Log24

Monday, December 11, 2017

The Diamond Theorem at SASTRA

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

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

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

DOI

Wednesday, August 23, 2017

The Diamond Theorem in Vancouver

Filed under: Uncategorized — Tags: — m759 @ 2:56 PM

A designer from New Zealand

Happy 10th birthday to the hashtag.

Tuesday, September 20, 2016

The Diamond Theorem

Filed under: Uncategorized — m759 @ 11:00 AM

As the Key to All Mythologies

For the theorem of the title, see "Diamond Theorem" in this journal.

"These were heavy impressions to struggle against,
and brought that melancholy embitterment which
is the consequence of all excessive claim: even his
religious faith wavered with his wavering trust in his
own authorship, and the consolations of the Christian
hope in immortality seemed to lean on the immortality
of the still unwritten Key to all Mythologies."

Middlemarch , by George Eliot, Ch. XXIX

Related material from Sunday's print New York Times

Sunday's Log24 sermon

See also the Lévi-Strauss "Key to all Mythologies" in this journal,
as well as the previous post.

Monday, August 5, 2013

Diamond Theorem in ArXiv

Filed under: Uncategorized — m759 @ 9:00 PM

The diamond theorem is now in the arXiv

IMAGE- The diamond theorem in the arXiv

Tuesday, July 2, 2013

Diamond Theorem Updates

Filed under: Uncategorized — Tags: — m759 @ 8:00 PM

My diamond theorem articles at PlanetMath and at 
Encyclopedia of Mathematics have been updated
to clarify the relationship between the graphic square
patterns of the diamond theorem and the schematic
square patterns of the Curtis Miracle Octad Generator.

Friday, May 10, 2013

Cullinane diamond theorem

Filed under: Uncategorized — m759 @ 3:00 PM

A page with the above title has been created at
the Encyclopedia of Mathematics.

How long it will stay there remains to be seen.

Friday, February 25, 2011

Diamond Theorem Exposition

Filed under: Uncategorized — m759 @ 11:00 PM

"THE DIAMOND THEOREM AND QUILT PATTERNS
Victoria Blumen, Mathematics, Junior, Benedictine University
Tim Comar, Benedictine University
Mathematics
Secondary Source Research
 
Let D be a 4 by 4 block quilt shape, where each of the 16 square blocks is consists of [sic ] two triangles, one of which is colored red and the other of which is colored blue.  Let G: D -> D_g be a mapping of D that interchanges a pair of columns, rows, or quadrants of D.  The diamond theorem states that G(D) = D_g has either ordinary or color-interchange symmetry.  In this talk, we will prove the diamond theorem and explore symmetries of quilt patterns of the form G(D)."

Exercise Correct the above statement of the theorem.

Background— This is from a Google search result at about 10:55 PM ET Feb. 25, 2011—

[DOC] THE DIAMOND THEOREM AND QUILT PATTERNS – acca.elmhurst.edu
File Format: Microsoft Word – 14 hours ago –
Let G: D -> D_g be a mapping of D that interchanges a pair of columns, rows, or quadrants of D. The diamond theorem states that G(D) = D_g has either
acca.elmhurst.edu/…/victoria_blumen9607_
THE%20DIAMOND%20THEOREM%20AND%20QUILT%20PATTERNS…

The document is from a list of mathematics abstracts for the annual student symposium of the ACCA (Associated Colleges of the Chicago Area) held on April 10, 2010.

Update of Feb. 26For a related remark quoted here  on the date of the student symposium, see Geometry for Generations.

Sunday, October 31, 2010

Diamond Theorem in Norway

Filed under: Uncategorized — m759 @ 10:00 PM

IMAGE- The 2x2 case of the diamond theorem as illustrated by Josefine Lyche, Oct. 2010

Click on above image for artist's page.

Click here for exhibit page.

Click here for underlying geometry.

Friday, October 28, 2016

Diamond-Theorem Application

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

Abstract:

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

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

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

Excerpts —

Related material — Posts tagged Diamond Theorem Correlation.

Friday, August 1, 2014

The Diamond-Theorem Correlation

Filed under: Uncategorized — Tags: , — m759 @ 2:00 AM

Click image for a larger, clearer version.

IMAGE- The symplectic correlation underlying Rosenhain and Göpel tetrads

Monday, March 12, 2018

“Quantum Tesseract Theorem?”

Filed under: Uncategorized — 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 —

     "Open the pod bay doors, HAL."

Monday, January 8, 2018

Raiders of the Lost Theorem

Filed under: Uncategorized — Tags: — m759 @ 11:15 PM
 

The Quantum Tesseract Theorem 

 


 

Raiders —

A Wrinkle in Time
starring Storm Reid,
Reese Witherspoon,
Oprah Winfrey &
Mindy Kaling

 

Time Magazine  December 25, 2017 – January 1, 2018

Thursday, August 31, 2017

A Conway-Norton-Ryba Theorem

Filed under: Uncategorized — m759 @ 1:40 PM

In a book to be published Sept. 5 by Princeton University Press,
John Conway, Simon Norton,  and Alex Ryba present the following
result on order-four magic squares —

A monograph published in 1976, "Diamond Theory," deals with 
more general 4×4 squares containing entries from the Galois fields
GF(2), GF(4), or GF(16).  These squares have remarkable, if not 
"magic," symmetry properties.  See excerpts in a 1977 article.

See also Magic Square and Diamond Theorem in this  journal.

Saturday, March 14, 2015

Unicode Diamonds

Filed under: Uncategorized — m759 @ 9:16 PM

The following figure, intended to display as
a black diamond, was produced with
HTML and Unicode characters. Depending
on the technology used to view it, the figure
may contain gaps or overlaps.

◢◣
◥◤

Some variations:

◤◥
◣◢

◤◥
◢◣

◤◣
◢◥

◤◣
◥◢

Such combined Unicode characters —

◢  black lower right triangle,
◣  black lower left triangle,
᭘  black upper left triangle,
᭙  black upper right triangle 

— might be used for a text-only version of the Diamond 16 Puzzle
that is more easily programmed than the current version.

The tricky part would be coding the letter-spacing and
line-height to avoid gaps or overlaps within the figures in
a variety of browsers. The w3.org visual formatting model
may or may not be helpful here.

Update of 11:20 PM ET March 15, 2015 — 
Seekers of simplicity should note that there is
a simple program in the Processing.js  language, not  using
such Unicode characters, that shows many random affine
permutations of a 4×4 diamond-theorem array when the
display window is clicked.

Monday, October 13, 2014

Raiders of the Lost Theorem

Filed under: Uncategorized — Tags: — m759 @ 12:05 PM

(Continued from Nov. 16, 2013.)

The 48 actions of GL(2,3) on a 3×3 array include the 8-element
quaternion group as a subgroup. This was illustrated in a Log24 post,
Hamilton’s Whirligig, of Jan. 5, 2006, and in a webpage whose
earliest version in the Internet Archive is from June 14, 2006.

One of these quaternion actions is pictured, without any reference
to quaternions, in a 2013 book by a Netherlands author whose
background in pure mathematics is apparently minimal:

In context (click to enlarge):

Update of later the same day —

Lee Sallows, Sept. 2011 foreword to Geometric Magic Squares —

“I first hit on the idea of a geometric magic square* in October 2001,**
and I sensed at once that I had penetrated some previously hidden portal
and was now standing on the threshold of a great adventure. It was going
to be like exploring Aladdin’s Cave. That there were treasures in the cave,
I was convinced, but how they were to be found was far from clear. The
concept of a geometric magic square is so simple that a child will grasp it
in a single glance. Ask a mathematician to create an actual specimen and
you may have a long wait before getting a response; such are the formidable
difficulties confronting the would-be constructor.”

* Defined by Sallows later in the book:

“Geometric  or, less formally, geomagic  is the term I use for
a magic square in which higher dimensional geometrical shapes
(or tiles  or pieces ) may appear in the cells instead of numbers.”

** See some geometric  matrices by Cullinane in a March 2001 webpage.

Earlier actual specimens — see Diamond Theory  excerpts published in
February 1977 and a brief description of the original 1976 monograph:

“51 pp. on the symmetries & algebra of
matrices with geometric-figure entries.”

— Steven H. Cullinane, 1977 ad in
Notices of the American Mathematical Society

The recreational topic of “magic” squares is of little relevance
to my own interests— group actions on such matrices and the
matrices’ role as models of finite geometries.

Thursday, March 27, 2014

Diamond Space

Filed under: Uncategorized — Tags: — m759 @ 2:28 PM

(Continued)

Definition:  A diamond space  — informal phrase denoting
a subspace of AG(6, 2), the six-dimensional affine space
over the two-element Galois field.

The reason for the name:

IMAGE - The Diamond Theorem, including the 4x4x4 'Solomon's Cube' case

Click to enlarge.

Sunday, February 2, 2014

Diamond Theory Roulette

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

ReCode Project program from Radamés Ajna of São Paulo —

At the program's webpage, click the image to
generate random permutations of rows, columns,
and quadrants
. Note the resulting image's ordinary
or color-interchange symmetry.

Tuesday, December 3, 2013

Diamond Space

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

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

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

Monday, February 11, 2013

The Penrose Diamond

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

IMAGE- The Penrose Diamond

Related material:

(Click to enlarge.)

See also remarks on Penrose linked to in Sacerdotal Jargon.

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

Thursday, May 31, 2012

Black Diamond

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

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

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

Saying less—

Sunday, March 18, 2012

Square-Triangle Diamond

Filed under: Uncategorized — Tags: — m759 @ 5:01 AM

The diamond shape of yesterday's noon post
is not wholly without mathematical interest …

The square-triangle theorem

"Every triangle is an n -replica" is true
if and only if n  is a square.

IMAGE- Square-to-diamond (rhombus) shear in proof of square-triangle theorem

The 16 subdiamonds of the above figure clearly
may be mapped by an affine transformation
to 16 subsquares of a square array.

(See the diamond lattice  in Weyl's Symmetry .)

Similarly for any square n , not just 16.

There is a group of 322,560 natural transformations
that permute the centers  of the 16 subsquares
in a 16-part square array. The same group may be
viewed as permuting the centers  of the 16 subtriangles
in a 16-part triangular array.

(Updated March 29, 2012, to correct wording and add Weyl link.)

Saturday, March 17, 2012

The Purloined Diamond

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

(Continued)

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

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

Geometry lesson: the vesica piscis in Finnegans Wake

The Diamond in the Football

Football-mandorla

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

Wednesday, December 21, 2011

The Purloined Diamond

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

Stephen Rachman on "The Purloined Letter"

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

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

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

Related material—

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

A Rhetorical Question

Osserman in 2004

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

Who bestowed the magic kiss on the mathematical frog?"

