The title is a useful search phrase:
Sunday, November 18, 2018
Diamond Theorem Symmetry
Monday, December 11, 2017
The Diamond Theorem at SASTRA
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 . . .
Wednesday, August 23, 2017
The Diamond Theorem in Vancouver
Tuesday, September 20, 2016
The Diamond Theorem …
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 LéviStrauss "Key to all Mythologies" in this journal,
as well as the previous post.
Monday, August 5, 2013
Diamond Theorem in ArXiv
The diamond theorem is now in the arXiv—
Tuesday, July 2, 2013
Diamond Theorem Updates
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
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
"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 colorinterchange 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. 26— For a related remark quoted here on the date of the student symposium, see Geometry for Generations.
Sunday, October 31, 2010
Diamond Theorem in Norway
Friday, October 28, 2016
DiamondTheorem Application
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 graylevel 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). 
Excerpts —
Related material — Posts tagged Diamond Theorem Correlation.
Friday, August 1, 2014
The DiamondTheorem Correlation
Click image for a larger, clearer version.
Thursday, May 7, 2020
Kant as Diamond Cutter
“He wished Kant were alive. Kant would have appreciated it.
That master diamond cutter.”
— Robert M. Pirsig, Zen and the Art of Motorcycle Maintenance , Part III.
Kant’s “category theory” —
“In the Transcendental Analytic, Kant deduces the table of twelve categories, or pure concepts of the understanding….
The categories must be ‘schematized’ because their nonempirical origin in pure understanding prevents their having the sort of sensible content that would connect them immediately to the objects of experience; transcendental schemata are mediating representations that are meant to establish the connection between pure concepts and appearances in a rulegoverned way. Mathematical concepts are discussed in this context since they are unique in being pure but also sensible concepts: they are pure because they are strictly a priori in origin, and yet they are sensible since they are constructed in concreto . ” — Shabel, Lisa, “Kant’s Philosophy of Mathematics”, The Stanford Encyclopedia of Philosophy (Spring 2016 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/spr2016/entries/kantmathematics/>. 
See also The Diamond Theorem and Octad.us.
Wednesday, April 17, 2019
The Royal Society Diamond
The phrase "pattern recognition" in a news story about the
April 13 death of Princeton neuroscientist Charles Gross,
and yesterday's post about a fanciful "purloined diamond,"
suggest a review of a less fanciful diamond.
See also earlier posts tagged Fitch
and my own, much earlier and very
different, approach to such patterns —
Monday, March 12, 2018
“Quantum Tesseract Theorem?”
Remarks related to a recent film and a notsorecent film.
For some historical background, see Dirac and Geometry in this journal.
Also (as Thas mentions) after Saniga and Planat —
The SanigaPlanat 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
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 ConwayNortonRyba Theorem
In a book to be published Sept. 5 by Princeton University Press,
John Conway, Simon Norton, and Alex Ryba present the following
result on orderfour 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
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 textonly version of the Diamond 16 Puzzle
that is more easily programmed than the current version.
The tricky part would be coding the letterspacing and
lineheight 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 diamondtheorem array when the
display window is clicked.
Monday, October 13, 2014
Raiders of the Lost Theorem
(Continued from Nov. 16, 2013.)
The 48 actions of GL(2,3) on a 3×3 array include the 8element
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 wouldbe 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 geometricfigure 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
Definition: A diamond space — informal phrase denoting
a subspace of AG(6, 2), the sixdimensional affine space
over the twoelement Galois field.
The reason for the name:
Click to enlarge.
Sunday, February 2, 2014
Diamond Theory Roulette
A 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 colorinterchange symmetry.
Tuesday, December 3, 2013
Diamond Space
A new website illustrates its URL.
See DiamondSpace.net.
Monday, February 11, 2013
The Penrose Diamond
Related material:
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
“To say more is to say less.”
― Harlan Ellison, as quoted at goodreads.com
Saying less—
 Home Schooling for the Hunger Games (last Thursday)
 Black diamonds in last Saturday's Talk Amongst Yourselves
 Last night's New York Times obituaries
Sunday, March 18, 2012
SquareTriangle Diamond
The diamond shape of yesterday's noon post
is not wholly without mathematical interest …
"Every triangle is an n replica" is true
if and only if n is a square.
