mathieu group M24.

For some related material that is more up-to-date, search the Web
for Mathieu + Kummer .

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.

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, June 27, 2011

Galois Cube Revisited

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

http://www.log24.com/log/pix11A/110427-Cube27.jpg
   The 3×3×3 Galois Cube

    See Unity and Multiplicity.

   This cube, unlike Rubik's, is a
    purely mathematical structure.

    Its properties may be compared
    with those of the order-2  Galois
    cube (of eight subcubes, or
    elements ) and the order-4  Galois
    cube (of 64 elements). The
    order-3  cube (of 27 elements)
    lacks, because it is based on
    an odd  prime, the remarkable
    symmetry properties of its smaller
    and larger cube neighbors.

Saturday, April 30, 2011

Crimson Walpurgisnacht

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

Part I — Unity and Multiplicity
              (Continued from The Talented and Galois Cube)

On Husserl's 'Philosophie der Arithmetik'- 'A feeling, an angel, the moon, and Italy'

Part II — "A feeling, an angel, the moon, and Italy"—

Click for details

Dean Martin and Peter Lawford in Crimson ad for 2011 Quincy House Q-Ball

Wednesday, October 9, 2019

The Joy of Six

Filed under: General — Tags: , , , , — m759 @ 11:07 PM

Note that in the pictures below of the 15 two-subsets of a six-set,
the symbols 1 through 6 in Hudson's square array of 1905 occupy the
same positions as the anticommuting Dirac matrices in Arfken's 1985
square array. Similarly occupying these positions are the skew lines
within a generalized quadrangle (a line complex) inside PG(3,2).

Anticommuting Dirac matrices as spreads of projective lines

Related narrative The "Quantum Tesseract Theorem."

Tuesday, August 27, 2019

Tiger Leaps

Filed under: General — Tags: , — m759 @ 10:33 AM

Walter Benjamin on 'a tiger's leap into the past'

See also Tiger in this journal, esp.

The image “http://www.log24.com/log/pix05/050123-Tiger.JPG” cannot be displayed, because it contains errors.   and . . .

other "Death and the Spirit" posts.

Monday, August 19, 2019

A Couple of Tots

Filed under: General — Tags: — m759 @ 10:15 PM

The title is from the post "Child's Play" of May 21, 2012 . . .

"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)."

Related material —

"Schoolgirl Space: 1984 Revisited" (July 9, 2019) and
posts tagged Tetrahedron vs. Square.

Sunday, August 18, 2019

Structure at Pergamon

Filed under: General — Tags: — m759 @ 2:35 AM

Some background for The Epstein Chronicles

"What modern painters
are trying to do,
if they only knew it,
is paint invariants."

— James J. Gibson, Leonardo,
    Vol. 11, pp. 227-235.
    Pergamon Press Ltd., 1978

See also Robert Maxwell,
Frank Oppenheimer,
and the history of Leonardo .

Click the above Pergamon Press image
for Pergamon-related material.

Wednesday, August 14, 2019

Double Veil

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

"If you want to win, you need to know just one thing and not to waste your time
on anything else: the pleasures of erudition are reserved for losers. The more
a person knows, the more things have gone wrong."

— Eco, Umberto. Numero Zero  (p. 9). HMH Books. Kindle Edition. 

"All this notwithstanding, I dreamed what all losers dream, about one day writing
a book that would bring me fame and fortune. To learn how to become a great
writer, I became what in the last century was called the nègre  (or ghostwriter,
as they say today, to be politically correct) for an author of detective stories who
gave himself an American name to improve sales, like the actors in spaghetti
westerns. But I enjoyed working in the shadows, hidden behind a double veil
(the Other’s and the Other’s other name)."

— Eco, Umberto. Numero Zero  (pp. 9-10). HMH Books. Kindle Edition. 

How's this for a double veil, Umberto?

Boeing/Being Meets the Flying Spaghetti Monster . . .

The Mask of Zero

Filed under: General — Tags: , — m759 @ 12:55 PM

For the title, see Zero: Both Real and Imaginary (a Log24 search).

The title was suggested by the previous post, by Zorro Ranch,
by the classic 1967 film The Producers , and by . . .

Related material —

Vanity Fair  on Sept. 8, 2017, celebrated the young actress
who played Beverly Marsh in the 2017 film version of
Stephen King's IT . See a post from her 12th birthday —
"Winter's Game" — that touches upon Maori themes.

More generally, see Bester + Deceivers in this  journal.

And for the Church of Synchronology . . .
See posts related to the above Vanity Fair  date.

Sunday, July 14, 2019

Old Pathways in Science:

Filed under: General — Tags: — m759 @ 12:37 PM

The Quantum Tesseract Theorem Revisited

From page 274 — 

"The secret  is that the super-mathematician expresses by the anticommutation
of  his operators the property which the geometer conceives as  perpendicularity
of displacements.  That is why on p. 269 we singled out a pentad of anticommuting
operators, foreseeing that they would have an immediate application in describing
the property of perpendicular directions without using the traditional picture of space.
They express the property of perpendicularity without the picture of perpendicularity.