A Rhetorical Answer

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

Above: Amy Adams in "Sunshine Cleaning"

Thursday, September 8, 2011

Starring the Diamond

Filed under: Uncategorized — m759 @ 2:02 PM

"In any geometry satisfying Pappus's Theorem,
the four pairs of opposite points of 83
are joined by four concurrent lines.
"
— H. S. M. Coxeter (see below)

Continued from Tuesday, Sept. 6

The Diamond Star

http://www.log24.com/log/pix11B/110905-StellaOctangulaView.jpg

The above is a version of a figure from Configurations and Squares.

Yesterday's post related the the Pappus configuration to this figure.

Coxeter, in "Self-Dual Configurations and Regular Graphs," also relates Pappus to the figure.

Some excerpts from Coxeter—

http://www.log24.com/log/pix11B/110908-Coxeter83.jpg

The relabeling uses the 8 superscripts
from the first picture above (plus 0).
The order of the superscripts is from
an 8-cycle in the Galois field GF(9).

The relabeled configuration is used in a discussion of Pappus—

http://www.log24.com/log/pix11B/110908-Coxeter83part2.jpg

(Update of Sept. 10, 2011—
Coxeter here has a note referring to page 335 of
G. A. Miller, H. F. Blichfeldt, and L. E. Dickson,
Theory and Applications of Finite Groups , New York, 1916.)

Coxeter later uses the the 3×3 array (with center omitted) again to illustrate the Desargues  configuration—

http://www.log24.com/log/pix11B/110908-Coxeter103.jpg

The Desargues configuration is discussed by Gian-Carlo Rota on pp. 145-146 of Indiscrete Thoughts

"The value  of Desargues' theorem and the reason  why the statement of this theorem has survived through the centuries, while other equally striking geometrical theorems have been forgotten, is in the realization that Desargues' theorem opened a horizon of possibilities  that relate geometry and algebra in unexpected ways."

Monday, August 8, 2011

Diamond Theory vs. Story Theory (continued)

Filed under: Uncategorized — m759 @ 5:01 PM

Some background

Richard J. Trudeau, a mathematics professor and Unitarian minister, published in 1987 a book, The Non-Euclidean Revolution , that opposes what he calls the Story Theory of truth [i.e., Quine, nominalism, postmodernism] to what he calls the traditional Diamond Theory of truth [i.e., Plato, realism, the Roman Catholic Church]. This opposition goes back to the medieval "problem of universals" debated by scholastic philosophers.

(Trudeau may never have heard of, and at any rate did not mention, an earlier 1976 monograph on geometry, "Diamond Theory," whose subject and title are relevant.)

From yesterday's Sunday morning New York Times

"Stories were the primary way our ancestors transmitted knowledge and values. Today we seek movies, novels and 'news stories' that put the events of the day in a form that our brains evolved to find compelling and memorable. Children crave bedtime stories…."

Drew Westen, professor at Emory University

From May 22, 2009

Poster for 'Diamonds' miniseries on ABC starting May 24, 2009

The above ad is by
  Diane Robertson Design—

Credit for 'Diamonds' miniseries poster: Diane Robertson Design, London

Diamond from last night’s
Log24 entry, with
four colored pencils from
Diane Robertson Design:

Diamond-shaped face of Durer's 'Melencolia I' solid, with  four colored pencils from Diane Robertson Design
 
See also
A Four-Color Theorem.

For further details, see Saturday's correspondences
and a diamond-related story from this afternoon's
online New York Times.

Sunday, July 10, 2011

Wittgenstein’s Diamond

Filed under: Uncategorized — Tags: — m759 @ 9:29 AM

Philosophical Investigations  (1953)

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

— Translation by G.E.M. Anscombe

5.5563

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

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

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

See also

Related language in Łukasiewicz (1937)—

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

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

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

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

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

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

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

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

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

Monday, December 27, 2010

Church Diamond

Filed under: Uncategorized — m759 @ 3:09 PM

IMAGE- The diamond property

Also known, roughly speaking, as confluence  or the Church-Rosser property.

From "NYU Lambda Seminar, Week 2" —

[See also the parent page Seminar in Semantics / Philosophy of Language or:
 What Philosophers and Linguists Can Learn From Theoretical Computer Science But Didn't Know To Ask)
]

A computational system is said to be confluent, or to have the Church-Rosser or diamond property, if, whenever there are multiple possible evaluation paths, those that terminate always terminate in the same value. In such a system, the choice of which sub-expressions to evaluate first will only matter if some of them but not others might lead down a non-terminating path.

The untyped lambda calculus is confluent. So long as a computation terminates, it always terminates in the same way. It doesn't matter which order the sub-expressions are evaluated in.

A computational system is said to be strongly normalizing if every permitted evaluation path is guaranteed to terminate. The untyped lambda calculus is not strongly normalizing: ω ω doesn't terminate by any evaluation path; and (\x. y) (ω ω) terminates only by some evaluation paths but not by others.

But the untyped lambda calculus enjoys some compensation for this weakness. It's Turing complete! It can represent any computation we know how to describe. (That's the cash value of being Turing complete, not the rigorous definition. There is a rigorous definition. However, we don't know how to rigorously define "any computation we know how to describe.") And in fact, it's been proven that you can't have both. If a computational system is Turing complete, it cannot be strongly normalizing.

There is no connection, apart from the common reference to an elementary geometric shape, between the use of "diamond" in the above Church-Rosser sense and the use of "diamond" in the mathematics of (Cullinane's) Diamond Theory.

Any attempt to establish such a connection would, it seems, lead quickly into logically dubious territory.

Nevertheless, in the synchronistic spirit of Carl Jung and Arthur Koestler, here are some links to such a territory —

 Link One — "Insane Symmetry"  (Click image for further details)—

http://www.log24.com/log/pix10B/101227-InsaneSymmetry.jpg

See also the quilt symmetry in this  journal on Christmas Day.

Link Two — Divine Symmetry

(George Steiner on the Name in this journal on Dec. 31 last year ("All about Eve")) —

"The links are direct between the tautology out of the Burning Bush, that 'I am' which accords to language the privilege of phrasing the identity of God, on the one hand, and the presumptions of concordance, of equivalence, of translatability, which, though imperfect, empower our dictionaries, our syntax, our rhetoric, on the other. That 'I am' has, as it were, at an overwhelming distance, informed all predication. It has spanned the arc between noun and verb, a leap primary to creation and the exercise of creative consciousness in metaphor. Where that fire in the branches has gone out or has been exposed as an optical illusion, the textuality of the world, the agency of the Logos in logic—be it Mosaic, Heraclitean, or Johannine—becomes 'a dead letter.'"

George Steiner, Grammars of Creation

(See also, from Hanukkah this year,  A Geometric Merkabah and The Dreidel is Cast.)

Link Three – Spanning the Arc —

Part A — Architect Louis Sullivan on "span" (see also Kindergarten at Stonehenge)

Part B — "Span" in category theory at nLab —

http://www.log24.com/log/pix10B/101227-nLabSpanImage.jpg

Also from nLab — Completing Spans to Diamonds

"It is often interesting whether a given span in some partial ordered set can be completed into a diamond. The property of a collection of spans to consist of spans which are expandable into diamonds is very useful in the theory of rewriting systems and producing normal forms in algebra. There are classical results e.g. Newman’s diamond lemma, Širšov-Bergman’s diamond lemma (Širšov is also sometimes spelled as Shirshov), and Church-Rosser theorem (and the corresponding Church-Rosser confluence property)."

The concepts in this last paragraph may or may not have influenced the diamond theory of Rudolf Kaehr (apparently dating from 2007).

They certainly have nothing to do with the Diamond Theory of Steven H. Cullinane (dating from 1976).

For more on what the above San Francisco art curator is pleased to call "insane symmetry," see this journal on Christmas Day.

For related philosophical lucubrations (more in the spirit of Kaehr than of Steiner), see the New York Times  "The Stone" essay "Span: A Remembrance," from December 22—

“To understand ourselves well,” [architect Louis] Sullivan writes, “we must arrive first at a simple basis: then build up from it.”

Around 300 BC, Euclid arrived at this: “A point is that which has no part. A line is breadthless length.”

See also the link from Christmas Day to remarks on Euclid and "architectonic" in Mere Geometry.

Wednesday, July 28, 2010

Without Diamond-Blazons

Filed under: Uncategorized — Tags: — m759 @ 6:29 PM

Excerpt from Wallace Stevens's
"The Pediment of Appearance"—

Young men go walking in the woods,
Hunting for the great ornament,
The pediment* of appearance.

They hunt for a form which by its form alone,
Without diamond—blazons or flashing or
Chains of circumstance,

By its form alone, by being right,
By being high, is the stone
For which they are looking:

The savage transparence.

* Pediments, triangular and curved—

http://www.log24.com/log/pix10B/100728-Pediments.jpg

— From "Stones and Their Stories," an article written
and illustrated by E.M. Barlow, copyright 1913.

Related geometry—

http://www.log24.com/log/pix10B/100728-SimplifiedPeds.gif

 (See Štefan Porubský: Pythagorean Theorem .)

A proof with  diamond-blazons—

http://www.log24.com/log/pix10B/100728-DiamondProof.gif

(See Ivars Peterson's "Square of the Hypotenuse," Nov. 27, 2000.)

Saturday, June 9, 2018

SASTRA paper

Filed under: Uncategorized — Tags: — m759 @ 11:14 PM

Now out from behind a paywall . . .

The diamond theorem at SASTRA —

Monday, May 7, 2018

Data

Filed under: Uncategorized — Tags: — m759 @ 10:32 AM

(Continued from yesterday's Sunday School Lesson Plan for Peculiar Children)

Novelist George Eliot and programming pioneer Ada Lovelace —

For an image that suggests a resurrected multifaceted 
(specifically, 759-faceted) Osterman Omega (as in Sunday's afternoon
Log24 post
), behold  a photo from today's NY Times  philosophy
column "The Stone" that was reproduced here in today's previous post

For a New York Times  view of George Eliot data, see a Log24 post 
of September 20, 2016, on the diamond theorem as the Middlemarch
"key to all mythologies."

Tuesday, April 24, 2018

Illustrators of the Word

Filed under: Uncategorized — m759 @ 1:30 AM

Tom Wolfe in The Painted Word  (1975) 

“I am willing (now that so much has been revealed!)
to predict that in the year 2000, when the Metropolitan
or the Museum of Modern Art puts on the great
retrospective exhibition of American Art 1945-75,
the three artists who will be featured, the three seminal
figures of the era, will be not Pollock, de Kooning, and
Johns-but Greenberg, Rosenberg, and Steinberg.
Up on the walls will be huge copy blocks, eight and a half
by eleven feet each, presenting the protean passages of
the period … a little ‘fuliginous flatness’ here … a little
‘action painting’ there … and some of that ‘all great art
is about art’ just beyond. Beside them will be small
reproductions of the work of leading illustrators of
the Word from that period….”

The above group of 322,560 permutations appears also in a 2011 book —

From 'Beautiful Mathematics,' by Martin Erickson, an excerpt on the Cullinane diamond theorem (with source not mentioned)

— and in 2013-2015 papers by Anne Taormina and Katrin Wendland:

Wednesday, April 11, 2018

Behance Post

Filed under: Uncategorized — m759 @ 9:17 PM

I just came across the 2010 web page
https://pantone.ccnsite.com/gallery/HELDER-visual-identity/603956,
associated with the Adobe site "Behance.net." That page suggested
I too should have a Behance web presence.

And so the diamond theorem now appears at . . .

https://www.behance.net/gallery/64334249/The-Diamond-Theorem.

Thursday, March 29, 2018

“Before Creation Itself . . .”

Filed under: Uncategorized — m759 @ 10:13 AM

 From the Diamond Theorem Facebook page —

A question three hours ago at that page

"Is this Time Cube?"

Notes toward an answer —

And from Six-Set Geometry in this journal . . .

Wednesday, March 28, 2018

On Unfairly Excluding Asymmetry

Filed under: Uncategorized — Tags: — m759 @ 12:28 AM

A comment on the the Diamond Theorem Facebook page



Those who enjoy asymmetry may consult the "Expert's Cube" —

For further details see the previous post.

Friday, February 16, 2018

Two Kinds of Symmetry

Filed under: Uncategorized — m759 @ 11:29 PM

The Institute for Advanced Study (IAS) at Princeton in its Fall 2015 Letter 
revived "Beautiful Mathematics" as a title:

This ugly phrase was earlier used by Truman State University
professor Martin Erickson as a book title. See below. 

In the same IAS Fall 2015 Letter appear the following remarks
by Freeman Dyson —

". . . a special case of a much deeper connection that Ian Macdonald 
discovered between two kinds of symmetry which we call modular and affine.
The two kinds of symmetry were originally found in separate parts of science,
modular in pure mathematics and affine in physics. Modular symmetry is
displayed for everyone to see in the drawings of flying angels and devils
by the artist Maurits Escher. Escher understood the mathematics and got the
details right. Affine symmetry is displayed in the peculiar groupings of particles
created by physicists with high-energy accelerators. The mathematician
Robert Langlands was the first to conjecture a connection between these and
other kinds of symmetry. . . ." (Wikipedia link added.)

The adjective "modular"  might aptly be applied to . . .

The adjective "affine"  might aptly be applied to . . .

From 'Beautiful Mathematics,' by Martin Erickson, an excerpt on the Cullinane diamond theorem (with source not mentioned)

The geometry of the 4×4 square combines modular symmetry
(i.e., related to theta functions) with the affine symmetry above.

Hudson's 1905 discussion of modular symmetry (that of Rosenhain
tetrads and Göpel tetrads) in the 4×4 square used a parametrization
of that square by the digit 0 and the fifteen 2-subsets of a 6-set, but 
did not discuss the 4×4 square as an affine space.

For the connection of the 15 Kummer modular 2-subsets with the 16-
element affine space over the two-element Galois field GF(2), see my note
of May 26, 1986, "The 2-subsets of a 6-set are the points of a PG(3,2)" —

— and the affine structure in the 1979 AMS abstract
"Symmetry invariance in a diamond ring" —

For some historical background on the symmetry investigations by
Dyson and Macdonald, see Dyson's 1972 article "MIssed Opportunities."

For Macdonald's own  use of the words "modular" and "affine," see
Macdonald, I. G., "Affine Lie algebras and modular forms," 
Séminaire N. Bourbaki , Vol. 23 (1980-1981), Talk no. 577, pp. 258-276.

Monday, December 18, 2017

Wheelwright and the Dance

Filed under: Geometry — m759 @ 1:00 PM

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

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

A graphic approach to this concept:

"The Bounded Space" —

'Space Cross' from the Cullinane diamond theorem

"The Fire, Air, Earth, and Water" —

Logo for 'Elements of Finite Geometry'

Thursday, November 30, 2017

The Matrix for Quantum Mystics

Filed under: 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.

Tuesday, October 24, 2017

Visual Insight

Filed under: Geometry — m759 @ 1:00 PM

The most recent post in the "Visual Insight" blog of the
American Mathematical Society was by John Baez on Jan. 1, 2017


A visually  related concept — See Solomon's Cube in this  journal.
Chronologically  related — Posts now tagged New Year's Day 2017.
Solomon's cube is the 4x4x4 case of the diamond theorem — 

Tuesday, October 10, 2017

Dueling Formulas

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

Continued from the previous post and from posts
now tagged Dueling Formulas

The four-diamond formula of Jung and
the four-dot "as" of Claude Lévi-Strauss:

   

Simplified versions of the diamonds and the dots
 

The Ring of the Diamond Theorem          ::

I prefer Jung. For those who prefer Lévi-Strauss —

     First edition, Cornell University Press, 1970.

A related tale — "A Meaning, Like."

Monday, June 26, 2017

Upgrading to Six

Filed under: Uncategorized — Tags: , , — m759 @ 9:00 PM

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.

Monday, June 19, 2017

Dead End

Filed under: Uncategorized — Tags: — m759 @ 11:24 PM

Group Actions on Partitions- 1985

Google Search- 'group actions on partitions'- June 19, 2017

The above 1985 note was an attempt to view the diamond theorem
in a more general context. I know no more about the note now than 
I did in 1985. The only item in the search results above that is not  
by me (the seventh) seems of little relevance.

Tuesday, June 13, 2017

Found in Translation

Filed under: Uncategorized — m759 @ 9:16 PM

The diamond theorem in Denmark —

Tuesday, May 2, 2017

Image Albums

Filed under: Uncategorized — Tags: — m759 @ 1:05 PM

Pinterest boards uploaded to the new m759.net/piwigo

Diamond Theorem 

Diamond Theorem Correlation

Miracle Octad Generator

The Eightfold Cube

Six-Set Geometry

Diamond Theory Cover

Update of May 2 —

Four-Color Decomposition

Binary Galois Spaces

The Galois Tesseract

Update of May 3 —

Desargues via Galois

The Tetrahedral Model

Solomon's Cube

Update of May 8 —

Art Space board created at Pinterest

Friday, April 7, 2017

Personal Identity

Filed under: Uncategorized — 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.

Saturday, March 4, 2017

Hidden Figure

Filed under: Uncategorized — m759 @ 3:17 AM

From this morning's New York Times  Wire

A cybersecurity-related image from Thursday evening

The diamond theorem correlation at the University of Bradford

Thursday, March 2, 2017

Raiders of the Lost Crucible Continues

Filed under: Uncategorized — m759 @ 9:59 PM

Cover, 2005 paperback edition of 'Refiner's Fire,' a 1977 novel by Mark Helprin

Mariner Books paperback, 2005

See, too, this evening's A Common Space
and earlier posts on Raiders of the Lost Crucible.

Also not without relevance —

The diamond theorem correlation at the University of Bradford

Sunday, January 8, 2017

A Theory of Everything

Filed under: Uncategorized — Tags: — m759 @ 7:11 PM

The title refers to the Chinese book the I Ching ,
the Classic of Changes .

The 64 hexagrams of the I Ching  may be arranged
naturally in a 4x4x4 cube. The natural form of transformations
("changes") of this cube is given by the diamond theorem.

A related post —

The Eightfold Cube, core structure of the I Ching

Tuesday, October 18, 2016

Parametrization

Filed under: Uncategorized — m759 @ 6:00 AM

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

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

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

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

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

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

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

Tuesday, September 27, 2016

Chomsky and Lévi-Strauss in China

Filed under: Uncategorized — Tags: — m759 @ 7:31 AM

Or:  Philosophy for Jews

From a New Yorker  weblog post dated Dec. 6, 2012 —

"Happy Birthday, Noam Chomsky" by Gary Marcus—

"… two titans facing off, with Chomsky, as ever,
defining the contest"

"Chomsky sees himself, correctly, as continuing
a conversation that goes back to Plato, especially
the Meno dialogue, in which a slave boy is
revealed by Socrates to know truths about
geometry that he hadn’t realized he knew."

Socrates and the slave boy discussed a rather elementary "truth
about geometry" — A diamond inscribed in a square has area 2
(and side the square root of 2) if the square itself has area 4
(and side 2).

Consider that not-particularly-deep structure from the Meno dialogue
in the light of the following…

The following analysis of the Meno diagram from yesterday's
post "The Embedding" contradicts the Lévi-Strauss dictum on
the impossibility of going beyond a simple binary opposition.
(The Chinese word taiji  denotes the fundamental concept in
Chinese philosophy that such a going-beyond is both useful
and possible.)

The matrix at left below represents the feminine yin  principle
and the diamond at right represents the masculine yang .

      From a post of Sept. 22,
  "Binary Opposition Illustrated" —

A symbol of the unity of yin and yang —

Related material:

A much more sophisticated approach to the "deep structure" of the
Meno diagram —

The larger cases —

The diamond theorem

Tuesday, September 20, 2016

Savage Logic

Filed under: Uncategorized — m759 @ 9:29 PM

From "The Cerebral Savage," by Clifford Geertz —

(Encounter, Vol. 28 No. 4 (April 1967), pp. 25-32.)

From http://www.diamondspace.net/about.html

The diamond theorem

Sunday, August 21, 2016

Review

Filed under: Uncategorized — m759 @ 2:00 PM

Fugue No. 21
B-Flat Major
Well-Tempered Clavier Book II
Johann Sebastian Bach
by Timothy A. Smith

Theme and Variations
by Steven H. Cullinane

The beginning of each —

Cullinane, 'Theme and Variations'

Some context —

The diamond theorem

Tuesday, June 21, 2016

The Central Structure

Filed under: Uncategorized — m759 @ 8:00 AM

"The central poem is the poem of the whole,
The poem of the composition of the whole"

— Wallace Stevens, "A Primitive like an Orb"

The symmetries of the central four squares in any pattern
from the 4×4 version of the diamond theorem  extend to
symmetries of the entire pattern.  This is true also of the
central eight cubes in the 4×4×4  Solomon's cube .

Friday, June 10, 2016

High Concept

Filed under: Uncategorized — m759 @ 9:00 AM

(Continued)

Orwell Meets Waugh

'Space Cross' from the Cullinane diamond theorem

Sunday, May 22, 2016

Sunday School

Filed under: Uncategorized — m759 @ 9:00 AM

A less metaphysical approach to a "pre-form" —

From Wallace Stevens, "The Man with the Blue Guitar":

IX

And the color, the overcast blue
Of the air, in which the blue guitar
Is a form, described but difficult,
And I am merely a shadow hunched
Above the arrowy, still strings,
The maker of a thing yet to be made . . . .

"Arrowy, still strings" from the diamond theorem

See also "preforming" and the blue guitar
in a post of May 19, 2010.

Update of 7:11 PM ET:
More generally, see posts tagged May 19 Gestalt.

Wednesday, May 18, 2016

Dueling Formulas

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

Jung's four-diamond formula vs. Levi-Strauss's 'canonical formula'

Note the echo of Jung's formula in the diamond theorem.