Thus far we have touched only the fringe of the structure of our set of sixteen E-operators.
Only by entering deeply into the theory of electrons could I show the whole structure
coming into evidence."

A related illustration, from posts tagged Dirac and Geometry —

Anticommuting Dirac matrices as spreads of projective lines

Compare and contrast Eddington's use of the word "perpendicular"
with a later use of the word by Saniga and Planat.

Wednesday, July 3, 2019

Best Meets Bester

Filed under: General — m759 @ 1:00 AM

See as well The Alchemist's Chessboard.

Tuesday, June 25, 2019

Analogy

Filed under: General — Tags: — m759 @ 6:36 AM

Monday, June 24, 2019

Ex Fano Apollinis

Filed under: General — Tags: — m759 @ 11:57 PM

Saturday, December 22, 2018

Cremona-Richmond

Filed under: General,Geometry — Tags: — m759 @ 12:34 PM

The following are some notes on the history of Clifford algebras
and finite geometry suggested by the "Clifford Modules" link in a
Log24 post of March 12, 2005

A more recent appearance of the configuration —

Wednesday, December 12, 2018

An Inscape for Douthat

Filed under: G-Notes,General,Geometry — Tags: , — m759 @ 9:41 AM

Some images, and a definition, suggested by my remarks here last night
on Apollo and Ross Douthat's remarks today on "The Return of Paganism" —

Detail of Feb. 20, 1986, note by Steven H. Cullinane on Weyl's 'relativity problem'

Kibler's 2008 'Variations on a theme' illustrated.

In finite geometry and combinatorics,
an inscape  is a 4×4 array of square figures,
each figure picturing a subset of the overall 4×4 array:


 

Related material — the phrase
"Quantum Tesseract Theorem" and  

A.  An image from the recent
      film "A Wrinkle in Time" — 

B.  A quote from the 1962 book —

"There's something phoney
in the whole setup, Meg thought.
There is definitely something rotten
in the state of Camazotz."

Sunday, November 18, 2018

Space Music

Filed under: General,Geometry — Tags: , , — m759 @ 9:27 AM

'The Eddington Song,' based on 'The Philosophy of Physical Science,' p. 141 (1939)

Update of Nov. 19 —

"Design is how it works." — Steve Jobs

See also www.cullinane.design.

Tuesday, September 25, 2018

Trinity

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

See some posts related to three names
associated with Trinity College, Cambridge —

Atiyah + Shaw + Eddington .

Saturday, September 1, 2018

Ron Shaw — D. 21 June 2016

The date of Ron Shaw's 2016 death appears to be June 21:

http://www.log24.com/log/pix18/180901-Ron_Shaw-d_21_June_2016-LMS-500w.jpg

All other Internet sources I have seen omit the June 21 date.

This  journal on that date —

http://www.log24.com/log/pix18/180901-The_Central_Structure-21_June_2016.jpg

Wednesday, August 22, 2018

Castle Rock

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

See the title in a TV review* from io9 this morning and in
this  journal.

http://www.log24.com/log/pix18/180822-io9-URL-and-logo.jpg

* Spoiler alert

Apple

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

http://www.log24.com/log/pix18/180822-Mirror_Mirror-ad-One_Bad_Apple-2012.jpg

Tuesday, August 21, 2018

Purported Endgame

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

http://www.log24.com/log/pix18/180821-NYT-Corrections-Japanese-IMA-NOW.jpg

A search for "IMA" yields, at ima.org.au . . .

"Since 1975, the Institute of Modern Art (IMA) has been Queensland’s 
leading independent forum for art and its discourses." And

http://www.log24.com/log/pix18/180821-IMA-Brisbane-Mirror_Mirror_Then_and_Now-text.jpg

http://www.log24.com/log/pix18/180821-IMA-Brisbane-Mirror_Mirror_Then_and_Now-500wide.jpg

"Clog, therefore, purple Jack and crimson Jill."

— Wallace Stevens

Available Light

Filed under: General — Tags: — m759 @ 12:16 PM

http://www.log24.com/log/pix18/180821-Jesse_Harris-AKA_Playland-co-writer.jpg

Click the image above for an interview dated Nov. 9, 2015.
Also on that date —

http://www.log24.com/log/pix18/180821-Nov_9_2015-Forever_Now-500w.jpg

Wednesday, August 1, 2018

Abschattungen*

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

“… I realized that to me, Gödel and Escher and Bach
were only shadows cast in different directions
by some central solid essence.
I tried to reconstruct the central object . . . ."

— Douglas Hofstadter (1979)

See also posts of July 23, 2007, and April 7, 2018.

* Term from a visual-culture lexicon —

“Ready Player One” Mnemonic:

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

O AS IS

 

See also other posts tagged "Is and As."

Tuesday, July 31, 2018

A is for Abschattungen

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

Max Bialystock discovers a new playwright

Lexicon

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

"A blank underlies the trials of device." — Wallace Stevens

IMAGE- The ninefold square .