An attempt by Lévi-Strauss to defend his  formula —

"… reducing the life of the mind to an abstract game . . . ." —

For a fictional version of such a game, see Das Glasperlenspiel .

Tuesday, May 3, 2016

Symmetry

Filed under: Uncategorized — m759 @ 12:00 PM

A note related to the diamond theorem and to the site
Finite Geometry of the Square and Cube —

The last link in the previous post leads to a post of last October whose
final link leads, in turn, to a 2009 post titled Summa Mythologica .

Webpage demonstrating symmetries of 'Solomon's Cube'

Some may view the above web page as illustrating the
Glasperlenspiel  passage quoted here in Summa Mythologica 

“"I suddenly realized that in the language, or at any rate
in the spirit of the Glass Bead Game, everything actually
was all-meaningful, that every symbol and combination of
symbols led not hither and yon, not to single examples,
experiments, and proofs, but into the center, the mystery
and innermost heart of the world, into primal knowledge.
Every transition from major to minor in a sonata, every
transformation of a myth or a religious cult, every classical
or artistic formulation was, I realized in that flashing moment,
if seen with a truly meditative mind, nothing but a direct route
into the interior of the cosmic mystery, where in the alternation
between inhaling and exhaling, between heaven and earth,
between Yin and Yang, holiness is forever being created.”

A less poetic meditation on the above web page* —

"I saw that in the alternation between front and back,
between top and bottom, between left and right,
symmetry is forever being created."

Update of Sept. 5, 2016 — See also a related remark
by Lévi-Strauss in 1955: "…three different readings
become possible: left to right, top to bottom, front 
to back."

* For the underlying mathematics, see a June 21, 1983, research note

Tuesday, April 26, 2016

Interacting

Filed under: Uncategorized — m759 @ 8:31 PM

"… I would drop the keystone into my arch …."

— Charles Sanders Peirce, "On Phenomenology"

" 'But which is the stone that supports the bridge?' Kublai Khan asks."

— Italo Calvino, Invisible Cities, as quoted by B. Elan Dresher.

(B. Elan Dresher. Nordlyd  41.2 (2014): 165-181,
special issue on Features edited by Martin Krämer,
Sandra Ronai and Peter Svenonius. University of Tromsø –
The Arctic University of Norway.
http://septentrio.uit.no/index.php/nordlyd)

Peter Svenonius and Martin Krämer, introduction to the
Nordlyd  double issue on Features —

"Interacting with these questions about the 'geometric' 
relations among features is the algebraic structure
of the features."

For another such interaction, see the previous post.

This  post may be viewed as a commentary on a remark in Wikipedia

"All of these ideas speak to the crux of Plato's Problem…."

See also The Diamond Theorem at Tromsø and Mere Geometry.

Monday, April 25, 2016

Seven Seals

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

 An old version of the Wikipedia article "Group theory"
(pictured in the previous post) —

"More poetically "

From Hermann Weyl's 1952 classic Symmetry

"Galois' ideas, which for several decades remained
a book with seven seals  but later exerted a more
and more profound influence upon the whole
development of mathematics, are contained in
a farewell letter written to a friend on the eve of
his death, which he met in a silly duel at the age of
twenty-one. This letter, if judged by the novelty and
profundity of ideas it contains, is perhaps the most
substantial piece of writing in the whole literature
of mankind."

The seven seals from the previous post, with some context —

These models of projective points are drawn from the underlying
structure described (in the 4×4 case) as part of the proof of the
Cullinane diamond theorem .

Monday, December 28, 2015

Mirrors, Mirrors, on the Wall

Filed under: Uncategorized — m759 @ 8:00 AM

The previous post quoted Holland Cotter's description of
the late Ellsworth Kelly as one who might have admired 
"the anonymous role of the Romanesque church artist." 

Work of a less anonymous sort was illustrated today by both
The New York Times  and The Washington Post

'Artist Who Shaped Geometries on a Bold Scale' - NY Times

'Ellsworth Kelly, the master of the deceptively simple' - Washington Post

The Post 's remarks are of particular interest:

Philip Kennicott in The Washington Post , Dec. 28, 2015,
on a work by the late Ellsworth Kelly —

“Sculpture for a Large Wall” consisted of 104 anodized aluminum panels, colored red, blue, yellow and black, and laid out on four long rows measuring 65 feet. Each panel seemed different from the next, subtle variations on the parallelogram, and yet together they also suggested a kind of language, or code, as if their shapes, colors and repeating patterns spelled out a basic computer language, or proto-digital message.

The space in between the panels, and the shadows they cast on the wall, were also part of the effect, creating a contrast between the material substance of the art, and the cascading visual and mental ideas it conveyed. The piece was playful, and serious; present and absent; material and imaginary; visually bold and intellectually diaphanous.

Often, with Kelly, you felt as if he offered up some ideal slice of the world, decontextualized almost to the point of absurdity. A single arc sliced out of a circle; a single perfect rectangle; one bold juxtaposition of color or shape. But when he allowed his work to encompass more complexity, to indulge a rhetoric of repetition, rhythmic contrasts, and multiple self-replicating ideas, it began to feel like language, or narrative. And this was always his best mode.

Compare and contrast a 2010 work by Josefine Lyche

IMAGE- The 2x2 case of the diamond theorem as illustrated by Josefine Lyche, Oct. 2010

Lyche's mirrors-on-the-wall installation is titled
"The 2×2 Case (Diamond Theorem)."

It is based on a smaller illustration of my own.

These  variations also, as Kennicott said of Kelly's,
"suggested a kind of language, or code."

This may well be the source of their appeal for Lyche.
For me, however, such suggestiveness is irrelevant to the
significance of the variations in a larger purely geometric
context.

This context is of course quite inaccessible to most art
critics. Steve Martin, however, has a phrase that applies
to both Kelly's and Lyche's installations: "wall power."
See a post of Dec. 15, 2010.

Friday, November 13, 2015

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

Filed under: Uncategorized — 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)—

Thursday, October 15, 2015

Contrapuntal Interweaving

Filed under: Uncategorized — Tags: — m759 @ 2:01 AM

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

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

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

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

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

 

Saturday, October 10, 2015

Nonphysical Entities

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

Norwegian Sculpture Biennial 2015 catalog, p. 70 —

" 'Ambassadørene' er fysiske former som presenterer
ikk-fysiske fenomener. "

Translation by Google —

" 'Ambassadors' physical forms presents
nonphysical phenomena. "

Related definition —

Are the "line diagrams" of the diamond theorem and
the analogous "plane diagrams" of the eightfold cube
nonphysical entities? Discuss.

Wednesday, August 26, 2015

“The Quality Without a Name”

Filed under: Uncategorized — Tags: , — m759 @ 8:00 AM

The title phrase, paraphrased without quotes in
the previous post, is from Christopher Alexander's book
The Timeless Way of Building  (Oxford University Press, 1979).

A quote from the publisher:

"Now, at last, there is a coherent theory
which describes in modern terms
an architecture as ancient as
human society itself."

Three paragraphs from the book (pp. xiii-xiv):

19. Within this process, every individual act
of building is a process in which space gets
differentiated. It is not a process of addition,
in which preformed parts are combined to
create a whole, but a process of unfolding,
like the evolution of an embryo, in which
the whole precedes the parts, and actualy
gives birth to then, by splitting.

20. The process of unfolding goes step by step,
one pattern at a time. Each step brings just one
pattern to life; and the intensity of the result
depends on the intensity of each one of these
individual steps.

21. From a sequence of these individual patterns,
whole buildings with the character of nature
will form themselves within your thoughts,
as easily as sentences.

Compare to, and contrast with, these illustrations of "Boolean space":

(See also similar illustrations from Berkeley and Purdue.)

Detail of the above image —

Note the "unfolding," as Christopher Alexander would have it.

These "Boolean" spaces of 1, 2, 4, 8, and 16 points
are also Galois  spaces.  See the diamond theorem —

Friday, August 14, 2015

Discrete Space

Filed under: Uncategorized — Tags: — m759 @ 7:24 AM

(A review)

Galois space:

Image-- examples from Galois affine geometry

Counting symmetries of  Galois space:
IMAGE - The Diamond Theorem

The reason for these graphic symmetries in affine Galois space —

symmetries of the underlying projective Galois space:

Monday, June 15, 2015

Omega Matrix

Filed under: Uncategorized — 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 Ω is the 4×4 grid above.

See the Cullinane diamond theorem.

Sunday, May 17, 2015

Moon Shadow

Filed under: Uncategorized — m759 @ 7:07 AM

IMAGE- The diamond theorem and umbral moonshine

"I'm being followed by a moon shadow…."  — Song lyric

Thursday, May 7, 2015

Paradigm for Pedagogues

Filed under: Uncategorized — Tags: — m759 @ 7:14 PM

Illustrations from a post of Feb. 17, 2011:

Plato’s paradigm in the Meno —

http://www.log24.com/log/pix11/110217-MenoFigure16bmp.bmp

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

http://www.log24.com/log/pix11/110217-MenoFigureColored16bmp.bmp

Saturday, April 4, 2015

Harrowing of Hell (continued)

Filed under: Uncategorized — m759 @ 3:28 PM

Holy Saturday is, according to tradition, the day of 
the harrowing of Hell.

Notes:

The above passage on "Die Figuren der vier Modi
im Magischen Quadrat 
" should be read in the context of
a Log24 post from last year's Devil's Night (the night of
October 30-31).  The post, "Structure," indicates that, using
the transformations of the diamond theorem, the notorious
"magic" square of Albrecht Dürer may be transformed
into normal reading order.  That order is only one of
322,560 natural reading orders for any 4×4 array of
symbols. The above four "modi" describe another.

Wednesday, April 1, 2015

Manifest O

Filed under: Uncategorized — Tags: , — m759 @ 4:44 AM

The title was suggested by
http://benmarcus.com/smallwork/manifesto/.

The "O" of the title stands for the octahedral  group.

See the following, from http://finitegeometry.org/sc/map.html —

83-06-21 An invariance of symmetry The diamond theorem on a 4x4x4 cube, and a sketch of the proof.
83-10-01 Portrait of O  A table of the octahedral group O using the 24 patterns from the 2×2 case of the diamond theorem.
83-10-16 Study of O  A different way of looking at the octahedral group, using cubes that illustrate the 2x2x2 case of the diamond theorem.
84-09-15 Diamonds and whirls Block designs of a different sort — graphic figures on cubes. See also the University of Exeter page on the octahedral group O.

Monday, March 2, 2015

Elements of Design

Filed under: Uncategorized — m759 @ 1:28 AM

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


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

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

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

Saturday, February 21, 2015

High and Low Concepts

Filed under: Uncategorized — Tags: — m759 @ 4:30 PM

Steven Pressfield on April 25, 2012:

What exactly is High Concept?

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

ZZZZZZZZ.