Wednesday, May 16, 2018

Review

Filed under: General — Tags: , — m759 @ 5:52 PM

The title of the previous post, "Church and Temple," together
with today's online New York Times  obituaries for singer 
Lara Saint Paul (d. May 8) and playwright Leah Rose Napolin
(d. May 13), suggests a review

See as well a Log24 search for Isaac Singer.

Same Old Story

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

. . . as time goes by.

Friday, April 13, 2018

Philosophy 101

Filed under: General — Tags: — m759 @ 2:48 PM

See also Log24 posts now tagged "Is and As."

Monday, March 12, 2018

“Quantum Tesseract Theorem?”

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

Remarks related to a recent film and a not-so-recent film.

For some historical background, see Dirac and Geometry in this journal.

Also (as Thas mentions) after Saniga and Planat —

The Saniga-Planat paper was submitted on December 21, 2006.

Excerpts from this  journal on that date —

A Halmos tombstone and the tale of HAL and the pod bay doors

     "Open the pod bay doors, HAL."

Monday, February 26, 2018

The Unfolding

Filed under: General,Geometry — Tags: — m759 @ 9:20 AM

From the website of Richard P. Gabriel

" As part of my studies, I came up with a 'theory of poetry' 
based loosely on Christopher Alexander’s 'Nature of Order.' "
[The Alexander link is mine, not Gabriel’s.]

A phrase from this  journal a year ago today — "poetic order" —
links to the theory of Gabriel —

From Gabriel's "The Nature of Poetic Order" —

Positive Space

• Positive space is the characteristic of a center
that moves outward from itself, seemingly oozing life
rather than collapsing on itself
• An image that resonates is showing positive space
• A word that has many connotations that fit with the
other centers in the poem is showing positive space
• It is an expansion outward rather than a contraction
inward, and it shows that the poem is unfolding
in front of us and not dying

Related material —

From a post of April 26, 2017

Sunday, February 25, 2018

The Dark, Seductive Art of Groton

Filed under: General — Tags: — m759 @ 9:26 AM

A sequel to Friday's "The Dark, Seductive Art of Phillips Exeter."

From Twitter today —

Groton School‏ @GrotonSchool  Feb 1

Groton School Theater Department Presents: CABARET
Friday, February 23rd at 7:30 pm
Saturday, February 24th at 7:30 pm
Sunday, February 25th at 3 pm
All performances are free to the public. Please make ticket
reservations in advance online: http://bit.ly/CABARET2018.

Friday, February 23, 2018

The Dark, Seductive Art of Phillips Exeter

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

From two pedagogues in Montana —

https://www.nctm.org/Publications/
Mathematics-Teaching-in-Middle-School/
2016/Vol22/Issue1/Quilt-Block-Symmetries/

Related material from author Dan Brown's father,
a pedagogue who taught at Phillips Exeter Academy :

Click the above image for some background.

Related material from Log24 —
 

Verwandlungslehre.
 

Compare and contrast with the above
Transformational Geometry  cover:

Related material from Vienna — The previous post and
Wittgenstein on Bewitchment.

See as well . . .

Click to enlarge.

Snow Games

Filed under: General — Tags: — m759 @ 9:42 AM

'Origin' (NOT by Dan Brown)

Related material — (Click to enlarge) —

"Risin' up to the challenge of our rival"

Eye of the Tiger

Detail —

Saturday, February 17, 2018

The Binary Revolution

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

Michael Atiyah on the late Ron Shaw

Phrases by Atiyah related to the importance in mathematics
of the two-element Galois field GF(2) —

  • "The digital revolution based on the 2 symbols (0,1)"
  • "The algebra of George Boole"
  • "Binary codes"
  • "Dirac's spinors, with their up/down dichotomy"

These phrases are from the year-end review of Trinity College,
Cambridge, Trinity Annual Record 2017 .

I prefer other, purely geometric, reasons for the importance of GF(2) —

  • The 2×2 square
  • The 2x2x2 cube
  • The 4×4 square
  • The 4x4x4 cube

See Finite Geometry of the Square and Cube.

See also today's earlier post God's Dice and Atiyah on the theology of 
(Boolean) algebra vs. (Galois) geometry:

God’s Dice

Filed under: General,Geometry — Tags: , — m759 @ 10:45 AM

On a Trinity classmate of Ian Macdonald (see previous post)—

Atiyah's eulogy of Shaw in Trinity Annual Record 2017 
is on pages 137 through 146.  The conclusion —

 

Sunday, December 10, 2017

Geometry

Filed under: G-Notes,General,Geometry — Tags: , — m759 @ 8:45 PM

Google search result for Plato + Statesman + interlacing + interweaving

See also Symplectic in this journal.

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  figure appears in remarks on
the diamond-theorem correlation in the webpage
Rosenhain and Göpel Tetrads in PG(3,2). See also
related remarks on the notion of  linear  (or line ) complex
in the finite projective space PG(3,2) —

Anticommuting Dirac matrices as spreads of projective lines

Ron Shaw on the 15 lines of the classical generalized quadrangle W(2), a general linear complex in PG(3,2)

Thursday, November 30, 2017

The Matrix for Quantum Mystics

Filed under: G-Notes,General,Geometry — Tags: , — m759 @ 10:29 PM

Scholia on the title — See Quantum + Mystic in this journal.