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

The Diamond Theorem Correlation:

From left to right

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

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

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

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

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

Wednesday, November 19, 2014

The Eye/Mind Conflict

Filed under: Uncategorized — Tags: — m759 @ 10:25 AM

Harold Rosenberg, "Art and Words," 
The New Yorker , March 29, 1969. From page 110:

"An advanced painting of this century inevitably gives rise
in the spectator to a conflict between his eye and his mind; 
as Thomas Hess has pointed out, the fable of the emperor's 
new clothes is echoed at the birth of every modemist art 
movement. If work in a new mode is to be accepted, the 
eye/mind conflict must be resolved in favor of the mind; 
that is, of the language absorbed into the work. Of itself, 
the eye is incapable of breaking into the intellectual system 
that today distinguishes between objects that are art and 
those that are not. Given its primitive function of 
discriminating among things in shopping centers and on 
highways, the eye will recognize a Noland as a fabric
design, a Judd as a stack of metal bins— until the eye's 
outrageous philistinism has been subdued by the drone of 
formulas concerning breakthroughs in color, space, and 
even optical perception (this, too, unseen by the eye, of 
course). It is scarcely an exaggeration to say that paintings 
are today apprehended with the ears. Miss Barbara Rose, 
once a promoter of striped canvases and aluminum boxes, 
confesses that words are essential to the art she favored 
when she writes, 'Although the logic of minimal art gained 
critical respect, if not admiration, its reductiveness allowed
for a relatively limited art experience.' Recent art criticism 
has reversed earlier procedures: instead of deriving principles 
from what it sees, it teaches the eye to 'see' principles; the 
writings of one of America's influential critics often pivot on 
the drama of how he failed to respond to a painting or 
sculpture the first few times he saw it but, returning to the 
work, penetrated the concept that made it significant and
was then able to appreciate it. To qualify as a member of the 
art public, an individual must be tuned to the appropriate 
verbal reverberations of objects in art galleries, and his 
receptive mechanism must be constantly adjusted to oscillate 
to new vocabularies."

New vocabulary illustrated:

Graphic Design and a Symplectic Polarity —

Background: The diamond theorem
and a zero system .

Monday, November 3, 2014

The Rhetoric of Abstract Concepts

Filed under: Uncategorized — m759 @ 12:48 PM

From a post of June 3, 2013:

New Yorker  editor David Remnick at Princeton today
(from a copy of his prepared remarks):

“Finally, speaking of fabric design….”

I prefer Tom and Harold:

Tom Wolfe in The Painted Word 

“I am willing (now that so much has been revealed!)
to predict that in the year 2000, when the Metropolitan
or the Museum of Modern Art puts on the great
retrospective exhibition of American Art 1945-75,
the three artists who will be featured, the three seminal
figures of the era, will be not Pollock, de Kooning, and
Johns-but Greenberg, Rosenberg, and Steinberg.
Up on the walls will be huge copy blocks, eight and a half
by eleven feet each, presenting the protean passages of
the period … a little ‘fuliginous flatness’ here … a little
‘action painting’ there … and some of that ‘all great art
is about art’ just beyond. Beside them will be small
reproductions of the work of leading illustrators of
the Word from that period….”

Harold Rosenberg in The New Yorker  (click to enlarge)

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….”

Symplectic :

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

— Steven H. Cullinane,
diamond theorem illustration

Sunday, November 2, 2014

A Singular Place

Filed under: Uncategorized — m759 @ 5:09 PM

"Macy’s Herald Square occupies a singular place
in American retailing." — NY Times  today, in print
on page BU1 of the New York edition with the headline:

Makeover on 34th Street .

A Singular Time:

See Remember Me to Herald Square, at noon on
August 21, 2014, and related earlier Log24 posts.

Also on Aug. 21, 2014: from a blog post, 'Tiles,' by
Theo Wright, a British textile designer —

The 24 tile patterns displayed by Wright may be viewed
in their proper mathematical context at …

http://www.diamondspace.net/about.html:

IMAGE - The Diamond Theorem

Friday, October 31, 2014

Structure

Filed under: Uncategorized — m759 @ 3:00 AM

On Devil’s Night

Introducing a group of 322,560 affine transformations of Dürer’s ‘Magic’ Square

IMAGE- Introduction to 322,560 Affine Transformations of Dürer's 'Magic' Square

The four vector-space substructures of digits in 1st, 2nd, 3rd, 4th place,
together with the diamond theorem, indicate that Dürer’s square “minus one”
can be transformed by permutations of rows, columns, and quadrants to a
square with (decimal) digits in the usual numerical order, increasing from
top left to bottom right. Such permutations form a group of order 322,560.

(Continued from Vector Addition in a Finite Field, Twelfth Night, 2013.)

Saturday, September 20, 2014

Symplectic Structure

Filed under: Uncategorized — Tags: — m759 @ 11:30 AM

(Continued)

The fictional zero theorem  of Terry Gilliam's current film
by that name should not be confused with the zero system
underlying the diamond theorem.

Sunday, September 14, 2014

Sensibility

Filed under: Uncategorized — Tags: , — m759 @ 9:26 AM

Structured gray matter:

Graphic symmetries of Galois space:
IMAGE - The Diamond Theorem

The reason for these graphic symmetries in affine  Galois space —

symmetries of the underlying projective  Galois space:

Tuesday, September 9, 2014

Smoke and Mirrors

Filed under: Uncategorized — 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.

Sunday, August 31, 2014

Sunday School

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

The Folding

Cynthia Zarin in The New Yorker , issue dated April 12, 2004—

“Time, for L’Engle, is accordion-pleated. She elaborated,
‘When you bring a sheet off the line, you can’t handle it
until it’s folded, and in a sense, I think, the universe can’t
exist until it’s folded — or it’s a story without a book.’”

The geometry of the 4×4 square array is that of the
3-dimensional projective Galois space PG(3,2).

This space occurs, notably, in the Miracle Octad Generator (MOG)
of R. T. Curtis (submitted to Math. Proc. Camb. Phil. Soc.  on
15 June 1974).  Curtis did not, however, describe its geometric
properties. For these, see the Cullinane diamond theorem.

Some history: 

Curtis seems to have obtained the 4×4 space by permuting,
then “folding” 1×8 binary sequences into 4×2 binary arrays.
The original 1×8 sequences came from the method of Turyn
(1967) described by van Lint in his book Coding Theory
(Springer Lecture Notes in Mathematics, No. 201 , first edition
published in 1971). Two 4×2 arrays form each 4×4 square array
within the MOG. This construction did not suggest any discussion
of the geometric properties of the square arrays.

[Rewritten for clarity on Sept. 3, 2014.]

Thursday, July 17, 2014

Paradigm Shift:

Filed under: Uncategorized — Tags: — m759 @ 11:01 AM
 

Continuous Euclidean space to discrete Galois space*

Euclidean space:

Point, line, square, cube, tesseract

From a page by Bryan Clair

Counting symmetries in Euclidean space:

Galois space:

Image-- examples from Galois affine geometry

Counting symmetries of  Galois space:
IMAGE - The Diamond Theorem

The reason for these graphic symmetries in affine Galois space —

symmetries of the underlying projective Galois space:

* For related remarks, see posts of May 26-28, 2012.

Sunday, June 8, 2014

Vide

Filed under: Uncategorized — Tags: , , — m759 @ 10:00 AM

Some background on the large Desargues configuration

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

— Gian-Carlo Rota discussing the theorem of Desargues

What space  tells us about the theorem :  

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

Vide  Classical Geometry in Light of Galois Geometry.

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

A coordinate-free approach to symplectic structure

Wednesday, May 14, 2014

Two -Year College

Filed under: Uncategorized — m759 @ 9:45 AM

See last night’s pentagram photo and a post from May 13, 2012.

That post links to a little-known video of a 1972 film.
A speech from the film was used by Oslo artist Josefine Lyche as a
voice-over in her  2011 golden-ratio video (with pentagrams) that she
exhibited along with a large, wall-filling copy of some of my own work.
The speech (see video below) is clearly nonsense.

The patterns* Lyche copied are not.

“Who are you, anyway?” 

— Question at 00:41 of 15:00, Rainbow Bridge (Part 5 of 9)
at YouTube, addressed to Baron Bingen as “Mr. Rabbit”

* Patterns exhibited again later, apparently without the Lyche pentagram video.
It turns out, by the way, that Lyche created that video by superimposing
audio from the above “Rainbow Bridge” film onto a section of Disney’s 1959
Donald in Mathmagic Land” (see 7:17 to 8:57 of the 27:33 Disney video).

Friday, March 21, 2014

Three Constructions of the Miracle Octad Generator

Filed under: Uncategorized — Tags: , , — m759 @ 12:24 PM

IMAGE- Two constructions, by Turyn/Curtis, and by Conway, of the Miracle Octad Generator

See also a Log24 post on this subject from Dec. 14, 2013,
especially (scroll down) the update of March 9, 2014.

Related material on the Turyn-Curtis construction
from the University of Cambridge —

— Slide by “Dr. Parker” — Apparently Richard A. Parker —
Lecture 4, “Discovering M24,” in slides for lectures 1-8 from lectures
at Cambridge in 2010-2011 on “Sporadic and Related Groups.”
See also the Parker lectures of 2012-2013 on the same topic.

A third construction of Curtis’s 35  4×6  1976 MOG arrays would use
Cullinane’s analysis of the 4×4 subarrays’ affine and projective structure,
and point out the fact that Conwell’s 1910 correspondence of the 35
4+4-partitions of an 8-set with the 35 lines of the projective 3-space
over the 2-element field, PG(3, 2), is essentially the same correspondence
as that constituting Curtis’s 1976 MOG.

See The Diamond Theorem,  Finite RelativityGalois Space,
Generating the Octad Generator, and The Klein Correspondence.

Update of March 22-March 23 —

Adding together as (0,1)-matrices over GF(2) the black parts (black
squares as 1’s, all other squares as 0’s) of the 35  4×6 arrays of the 1976
Curtis MOG would then reveal*  the symmetric role played in octads
by what Curtis called the heavy brick , and so reveal also the action of
S3 on the three Curtis bricks that leaves invariant the set of all 759
octads of the S(5, 8, 24) constructed from the 35  MOG arrays.  For more
details of this “by-hand” construction, see Geometry of the 4×4 Square.
For the mathematical properties of the S(5, 8, 24), it is convenient to
have a separate construction,  not  by hand (such as Turyn’s), of the
extended binary Golay code. See the Brouwer preprint quoted above.

* “Then a miracle occurs,” as in the classic 1977 Sidney Harris cartoon.

Illustration of array addition from March 23 —

IMAGE- Discovery of the S_3 action on bricks in the Conwell-Cullinane 'by-hand' approach to octad-building

Saturday, March 15, 2014

Language Game

Filed under: Uncategorized — m759 @ 3:00 PM

Cullinane = Cullinan + e

Greene = Green + e

Tuesday, March 11, 2014

Depth

Filed under: Uncategorized — Tags: , — m759 @ 11:16 AM

"… this notion of ‘depth’ is an elusive one
even for a mathematician who can recognize it…."