The Matrix of Lévi-Strauss

"In Vol. I of Structural Anthropology , p. 209, I have shown that
this analysis alone can account for the double aspect of time
representation in all mythical systems: the narrative is both
'in time' (it consists of a succession of events) and 'beyond'
(its value is permanent)." — Claude Lévi-Strauss, 1976

I prefer the earlier, better-known, remarks on time by T. S. Eliot
in Four Quartets , and the following four quartets (from
The Matrix Meets the Grid) —

.

From a Log24 post of June 26-27, 2017:

A work of Eddington cited in 1974 by von Franz

See also Dirac and Geometry and Kummer in this journal.

Ron Shaw on Eddington's triads "associated in conjugate pairs" —

For more about hyperbolic  and isotropic  lines in PG(3,2),
see posts tagged Diamond Theorem Correlation.

For Shaw, in memoriam — See Contrapuntal Interweaving and The Fugue.

Friday, November 24, 2017

The Matrix Meets the Grid

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

The Matrix —

  The Grid —

  Picturing the Witt Construction

     "Read something that means something." — New Yorker  ad

Thursday, November 23, 2017

The Matrix

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

David Brooks in The New York Times  today

"We once had a unifying national story, celebrated each Thanksgiving.
It was an Exodus story. Americans are the people who escaped oppression,
crossed a wilderness and are building a promised land. The Puritans brought
this story with them. Each wave of immigrants saw themselves in this story.
The civil rights movement embraced this story.

But we have to admit that many today do not resonate with this story. . . .

Today, we have no common national narrative, no shared way
of interpreting the flow of events. Without a common story,
we don’t know what our national purpose is. We have no
common set of goals or ideals.

We need a new national narrative."

From a post of August 15, 2010

http://www.log24.com/log/pix10B/100815-NeoAndOracle.jpg

For some background, see Java Jive and Today's Theology.

Related readings —

From 1928:

From the previous post:

"Thus, instead of Propp's chronological scheme,
in which the order of succession of events
is a feature of the structure . . .
another scheme should be adopted, which would present
a structural model defined as the group of transformations
of a small number of elements. This scheme would appear
as a matrix . . . ."

Claude Lévi-Strauss, 1960 

Lévi-Strauss vs. Propp

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

​Claude Lévi-Strauss

From his "Structure and Form:
Reflections on a Work by Vladimir Propp
" *

To maintain. as I have done. that the permutability of contents is not arbitrary amounts to saying that, if the analysis is carried to a sufficiently deep level, behind diversity we will discover constancy. And, of course. the avowed constancy of form must not hide from us that functions are also permutable.

The structure of the folktale as it is illustrated by Propp presents a chronological succession of qualitatively distinct functions. each constituting an independent genre. One can wonder whether—as with dramatis personae and their attributes— Propp does not stop too soon, seeking the form too close to the level of empirical observation. Among the thirty-one functions that he distinguishes, several are reducible to the same  function reappearing at different  moments of the narrative but after undergoing one or a number of transformations . I have already suggested that this could be true of the false hero (a transformation of the villain), of assigning a difficult task (a transformation of the test), etc. (see p. 181 above), and that in this case the two parties  constituting the fundamental tale would themselves be transformations of each other.

Nothing prevents pushing this reduction even further and analyzing each separate partie  into a small number of recurrent functions, so that several of Propp's functions would constitute groups of transformations of one and the same function. We could treat the "violation" as the reverse of the "prohibition" and the latter as a negative transformation of the "injunction." The "departure" of the hero and his "return" would appear as the negative and positive expressions of the same disjunctive function. The "quest" of the hero (hero pursues someone or something) would become the opposite of "pursuit" (hero is pursued by something or someone), etc.

In Vol. I of Structural Anthropology , p. 209, I have shown that this analysis alone can account for the double aspect of time representation in all mythical systems: the narrative is both "in time" (it consists of a succession of events) and "beyond" (its value is permanent). With regard to Propp's theories my analysis offers another advantage: I can reconcile much better than Propp himself  his principle of a permanent order of wondertale elements with the fact that certain functions or groups of functions are shifted from one tale to the next (pp. 97-98. p. 108) If my view is accepted, the chronological succession will come to be absorbed into an atemporal matrix structure whose form is indeed constant. The shifting of functions is then no more than a mode of permutation (by vertical columns or fractions of columns).

These critical remarks are certainly valid for the method used by Propp and for his conclusions. However. it cannot be stressed enough that Propp envisioned them and in several places formulated with perfect clarity the solutions I have just suggested. Let us take up again from this viewpoint the two essential themes of our discussion: constancy of the content (in spite of its permutability) and permutability of functions (in spite of their constancy).

* Translated from a 1960 work in French.  It appeared in English as Chapter VIII
of Structural Anthropology, Volume 2  (U. of Chicago Press, 1976).  Chapter VIII
was originally published in Cahiers de l'Institut de Science 
Économique Appliquée , 
No. 9 (Series M, No. 7) (Paris: ISEA, March 1960).

See also "Lévi-Strauss" + Formula  in this journal.

Some background related to the previous post

Wednesday, November 22, 2017

Goethe on All Souls’ Day

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

David E. Wellbery on Goethe

From an interview published on 2 November 2017 at

http://literaturwissenschaft-berlin.de/interview-with-david-wellbery/

as later republished in 

https://thepointmag.com/2017/dialogue/
irreducible-significance-david-wellbery-literature-goethe-cavell
 —

 

The logo at left above is that of The Point .
The menu icon at right above is perhaps better
suited to illustrate Verwandlungslehre .

Weyl on symmetry, the eightfold cube, the Fano plane, and trigrams of the I Ching

Sunday, October 29, 2017

Sunday in the Park

Filed under: General — Tags: — m759 @ 1:18 PM

http://www.abc.net.au/news/2017-10-29/
mexico-city-day-of-dead-parade-honours-quake-rescuers/
9097134

Scholium —

Related material —  Sunday in the Park  in this  journal.

Damnation… Or Not?

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

Related material —

Faust Vivifies Death with Wit and Humor
by April H. N. Yee, Harvard Crimson , Feb. 7, 2008.

See as well all posts now tagged Willow and Mandorla.

Tuesday, October 17, 2017

The Movement of Analogy

Filed under: General,Geometry — Tags: , — m759 @ 4:20 PM

The title is a phrase by Octavio Paz from today's post
"Status Symbols."

Other phrases from a link target in Sunday's post 
The Strength at the Centre

                                a single world
In which he is and as and is are one.

See also Four Dots in this journal.

Status Symbols

Filed under: General,Geometry — Tags: — m759 @ 2:02 PM

"Status: Defunct"  

As is now its owner, who reportedly
died at 80 on Sunday, October 15, 2017.

In memoriam —

Excerpts from Log24 posts on Sunday night 
and yesterday evening

http://www.log24.com/log/pix11/110203-Scholia.jpg.

" … listen: there's a hell
of a good universe next door; let's go"

— e. e. cummings

Some literary background —

"At the point of convergence
the play of similarities and differences
cancels itself out in order that 
identity alone may shine forth
The illusion of motionlessness,
the play of mirrors of the one: 
identity is completely empty;
it is a crystallization and
in its transparent core
the movement of analogy 
begins all over once again."

— The Monkey Grammarian 

by Octavio Paz, translated by Helen Lane 

Sunday, October 15, 2017

The Strength at the Centre

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

The title, a phrase from a poem by Wallace Stevens,
was suggested by the previous post, "Center."

See posts tagged May 19 Gestalt in particular, 
May 19, 2007 — "Point of View."

Saturday, October 14, 2017

Center

Rosalind Krauss in 1978

"To get inside the systems of this work,
whether LeWitt's or Judd's or Morris's,
is precisely to enter
a world without a center,
a world of substitutions and transpositions
nowhere legitimated by the revelations
of a transcendental subject. This is the strength
of this work, its seriousness, and its claim to modernity." 

Wikipedia

"The center of
the quaternion group,
Q8 = {1, −1, i, −i, j, −j, k, −k} ,
is {1, −1}."

Illustration from a post of Feb. 3,  2011

http://www.log24.com/log/pix11/110203-Scholia.jpg.

Tuesday, October 10, 2017

Another 35-Year Wait

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 9:00 PM

The title refers to today's earlier post "The 35-Year Wait."

A check of my activities 35 years ago this fall, in the autumn
of 1982, yields a formula I prefer to the nonsensical, but famous,
"canonical formula" of Claude Lévi-Strauss.

The Lévi-Strauss formula

My "inscape" formula, from a note of Sept. 22, 1982 —

S = f ( f ( X ) ) .

Some mathematics from last year related to the 1982 formula —

Koen Thas, 'Unextendible Mututally Unbiased Bases' (2016)

See also Inscape in this  journal and posts tagged Dirac and Geometry.

Monday, June 26, 2017

Upgrading to Six

This post was suggested by the previous post — Four Dots —
and by the phrase "smallest perfect" in this journal.

Related material (click to enlarge) —

Detail —

From the work of Eddington cited in 1974 by von Franz —

See also Dirac and Geometry and Kummer in this journal.

Updates from the morning of June 27 —

Ron Shaw on Eddington's triads "associated in conjugate pairs" —

For more about hyperbolic  and isotropic  lines in PG(3,2),
see posts tagged Diamond Theorem Correlation.

For Shaw, in memoriam — See Contrapuntal Interweaving and The Fugue.

Four Dots

Analogies — "A : B  ::  C : D"  may be read  "A is to B  as  C is to D."

Gian-Carlo Rota on Heidegger…

"… The universal as  is given various names in Heidegger's writings….

The discovery of the universal as  is Heidegger's contribution to philosophy….

The universal 'as' is the surgence of sense in Man, the shepherd of Being.