— G. H. Hardy,  A Mathematician's Apology

Part I:  An Inch Deep

IMAGE- Catch-phrase 'a mile wide and an inch deep' in mathematics education

Part II:  An Inch Wide

See a search for "square inch space" in this journal.

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

 

See also recent posts with the tag depth.

Thursday, February 6, 2014

The Representation of Minus One

Filed under: Uncategorized — Tags: , — m759 @ 6:24 AM

For the late mathematics educator Zoltan Dienes.

"There comes a time when the learner has identified
the abstract content of a number of different games
and is practically crying out for some sort of picture
by means of which to represent that which has been
gleaned as the common core of the various activities."

— Article by "Melanie" at Zoltan Dienes's website

Dienes reportedly died at 97 on Jan. 11, 2014.

From this journal on that date —

http://www.log24.com/log/pix11/110219-SquareRootQuaternion.jpg

A star figure and the Galois quaternion.

The square root of the former is the latter.

Update of 5:01 PM ET Feb. 6, 2014 —

An illustration by Dienes related to the diamond theorem —

See also the above 15 images in

http://www.log24.com/log/pix11/110220-relativprob.jpg

and versions of the 4×4 coordinatization in  The 4×4 Relativity Problem
(Jan. 17, 2014).

Friday, January 17, 2014

The 4×4 Relativity Problem

Filed under: Uncategorized — Tags: , , — m759 @ 11:00 PM

The sixteen-dot square array in yesterday’s noon post suggests
the following remarks.

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

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

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

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

The 1977 matrix Q is echoed in the following from 2002—

IMAGE- Dolgachev and Keum, coordinatization of the 4x4 array in 'Birational Automorphisms of Quartic Hessian Surfaces,' AMS Transactions, 2002

A different representation of Cullinane’s 1977 square model of the
16-point affine geometry over the two-element Galois field GF(2)
is supplied by Conway and Sloane in Sphere Packings, Lattices and Groups   
(first published in 1988) :

IMAGE- The Galois tesseract as a four-dimensional vector space, from a diagram by Conway and Sloane in 'Sphere Packings, Lattices, and Groups'

Here a, b, c, d   are basis vectors in the vector 4-space over GF(2).
(For a 1979 version of this vector space, see AMS Abstract 79T-A37.)

See also a 2011 publication of the Mathematical Association of America —

From 'Beautiful Mathematics,' by Martin Erickson, an excerpt on the Cullinane diamond theorem (with source not mentioned)

Wednesday, January 8, 2014

Occupy Space

Filed under: Uncategorized — m759 @ 9:00 PM

(Continued

Three Notes on Design

1.  From the Museum of Modern Art  today—

“It’s a very nice gesture of a kind of new ethos:
To make publicly accessible, unticketed space
that is attractive and has cultural programming,”
Glenn D. Lowry, MoMA’s director, said.

2.  From The New York Times  today—

3.  From myself  last December

IMAGE- Summary of the diamond theorem at 'Diamond Space' website

Thursday, December 5, 2013

Fields

Filed under: Uncategorized — Tags: , , — m759 @ 1:20 AM

Edward Frenkel recently claimed for Robert Langlands
the discovery of a link between two "totally different"
fields of mathematics— number theory and harmonic analysis.
He implied that before Langlands, no relationship between
these fields was known.

See his recent book, and his lecture at the Fields Institute
in Toronto on October 24, 2013.

Meanwhile, in this journal on that date, two math-related
quotations for Stephen King, author of Doctor Sleep

"Danvers is a town in Essex County, Massachusetts, 
United States, located on the Danvers River near the
northeastern coast of Massachusetts. Originally known
as Salem Village, the town is most widely known for its
association with the 1692 Salem witch trials. It is also
known for the Danvers State Hospital, one of the state's
19th-century psychiatric hospitals, which was located here." 

"The summer's gone and all the roses fallin' "

For those who prefer their mathematics presented as fact, not fiction—

(Click for a larger image.)

The arrows in the figure at the right are an attempt to say visually that 
the diamond theorem is related to various fields of mathematics.
There is no claim that prior to the theorem, these fields were not  related.

See also Scott Carnahan on arrow diagrams, and Mathematical Imagery.

Wednesday, October 16, 2013

Theme and Variations

Filed under: Uncategorized — m759 @ 6:16 PM

(Continued)

IMAGE- The Diamond Theorem

Josefine Lyche’s large wall version of the twenty-four 2×2 variations
above was apparently offered for sale today in Norway —

Click image for more details and click here for a translation.

Monday, September 30, 2013

Interview with Josefine Lyche

Filed under: Uncategorized — m759 @ 10:00 PM

For those who understand spoken Norwegian.

I do not. The interview apparently gives some

background on Lyche’s large wall version of

The 2×2 Case (Diamond Theorem) II.

(After Steven H. Cullinane)” 2012

Size: 260 x 380 cm

See also this work as displayed at a Kjærlighet til Oslo page.

(Updated March 30, 2014, to replace dead Kjaerlighet link.)

Tuesday, September 3, 2013

“The Stone” Today Suggests…

Filed under: Uncategorized — m759 @ 12:31 PM

A girl's best friend?

The Philosopher's Gaze , by David Michael Levin,
U. of California Press, 1999, in III.5, "The Field of Vision," pp. 174-175—

The post-metaphysical question—question for a post-metaphysical phenomenology—is therefore: Can the perceptual field, the ground of perception, be released  from our historical compulsion to represent it in a way that accommodates our will to power and its need to totalize and reify the presencing of being? In other words: Can the ground be experienced as  ground? Can its hermeneutical way of presencing, i.e., as a dynamic interplay of concealment and unconcealment, be given appropriate  respect in the receptivity of a perception that lets itself  be appropriated by  the ground and accordingly lets  the phenomenon of the ground be  what and how it is? Can the coming-to-pass of the ontological difference that is constitutive of all the local figure-ground differences taking place in our perceptual field be made visible hermeneutically, and thus without violence to its withdrawal into concealment? But the question concerning the constellation of figure and ground cannot be separated from the question concerning the structure of subject and object. Hence the possibility of a movement beyond metaphysics must also think the historical possibility of breaking out of this structure into the spacing of the ontological difference: différance , the primordial, sensuous, ekstatic écart . As Heidegger states it in his Parmenides lectures, it is a question of "the way historical man belongs within the bestowal of being (Zufügung des Seins ), i.e., the way this order entitles him to acknowledge being and to be the only being among all beings to see  the open" (PE* 150, PG** 223. Italics added). We might also say that it is a question of our response-ability, our capacity as beings gifted with vision, to measure up to the responsibility for perceptual responsiveness laid down for us in the "primordial de-cision" (Entscheid ) of the ontological difference (ibid.). To recognize the operation of the ontological difference taking place in the figure-ground difference of the perceptual Gestalt  is to recognize the ontological difference as the primordial Riß , the primordial Ur-teil  underlying all our perceptual syntheses and judgments—and recognize, moreover, that this rift, this  division, decision, and scission, an ekstatic écart  underlying and gathering all our so-called acts of perception, is also the only "norm" (ἀρχή ) by which our condition, our essential deciding and becoming as the ones who are gifted with sight, can ultimately be judged.

* PE: Parmenides  of Heidegger in English— Bloomington: Indiana University Press, 1992

** PG: Parmenides  of Heidegger in German— Gesamtausgabe , vol. 54— Frankfurt am Main: Vittorio Klostermann, 1992

Examples of "the primordial Riß " as ἀρχή  —

For an explanation in terms of mathematics rather than philosophy,
see the diamond theorem. For more on the Riß  as ἀρχή , see
Function Decomposition Over a Finite Field.

Monday, August 12, 2013

Form

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

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

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

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

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

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

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

Monday, August 5, 2013

Wikipedia Updates

Filed under: Uncategorized — m759 @ 12:30 PM

I added links today in the following Wikipedia articles:

The links will probably soon be deleted,
but it seemed worth a try.

Tuesday, July 9, 2013

Vril Chick

Filed under: Uncategorized — m759 @ 4:30 AM

Profile picture of "Jo Lyxe" (Josefine Lyche) at Vimeo

Profile picture for "Jo Lyxe" (Josefine Lyche) at Vimeo

Compare to an image of Vril muse Maria Orsitsch.

From the catalog of a current art exhibition
(25 May – 31 August, 2013) in Norway,
I DE LANGE NÆTTER —

Josefine Lyche
Born in 1973 in Bergen, Norway.
Lives and works in Oslo and Berlin.

Keywords (to help place my artwork in the
proper context): Aliens, affine geometry, affine
planes, affine spaces, automorphisms, binary
codes, block designs, classical groups, codes,
coding theory, collineations, combinatorial,
combinatorics, conjugacy classes, the Conwell
correspondence, correlations, Cullinane,
R. T. Curtis, design theory, the diamond theorem,
diamond theory, duads, duality, error correcting
codes, esoteric, exceptional groups,
extraterrestrials, finite fields, finite geometry, finite
groups, finite rings, Galois fields, generalized
quadrangles, generators, geometry, GF(2),
GF(4), the (24,12) Golay code, group actions,
group theory, Hadamard matrices, hypercube,
hyperplanes, hyperspace, incidence structures,
invariance, Karnaugh maps, Kirkman’s schoolgirls
problem, Latin squares, Leech lattice, linear
groups, linear spaces, linear transformations,
Magick, Mathieu groups, matrix theory, Meno,
Miracle Octad Generator, MOG, multiply transitive
groups, occultism, octahedron, the octahedral
group, Orsic, orthogonal arrays, outer automorphisms,
parallelisms, partial geometries,
permutation groups, PG(3,2), Plato, Platonic
solids, polarities, Polya-Burnside theorem, projective
geometry, projective planes, projective
spaces, projectivities, Pythagoras, reincarnation,
Reed-Muller codes, the relativity problem,
reverse engineering, sacred geometry, Singer
cycle, skew lines, Socrates, sporadic simple
groups, Steiner systems, Sylvester, symmetric,
symmetry, symplectic, synthemes, synthematic,
Theosophical Society tesseract, Tessla, transvections,
Venn diagrams, Vril society, Walsh
functions, Witt designs.

(See also the original catalog page.)

Clearly most of this (the non-highlighted parts) was taken
from my webpage Diamond Theory. I suppose I should be
flattered, but I am not thrilled to be associated with the
(apparently fictional) Vril Society.

For some background, see (for instance) 
Conspiracy Theories and Secret Societies for Dummies .

Tuesday, June 25, 2013

Lexicon (continued)

Filed under: Uncategorized — m759 @ 7:20 PM

Online biography of author Cormac McCarthy—

" he left America on the liner Sylvania, intending to visit
the home of his Irish ancestors (a King Cormac McCarthy
built Blarney Castle)." 