The disclosure of the primordial as  is the end of a search that began with Plato….
This search comes to its conclusion with Heidegger."

— "Three Senses of 'A is B' in Heideggger," Ch. 17 in Indiscrete Thoughts
 

See also Four Dots in this journal. 

Some context:  McLuhan + Analogy.

Friday, June 23, 2017

A Kind of Cross

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

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

Gravity's Rainbow

See also Heidegger + Rift in this  journal.

“Information from the Middle of the Night”

Filed under: General — Tags: — m759 @ 2:02 AM

The title is from an obituary in tonight's online New York Times.

Information —

See also another art publication cover from 1976 —

Thursday, May 11, 2017

Reopening the Tesseract

Filed under: General,Geometry — Tags: — m759 @ 11:27 AM

Dialogue from the film "Interstellar" —

Cooper: Did it work?

TARS: I think it might have.

Cooper: How do you know?

TARS: Because the bulk beings
            are closing the tesseract.

Related material — "Bulk apperception"
in this journal, and

Monday, May 8, 2017

New Pinterest Board

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

https://www.pinterest.com/stevenhcullinane/art-space/

Sunday, May 7, 2017

Art Space

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

Detail of an image in the previous post

This suggests a review of a post on a work of art by fashion photographer
Peter Lindbergh, made when he was younger and known as "Sultan."

The balls in the foreground relate Sultan's work to my own.

Linguistic backstory —

The art space where the pieces by Talman and by Lindbergh
were displayed is Museum Tinguely in Basel.

As the previous post notes, the etymology of "glamour" (as in
fashion photography) has been linked to "grammar" (as in 
George Steiner's Grammars of Creation ). A sculpture by 
Tinguely (fancifully representing Heidegger) adorns one edition
of Grammars .

Yale University Press, 2001:

Tinguely, "Martin Heidegger,
Philosopher," sculpture, 1988

Saturday, April 8, 2017

A Perfect Nonentity

Filed under: General,Geometry — Tags: , — m759 @ 8:25 AM

The title is from Hume:

 "And were all my perceptions removed by death,
and could I neither think, nor feel, nor see, nor love,
nor hate, after the dissolution of my body, I should
be entirely annihilated, nor do I conceive what is
further requisite to make me a perfect nonentity."

— Book I, Part IV, Section vi  of  
    A Treatise of Human Nature

"What is further requisite" — Perhaps  

This four-dot notation ("as") is from a search for Lévi-Strauss in this journal.

See also "That I Am."

Friday, April 7, 2017

Personal Identity

Filed under: General,Geometry — Tags: , — m759 @ 2:40 PM

From "The Most Notorious Section Phrases," by Sophie G. Garrett
in The Harvard Crimson  on April 5, 2017 —

This passage reminds me of (insert impressive philosophy
that was not in the reading).

This student is just being a show off. We get that they are smart
and well read. Congrats, but please don’t make the rest of the us
look bad in comparison. It should be enough to do the assigned
reading without making connections to Hume’s theory of the self.

Hume on personal identity (the "self")

For my part, when I enter most intimately into what I call myself, I always stumble on some particular perception or other, of heat or cold, light or shade, love or hatred, pain or pleasure. I never can catch myself at any time without a perception, and never can observe any thing but the perception. When my perceptions are removed for any time, as by sound sleep, so long am I insensible of myself, and may truly be said not to exist. And were all my perceptions removed by death, and could I neither think, nor feel, nor see, nor love, nor hate, after the dissolution of my body, I should be entirely annihilated, nor do I conceive what is further requisite to make me a perfect nonentity.
. . . .

I may venture to affirm of the rest of mankind, that they are nothing but a bundle or collection of different perceptions, which succeed each other with an inconceivable rapidity, and are in a perpetual flux and movement. Our eyes cannot turn in their sockets without varying our perceptions. Our thought is still more variable than our sight; and all our other senses and faculties contribute to this change: nor is there any single power of the soul, which remains unalterably the same, perhaps for one moment. The mind is a kind of theatre, where several perceptions successively make their appearance; pass, repass, glide away, and mingle in an infinite variety of postures and situations. There is properly no simplicity in it at one time, nor identity in different, whatever natural propension we may have to imagine that simplicity and identity. The comparison of the theatre must not mislead us. They are the successive perceptions only, that constitute the mind; nor have we the most distant notion of the place where these scenes are represented, or of the materials of which it is composed.

Related material —
Imago Dei  in this journal.

The Ring of the Diamond Theorem

Backstory —
The previous post
and The Crimson Abyss.

Tuesday, March 14, 2017

Review

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

Saturday, February 11, 2017

The God Identity

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

The previous post suggests a related concept, in which
"I Am that I Am" is replaced by "I am as  I am." —

Monday, October 17, 2016

Groundhog Day for Hindus

Filed under: General — Tags: — m759 @ 11:25 AM

Live, Die, Repeat.

Groundhog Day Tablet

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

This  journal on that date —

The Los Angeles Times  this morning reported that poet
David Antin died at 84 last Tuesday, October 11.

From this  journal on that date

Sunday, October 16, 2016

Weiner* Mantra

Filed under: General — Tags: — m759 @ 11:09 PM

* For the title, see the Wikipedia article
  on the creator of the TV drama Mad Men .
  For another illustration of the mantra,
  see the previous post.

Facebook Image

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

Thursday, September 8, 2016

Hope of Heaven, Oslo Style

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

The previous post deals with the theory, now becoming widely known,
that the musical "Grease" is really about Sandy's dying dream of heaven.

Another such dream in Oslo, described by the Vigeland Museum —

The Monolith

"The Monolith was carved from one single granite block, hence the name (mono: one, litho: stone). Whereas the melancholy theme in the fountain is the eternal life cycle, the column gives room to a totally different interpretation: Man's longing and yearning for the spiritual and divine. Is the column to be understood as man's resurrection? The people are drawn towards heaven, not only characterised by sadness and controlled despair, but also delight and hope, next to a feeling of togetherness, carefully holding one another tight in this strange sense of salvation."

I prefer a different monolith.

Afternoon Delight

Filed under: General — Tags: — m759 @ 3:26 PM

The notation "O" in the previous post suggests
a review of the new "Grease" theory in light of
a phrase from a May 2014 Oslo art exhibition

"a desperate sense of imagined community."

Illustration (click for a video) —

"I'll have what she's having."

See as well Olivia Newton-John in this journal as the Muse of Dance

Tuesday, August 30, 2016

The Mystic Egress

Filed under: General — Tags: — m759 @ 8:16 PM

A detail from the previous post

"Last thing I remember,
 I was running for the door . . . ."

— Eagles, "Hotel California"

The Eightfold Cube in Oslo

Filed under: General,Geometry — Tags: — m759 @ 8:04 PM

A KUNSTforum.as article online today (translation by Google) —

The eightfold cube at the Vigeland Museum in Oslo

Update of Sept. 7, 2016: The corrections have been made,
except for the misspelling "Cullinan," which was caused by 
Google translation, not by KUNSTforum.

Shema, Faust

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

"The quotes create the illusion
that the dead are still speaking
to the reader. Faust writes about
the efforts of spiritualists to believe
in an afterlife for their slain kin, but
she’s the one summoning spirits."

April Yee, Harvard Crimson
     staff writer, February 7, 2008

"0! = 1" 

Quine's Shema

See also yesterday's Into the Woods 
and posts now tagged Willow and Mandorla.

Monday, August 29, 2016

Into the Woods

Filed under: General — Tags: , — m759 @ 10:19 AM

This just in:

Headline- 'Clown tries to lure kids into woods'

See also Cinderella in yesterday's post "As" —

The James Lapine version —

Friday, August 19, 2016

Princeton University Press in 1947

Filed under: General — Tags: , , — m759 @ 11:17 AM

From a review, in the context of Hollywood, of a Princeton
University Press book on William Blake from 1947 —

Wednesday, August 10, 2016

Analogies Test

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

Obituary for Wilford Stanton Miller, author in 1926
of the Miller Analogies Test  —  
         

Marshall McLuhan writing to Ezra Pound on Dec. 21, 1948—

"The American mind is not even close to being amenable
to the ideogram principle as yet.  The reason is simply this.
America is 100% 18th Century. The 18th century had
chucked out the principle of metaphor and analogy—
the basic fact that as A is to B so is C to D.  AB:CD.   
It can see AB relations.  But relations in four terms are still
verboten.  This amounts to deep occultation of nearly all
human thought for the U.S.A.

I am trying to devise a way of stating this difficulty as it exists.  
Until stated and publicly recognized for what it is, poetry and
the arts can’t exist in America."

A line for W. S. Miller, taken from "Annie Hall" —

"You know nothing of my work."

Friday, July 29, 2016

The Cumberbatch Question

Filed under: General — Tags: — m759 @ 2:29 PM

"How do I get from here to there?"

See also Compass in this journal.

Portal

Filed under: General — Tags: — m759 @ 1:29 PM

Darkness at Noon

 meets Midnight Special

and the result is

Thursday, June 2, 2016

Bullshit Studies

Filed under: General,Geometry — Tags: — m759 @ 11:30 AM

(Continued)

"The allusion to 'the most precious square of sense' shows
Shakespeare doing an almost scholastic demonstration of
the need for a ratio and interplay among the senses as
the very constitution of rationality."

— Marshall McLuhan, The Gutenberg Galaxy ,
University of Toronto Press, 1962, page 13

"What Shakespeare refers to in Lear  as the 'precious
square of sense' probably has reference to the traditional
'square of opposition' in logic and to that four-part analogy
of proportionality which is the interplay of sense and reason."     

— McLuhan, ibid. , page 241

This is of course nonsense, and, in view of McLuhan's pose
as a defender of the Catholic faith, damned  nonsense.

Epigraph by McLuhan —

"The Gutenberg Galaxy  develops a mosaic or field
approach to its problems."

I prefer a different "mosaic or field" related to the movable
blocks  of Fröbel, not the movable type  of Gutenberg.