Two Years Ago:

Blarney in The Harvard Crimson

Melissa C. Wong, illustration for "Atlas to the Text,"
by Nicholas T. Rinehart:

Thirty Years Ago:

Non-Blarney from a rural outpost—

Illustration for the generalized diamond theorem,
by Steven H. Cullinane: 

See also Barry's Lexicon .

Sunday, June 9, 2013

Sicilian Reflections

Filed under: Uncategorized — m759 @ 9:00 AM

(Continued from Sept. 22, 2011)

See Taormina in this journal, and the following photo of "Anne Newton"—

Click photo for context.

Related material:

"Super Overarching" in this journal,
  a group of order 322,560, and

See also the MAA Spectrum  program —

— and an excerpt from the above book:

From 'Beautiful Mathematics,' by Martin Erickson, an excerpt on the Cullinane diamond theorem (with source not mentioned)

Backstory

Thursday, June 6, 2013

Review Comment Submitted

Filed under: Uncategorized — m759 @ 2:19 AM

The Mathematical Association of America has a
submit-a-review form that apparently allows readers
to comment on previously reviewed books.

This morning I submitted the following comment on
Alexander Bogomolny's March 16, 2012, review of 
Martin J. Erickson's Beautiful Mathematics :

In section 5.17, pages 106-108, "A Group of Operations,"
Erickson does not acknowledge any source. This section
is based on the Cullinane diamond theorem. See that
theorem (published in an AMS abstract in 1979) at
PlanetMath.org and EncyclopediaOfMath.org, and
elsewhere on the Web. Details of the proof given by
Erickson may be found in "Binary Coordinate Systems,"
a 1984 article on the Web at
http://finitegeometry.org/sc/gen/coord.html.

If and when the comment may be published, I do not know.

Update of about 6:45 PM ET June 7:

The above comment is now online at the MAA review site.

Update of about 7 PM ET July 29:

The MAA review site's web address was changed, and the 
above comment was omitted from the page at the new address.
This has now been corrected.

Tuesday, June 4, 2013

Cover Acts

Filed under: Uncategorized — m759 @ 11:00 AM

The Daily Princetonian  today:

IMAGE- 'How Jay White, a Neil Diamond cover act, duped Princeton'

A different cover act, discussed here  Saturday:

IMAGE- The diamond theorem affine group of order 322,560, published without acknowledgment of its source by the Mathematical Association of America in 2011

See also, in this journal, the Galois tesseract and the Crosswicks Curse.

"There is  such a thing as a tesseract." — Crosswicks saying

Saturday, June 1, 2013

Permanence

Filed under: Uncategorized — Tags: , — m759 @ 4:00 PM

"What we do may be small, but it has
  a certain character of permanence."

— G. H. Hardy, A Mathematician's Apology

The diamond theorem  group, published without acknowledgment
of its source by the Mathematical Association of America in 2011—

IMAGE- The diamond-theorem affine group of order 322,560, published without acknowledgment of its source by the Mathematical Association of America in 2011

Tuesday, May 28, 2013

Codes

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

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

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

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

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

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

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

IMAGE- The 35 square patterns within the Curtis MOG

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

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

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

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

Update of May 29, 2013:

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

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

Tuesday, April 2, 2013

Baker on Configurations

Filed under: Uncategorized — Tags: , — m759 @ 11:11 AM

The geometry posts of Sunday and Monday have been
placed in finitegeometry.org as

Classical Geometry in Light of Galois Geometry.

Some background:

See Baker, Principles of Geometry , Vol. II, Note I
(pp. 212-218)—

On Certain Elementary Configurations, and
on the Complete Figure for Pappus's Theorem

and Vol. II, Note II (pp. 219-236)—

On the Hexagrammum Mysticum  of Pascal.

Monday's elucidation of Baker's Desargues-theorem figure
treats the figure as a 15420configuration (15 points, 
4 lines on each, and 20 lines, 3 points on each).

Such a treatment is by no means new. See Baker's notes
referred to above, and 

"The Complete Pascal Figure Graphically Presented,"
a webpage by J. Chris Fisher and Norma Fuller.

What is new in the Monday Desargues post is the graphic
presentation of Baker's frontispiece figure using Galois geometry :
specifically, the diamond theorem square model of PG(3,2).

See also Cremona's kernel, or nocciolo :

Baker on Cremona's approach to Pascal—

"forming, in Cremona's phrase, the nocciolo  of the whole."

IMAGE- Definition of 'nocciolo' as 'kernel'

A related nocciolo :

IMAGE- 'Nocciolo': A 'kernel' for Pascal's Hexagrammum Mysticum: The 15 2-subsets of a 6-set as points in a Galois geometry.

Click on the nocciolo  for some
geometric background.

Tuesday, March 19, 2013

Mathematics and Narrative (continued)

Filed under: Uncategorized — 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.

Saturday, March 16, 2013

The Crosswicks Curse

Filed under: Uncategorized — Tags: — m759 @ 4:00 PM

Continues.

From the prologue to the new Joyce Carol Oates
novel Accursed

"This journey I undertake with such anticipation
is not one of geographical space but one of Time—
for it is the year 1905 that is my destination.

1905!—the very year of the Curse."

Today's previous post supplied a fanciful link
between the Crosswicks Curse of Oates and
the Crosswicks tesseract  of Madeleine L'Engle.

The Crosswicks Curse according to L'Engle
in her classic 1962 novel A Wrinkle in Time —

"There is  such a thing as a tesseract."

A tesseract is a 4-dimensional hypercube that
(as pointed out by Coxeter in 1950) may also 
be viewed as a 4×4 array (with opposite edges
identified).

Meanwhile, back in 1905

For more details, see how the Rosenhain and Göpel tetrads occur naturally
in the diamond theorem model of the 35 lines of the 15-point projective
Galois space PG(3,2).

See also Conwell in this journal and George Macfeely Conwell in the
honors list of the Princeton Class of 1905.

Tuesday, March 12, 2013

Smoke and Mirrors

Filed under: Uncategorized — Tags: , — m759 @ 7:00 AM

Sistine Chapel Smoke

Tromso Kunsthall Mirrors

Background for the smoke  image:
A remark by Michelangelo in a 2007 post,  High Concept.

Background for the mirrors  image:
Note the publication date— Mar. 10, 2013.

See that date in this journal and related material.

Tuesday, February 19, 2013

Configurations

Filed under: Uncategorized — Tags: — m759 @ 12:24 PM

Yesterday's post Permanence dealt with the cube
as a symmetric model of the finite projective plane
PG(2,3), which has 13 points and 13 lines. The points
and lines of the finite geometry occur in the cube as
the 13 axes of symmetry and the 13 planes through
the center perpendicular to those axes. If the three
axes lying in  a plane that cuts the cube in a hexagon
are supplemented by the axis perpendicular  to that
plane, each plane is associated with four axes and,
dually, each axis is associated with four planes.

My web page on this topic, Cubist Geometries, was
written on February 27, 2010, and first saved to the
Internet Archive on Oct. 4, 2010

For a more recent treatment of this topic that makes
exactly the same points as the 2010 page, see p. 218
of Configurations from a Graphical Viewpoint , by
Tomaž Pisanski and Brigitte Servatius, published by
Springer on Sept. 23, 2012 (date from both Google
Books
and Amazon.com):

For a similar 1998 treatment of the topic, see Burkard Polster's 
A Geometrical Picture Book  (Springer, 1998), pp. 103-104.

The Pisanski-Servatius book reinforces my argument of Jan. 13, 2013,
that the 13 planes through the cube's center that are perpendicular
to the 13 axes of symmetry of the cube should be called the cube's 
symmetry planes , contradicting the usual use of of that term.

That argument concerns the interplay  between Euclidean and
Galois geometry. Pisanski and Servatius (and, in 1998, Polster)
emphasize the Euclidean square and cube as guides* to
describing the structure of a Galois space. My Jan. 13 argument
uses Galois  structures as a guide to re-describing those of Euclid .
(For a similar strategy at a much more sophisticated level,
see a recent Harvard Math Table.)

Related material:  Remarks on configurations in this journal
during the month that saw publication of the Pisanski-Servatius book.

* Earlier guides: the diamond theorem (1978), similar theorems for
  2x2x2 (1984) and 4x4x4 cubes (1983), and Visualizing GL(2,p)
  (1985). See also Spaces as Hypercubes (2012).

Wednesday, January 2, 2013

PlanetMath link

Filed under: Uncategorized — m759 @ 2:00 PM

Update of May 27, 2013:
The post below is now outdated. See
http://planetmath.org/cullinanediamondtheorem .

__________________________________________________________________

The brief note on the diamond theorem at PlanetMath
disappeared some time ago. Here is a link to its
current URL: http://planetmath.org/?op=getobj;from=lec;id=49.

Update of 3 PM ET Jan. 2, 2013—

Another item recovered from Internet storage:

IMAGE- Miscellanea, 129: 'Triangles are square'- Amer. Math. Monthly, Vol. 91, No. 6, June-July 1984, p. 382

Click on the Monthly  page for some background.

Monday, December 10, 2012

Review of Leonardo Article

Filed under: Uncategorized — m759 @ 12:00 PM

Review of an often-cited Leonardo  article that is
now available for purchase online

The Tiling Patterns of Sebastien Truchet 
and the Topology of Structural Hierarchy

Authors: Cyril Stanley Smith and Pauline Boucher

Source: Leonardo , Vol. 20, No. 4,
20th Anniversary Special Issue:
Art of the Future: The Future of Art (1987),
pp. 373-385

Published by: The MIT Press

Stable URL: http://www.jstor.org/stable/1578535 .

Smith and Boucher give a well-illustrated account of
the early history of Truchet tiles, but their further remarks
on the mathematics underlying patterns made with
these tiles (see the diamond theorem* of 1976) are
worthless.

For instance

Excerpt from pages 383-384—

"A detailed analysis of Truchet's
patterns touches upon the most fundamental
questions of the relation between
mathematical formalism and the structure
of the material world. Separations
between regions differing in density
require that nothing  be as important as
something  and that large and small cells of
both must coexist. The aggregation of
unitary choice of directional distinction
at interfaces lies at the root of all being
and becoming."

* This result is about Truchet-tile patterns, but the
    underlying mathematics was first discovered by
    investigating superimposed patterns of half-circles .
    See Half-Circle Patterns at finitegeometry.org.

Saturday, December 8, 2012

Defining the Contest…

Filed under: Uncategorized — Tags: , , , , — m759 @ 5:48 AM

Chomsky vs. Santa

From a New Yorker  weblog yesterday—

"Happy Birthday, Noam Chomsky." by Gary Marcus—

"… two titans facing off, with Chomsky, as ever,
defining the contest"

"Chomsky sees himself, correctly, as continuing
a conversation that goes back to Plato, especially
the Meno dialogue, in which a slave boy is
revealed by Socrates to know truths about
geometry that he hadn’t realized he knew."