Sunday, May 29, 2016

The Ideogram Principle …

Filed under: General,Geometry — Tags: , — m759 @ 4:23 PM

According to McLuhan

Marshall McLuhan writing to Ezra Pound on Dec. 21, 1948—

"The American mind is not even close to being amenable
to the ideogram principle as yet.  The reason is simply this.
America is 100% 18th Century. The 18th century had
chucked out the principle of metaphor and analogy—
the basic fact that as A is to B so is C to D.  AB:CD.   
It can see AB relations.  But relations in four terms are still
verboten.  This amounts to deep occultation of nearly all
human thought for the U.S.A.

I am trying to devise a way of stating this difficulty as it exists.  
Until stated and publicly recognized for what it is, poetry and
the arts can’t exist in America."

For context, see Cameron McEwen,
"Marshall McLuhan, John Pick, and Gerard Manley Hopkins."
(Renascence , Fall 2011, Vol. 64 Issue 1, 55-76)

A relation in four terms

A : B  ::  C : D   as   Model : Crutch  ::  Metaphor : Ornament —

See also Dueling Formulas and Symmetry.

Saturday, May 28, 2016

Plan 4

Filed under: General — Tags: , — m759 @ 2:45 AM

See a search in this journal for "As Is."

Wednesday, May 25, 2016

Framework

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

"Studies of spin-½ theories in the framework of projective geometry
have been undertaken before." — Y. Jack Ng  and H. van Dam
February 20, 2009

For one such framework,* see posts from that same date 
four years earlier — February 20, 2005.

* A 4×4 array. See the 19771978and 1986 versions by 
Steven H. Cullinane,   the 1987 version by R. T. Curtis, and
the 1988 Conway-Sloane version illustrated below —

Cullinane, 1977

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

Cullinane, 1978

Cullinane, 1986

Curtis, 1987

Update of 10:42 PM ET on Sunday, June 19, 2016 —

The above images are precursors to

Conway and Sloane, 1988

Update of 10 AM ET Sept. 16, 2016 — The excerpt from the
1977 "Diamond Theory" article was added above.

Tuesday, April 5, 2016

Spring Play

Filed under: General — Tags: , , — m759 @ 4:24 PM

The spring play this March at Princeton's McCarter Theatre Center
was Agatha Christie's "The Mousetrap."

In related news —

See as well, in this  journal, a post from the date pictured above,
that of the Disneyland Diamond Celebration on May 22, 2015 —

Donald Duck with pentagram

Thursday, March 17, 2016

On the Eightfold Cube

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

The following page quotes "Raiders of the Lost Crucible,"
a Log24 post from Halloween 2015.

Discussion of Cullinane's eightfold cube as exhibited by Josefine Lyche at the Vigeland Museum in Oslo

From KUNSTforum.as, a Norwegian art quarterly, issue no. 1 of 2016.

Related posts — See Lyche Eightfold.

Tuesday, March 15, 2016

15 Projective Points Revisited

Filed under: General,Geometry — Tags: — m759 @ 11:59 PM

A March 10, 2016, Facebook post from KUNSTforum.as,
a Norwegian art quarterly —

Article on Josefine Lyche's Vigeland Museum exhibit, which included Cullinane's eightfold cube

Click image above for a view of pages 50-51 of a new KUNSTforum 
article showing two photos relevant to my own work — those labeled
"after S. H. Cullinane."

(The phrase "den pensjonerte Oxford-professoren Stephen H. Cullinane"
on page 51 is almost completely wrong. I have never been a professor,
I was never at Oxford, and my first name is Steven, not Stephen.)

For some background on the 15 projective points at the lower left of
the above March 10 Facebook post, see "The Smallest Projective Space."

Saturday, February 13, 2016

The Path

Filed under: General — Tags: — m759 @ 5:45 PM

(The title is a reference to the previous post.)

"Pope Reminds Us The Devil Is Real"
Catholic Online , Oct. 14, 2013

Related material:

The reported death of Justice Scalia

and posts on Scalia in this  journal.

Friday, February 12, 2016

The Game

Filed under: General,Geometry — Tags: — m759 @ 8:14 PM

"Right through hell
 there is a path…."
 — Malcolm Lowry,
Under the Volcano

Monday, February 8, 2016

A Game with Four Letters

Filed under: General,Geometry — Tags: — m759 @ 2:56 PM

Related material — Posts tagged Dirac and Geometry.

For an example of what Eddington calls "an open mind,"
see the 1958 letters of Nanavira Thera.
(Among the "Early Letters" in Seeking the Path ).

Thursday, December 3, 2015

Design Wars

Filed under: General,Geometry — Tags: — m759 @ 4:04 PM

"… if your requirement for success is to be like Steve Jobs,
good luck to you." 

— "Transformation at Yahoo Foiled by Marissa Mayer’s 
Inability to Bet the Farm," New York Times  online yesterday

"Design is how it works." — Steve Jobs

Related material:  Posts tagged Ambassadors.
 

Sculpture by Josefine Lyche of Cullinane's eightfold cube at Vigeland Museum in Oslo

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