See Meno Diamond in this journal. For instance, from 
the Feast of Saint Nicholas (Dec. 6th) this year—

The Meno Embedding

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

For related truths about geometry, see the diamond theorem.

For a related contest of language theory vs. geometry,
see pattern theory (Sept. 11, 16, and 17, 2012).

See esp. the Sept. 11 post,  on a Royal Society paper from July 2012
claiming that

"With the results presented here, we have taken the first steps
in decoding the uniquely human  fascination with visual patterns,
what Gombrich* termed our ‘sense of order.’ "

The sorts of patterns discussed in the 2012 paper —

IMAGE- Diamond Theory patterns found in a 2012 Royal Society paper

"First steps"?  The mathematics underlying such patterns
was presented 35 years earlier, in Diamond Theory.

* See Gombrich-Douat in this journal.

Thursday, November 29, 2012

Conceptual Art

Filed under: Uncategorized — m759 @ 12:09 PM

Quotes from the Bremen site
http://dada.compart-bremen.de/ 
 

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

" 'compArt | center of excellence digital art' is a project
at the University of Bremen, Germany. It is dedicated
to research and development in computing, design,
and teaching. It is supported by Rudolf Augstein Stiftung,
the University of Bremen, and Karin und Uwe Hollweg Stiftung."

See also Stiftung in this journal.

Sunday, November 18, 2012

Sermon

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

Happy birthday to

IMAGE- Margaret Atwood, Kim Wilde, Peta Wilson

Today's sermon, by Marie-Louise von Franz

Number and Time, by Marie-Louise von Franz

For more on the modern physicist analyzed by von Franz,
see The Innermost Kernel , by Suzanne Gieser.

Another modern physicist, Niels Bohr, died
on this date in 1962

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

The circle above is marked with a version
of the classic Chinese symbol
adopted as a personal emblem
by Danish physicist Niels Bohr,
leader of the Copenhagen School.

For the square, see the diamond theorem.

"Two things of opposite natures seem to depend
On one another, as a man depends
On a woman, day on night, the imagined
On the real. This is the origin of change.
Winter and spring, cold copulars, embrace
And forth the particulars of rapture come."

— Wallace Stevens,
  "Notes Toward a Supreme Fiction,"
  Canto IV of "It Must Change"

Wednesday, October 10, 2012

Ambiguation

Filed under: Uncategorized — m759 @ 1:00 AM

Wikipedia disambiguation page—

IMAGE- Wikipedia disambiguation page for 'Da Milano'

"When you come to a fork in the road…"

IMAGE- Alyssa Milano as a child, with fork

IMAGE- Ambiguation therapy in Milan

For another "shifting reality that shimmered
in a multiplicity of facets," see The Diamond Theorem.

Midnight

Filed under: Uncategorized — m759 @ 12:00 AM

Disambiguation

A new Wikipedia disambiguation page for "Diamond theorem"—

History of the above new Wikipedia page—

See also a Google search for "diamond theorem."

Friday, August 24, 2012

Formal Pattern

Filed under: Uncategorized — m759 @ 4:28 PM

(Continued from In Memoriam (Aug. 22), Chapman's Homer (Aug. 23),
and this morning's Colorful Tale)

An informative, but undated, critique of the late Marvin W. Meyer
by April D. DeConick at the website of the Society of Biblical Literature
appeared in more popular form in an earlier New York Times
op-ed piece, "Gospel Truth," dated Dec. 1, 2007.

A check, in accord with Jungian synchronicity, of this  journal
on that date yields a quotation from Plato's Phaedrus  —

"The soul or animate being has the care of the inanimate."

Related verses from T. S. Eliot's Four Quartets

"The detail of the pattern is movement."

"So we moved, and they, in a formal pattern."

Some background from pure mathematics (what the late
William P. Thurston called "the theory of formal patterns")—

The Animated Diamond Theorem.

Sunday, July 29, 2012

The Galois Tesseract

Filed under: Uncategorized — Tags: , — m759 @ 11:00 PM

(Continued)

The three parts of the figure in today's earlier post "Defining Form"—

IMAGE- Hyperplanes (square and triangular) in PG(3,2), and coordinates for AG(4,2)

— share the same vector-space structure:

   0     c     d   c + d
   a   a + c   a + d a + c + d
   b   b + c   b + d b + c + d
a + b a + b + c a + b + d   a + b + 
  c + d

   (This vector-space a b c d  diagram is from  Chapter 11 of 
    Sphere Packings, Lattices and Groups , by John Horton
    Conway and N. J. A. Sloane, first published by Springer
    in 1988.)

The fact that any  4×4 array embodies such a structure was implicit in
the diamond theorem (February 1979). Any 4×4 array, regarded as
a model of the finite geometry AG(4, 2), may be called a Galois tesseract.
(So called because of the Galois geometry involved, and because the
16 cells of a 4×4 array with opposite edges identified have the same
adjacency pattern as the 16 vertices of a tesseract (see, for instance,
Coxeter's 1950 "Self-Dual Configurations and Regular Graphs," figures
5 and 6).)

A 1982 discussion of a more abstract form of AG(4, 2):

Source:

The above 1982 remarks by Brouwer may or may not have influenced
the drawing of the above 1988 Conway-Sloane diagram.

Sunday, May 13, 2012

Children of Light*

Filed under: Uncategorized — m759 @ 8:28 AM

IMAGE- Nassau Presbyterian scripture for May 13, 2012- 1 John 5:1-5.

An earlier verse in 1 John—

1 John 1:5 "This then is the message
which we have heard of him,
and declare unto you, that God is light,
and in him is no darkness at all."

Catechism from a different cult—

"Who are you, anyway?" 

— Question at 00:41 of 15:01,
Rainbow Bridge (Part 5 of 9) at YouTube

See also the video accompanying artist Josefine Lyche's version
of the 2×2 case of the diamond theorem.

* Title of a Robert Stone novel

Thursday, February 9, 2012

Psycho

Filed under: Uncategorized — m759 @ 7:59 PM

Psychophysics

See …

  1. The Doors of Perception,
  2. The Diamond Theorem,
  3. Walsh Function Symmetry, and
  4. Yodogawa, 1982.

Related literary material—

Enda's Game  and Tesseract .

ART WARS continued

Filed under: Uncategorized — m759 @ 1:06 PM

On the Complexity of Combat—

(Click to enlarge.)

The above article (see original pdf), clearly of more 
theoretical than practical interest, uses the concept
of "symmetropy" developed by some Japanese
researchers.

For some background from finite geometry, see
Symmetry of Walsh Functions. For related posts
in this journal, see Smallest Perfect Universe.

Update of 7:00 PM EST Feb. 9, 2012—

Background on Walsh-function symmetry in 1982—

(Click image to enlarge. See also original pdf.)

Note the somewhat confusing resemblance to
a four-color decomposition theorem
used in the proof of the diamond theorem

Saturday, December 31, 2011

The Uploading

Filed under: Uncategorized — Tags: — m759 @ 4:01 PM

(Continued)

"Design is how it works." — Steve Jobs

From a commercial test-prep firm in New York City—

http://www.log24.com/log/pix11C/111231-TeachingBlockDesign.jpg

From the date of the above uploading—

http://www.log24.com/log/pix11B/110708-ClarkeSm.jpg

After 759

m759 @ 8:48 AM
 

Childhood's End

From a New Year's Day, 2012, weblog post in New Zealand

http://www.log24.com/log/pix11C/111231-Pyramid-759.jpg

From Arthur C. Clarke, an early version of his 2001  monolith

"So they left a sentinel, one of millions they have scattered
throughout the Universe, watching over all worlds with the
promise of life. It was a beacon that down the ages has been
patiently signaling the fact that no one had discovered it.
Perhaps you understand now why that crystal pyramid was set…."

The numerical  (not crystal) pyramid above is related to a sort of
mathematical  block design known as a Steiner system.

For its relationship to the graphic  block design shown above,
see the webpages Block Designs and The Diamond Theorem
as well as The Galois Tesseract and R. T. Curtis's classic paper
"A New Combinatorial Approach to M24," which contains the following
version of the above numerical pyramid—

http://www.log24.com/log/pix11C/111231-LeechTable.jpg

For graphic  block designs, I prefer the blocks (and the parents)
of Grand Rapids to those of New York City.

For the barbed tail  of Clarke's "Angel" story, see the New Zealand post
of New Year's Day mentioned above.

Monday, December 5, 2011

The Shining (Norwegian Version)

Filed under: Uncategorized — m759 @ 4:01 AM

A check tonight of Norwegian artist Josefine Lyche's recent activities
shows she has added a video to her web page that has for some time
contained a wall piece based on the 2×2 case of the diamond theorem

http://www.log24.com/log/pix11C/111205-Lyche-DiamondTheoremPage.jpg

The video (top left in screenshot above) is a tasteless New-Age discourse
that sounds frighteningly like the teachings of the late Heaven's Gate cult.

Investigating the source of the video on vimeo.com, I found the account of one "Jo Lyxe,"
who joined vimeo in September 2011. This is apparently a variant of Josefine Lyche's name.

The account has three videos—

  1. "High on RAM (OverLoad)"– Fluid running through a computer's innards
  2. "Death 2 Everyone"– A mystic vision of the afterlife
  3. "Realization of the Ultimate Reality (Beyond Form)"– The Blue Star video above

Lyche has elsewhere discussed her New-Age interests, so the contents of the videos
were not too surprising… except for one thing. Vimeo.com states that all three videos
were uploaded "2 months ago"— apparently when "Lyxe" first set up an account.*

I do not know, or particularly care, where she got the Blue Star video, but the other
videos interested me considerably when I found them tonight… since they are
drawn from films I discussed in this journal much more recently than "2 months ago."

"High on RAM (OverLoad)" is taken from the 1984 film "Electric Dreams" that I came across
and discussed here yesterday afternoon, well before  re-encountering it again tonight.

http://www.log24.com/log/pix11C/111205-Lyxe-HighOnRam.jpg

http://www.log24.com/log/pix11C/111205-ElectricDreamsTrailer.jpg

And "Death 2 Everyone" (whose title** is perhaps a philosophical statement about inevitable mortality
rather than a mad terrorist curse) is taken from the 1983 Natalie Wood film "Brainstorm."

http://www.log24.com/log/pix11C/111205-Lyxe-Death2.jpg

http://www.log24.com/log/pix11C/111205-Brainstorm-FreakyPart.jpg

"Brainstorm" was also discussed here recently… on November 18th, in a post suggested by the
reopening of the investigation into Wood's death.

I had no inkling that these "Jo Lyxe" videos existed until tonight.

The overlapping content of Lyche's mental ramblings and my own seems rather surprising.
Perhaps it is a Norwegian mind-meld, perhaps just a coincidence of interests.

* Update: Google searches by the titles  on Dec. 5 show that all three "Lyxe" videos
                 were uploaded on September 20 and 21, 2011.

** Update: A search shows a track with this title on a Glasgow band's 1994 album.

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