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 16part square array. The same group may be
viewed as permuting the centers of the 16 subtriangles
in a 16part triangular array.
(Updated March 29, 2012, to correct wording and add Weyl link.)
Saturday, March 17, 2012
The Purloined Diamond
The diamond from the Chirho 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) —
“He pointed at the football
on his desk. ‘There it is.’”
– Glory Road
Wednesday, December 21, 2011
The Purloined Diamond
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 doublecross 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
"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 whowouldwanttokissthat 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
Above: Amy Adams in "Sunshine Cleaning"
Thursday, September 8, 2011
Starring the Diamond
"In any geometry satisfying Pappus's Theorem,
the four pairs of opposite points of 8_{3}
are joined by four concurrent lines."
— H. S. M. Coxeter (see below)
Continued from Tuesday, Sept. 6—
The Diamond Star
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 "SelfDual Configurations and Regular Graphs," also relates Pappus to the figure.
Some excerpts from Coxeter—
The relabeling uses the 8 superscripts
from the first picture above (plus 0).
The order of the superscripts is from
an 8cycle in the Galois field GF(9).
The relabeled configuration is used in a discussion of Pappus—
(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—
The Desargues configuration is discussed by GianCarlo Rota on pp. 145146 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)
Richard J. Trudeau, a mathematics professor and Unitarian minister, published in 1987 a book, The NonEuclidean 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—
The above ad is by Diamond from last night’s

For further details, see Saturday's correspondences 
Sunday, July 10, 2011
Wittgenstein’s Diamond
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 LogicoPhilosophicus No. 5.5563).
— Translation by G.E.M. Anscombe
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.)
Related language in Łukasiewicz (1937)—
* 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 (square) for necessarily
and (diamond) for possibly. Then, for example, P 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
Also known, roughly speaking, as confluence or the ChurchRosser 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 ChurchRosser 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 subexpressions to evaluate first will only matter if some of them but not others might lead down a nonterminating 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 subexpressions 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 ChurchRosser 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)—
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 —
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, ŠiršovBergman’s diamond lemma (Širšov is also sometimes spelled as Shirshov), and ChurchRosser theorem (and the corresponding ChurchRosser 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 DiamondBlazons
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—
— From "Stones and Their Stories," an article written
and illustrated by E.M. Barlow, copyright 1913.
Related geometry—
(See Štefan Porubský: Pythagorean Theorem .)
A proof with diamondblazons—
(See Ivars Peterson's "Square of the Hypotenuse," Nov. 27, 2000.)
Thursday, August 6, 2020
After Personalities . . . Principles
In memory of New York personality Pete Hamill ,
who reportedly died yesterday —
Seven years ago yesterday —
In memory of another New York personality, a parkinggarage mogul
who reportedly died on August 9, 2005 —
Icon Parking posts and . . .
Thursday, July 30, 2020
A Picture Show for Quanta Magazine
An article yesterday at Quanta Magazine suggests a review . . .
From Diamond Theorem images at Pinterest —
Some background —
Thursday, July 9, 2020
The Enigma Glyphs
For those who prefer fiction —
“Twentyfour glyphs, each one representing not a letter, not a word,
but a concept, arranged into four groups, written in Boris’s own hand,
an artifact that seemed to have resurrected him from the dead. It was
as if he were sitting across from Bourne now, in the dim antiquity of
the museum library.
This was what Bourne was staring at now, written on the unfolded
bit of onionskin.”
— “Robert Ludlum’s” The Bourne Enigma , published on June 21, 2016
Passing, on June 21, 2016, into a higher dimension —
Friday, June 12, 2020
Bullshit Studies: “Hyperseeing”
In memoriam —
Friedman coedited the ISAMA journal Hyperseeing . See also . . .
See too the other articles in Volume 40 of Kybernetes .
Related material —
Compare and contrast the discussion of the geometry
of the 4×4 square in the diamond theorem (1976) with
Nat Friedman’s treatment of the same topic in 2001 —
Saturday, March 7, 2020
The “Octad Group” as Symmetries of the 4×4 Square
From “Mathieu Moonshine and Symmetry Surfing” —
(Submitted on 29 Sep 2016, last revised 22 Jan 2018)
by Matthias R. Gaberdiel (1), Christoph A. Keller (2),
and Hynek Paul (1)
(1) Institute for Theoretical Physics, ETH Zurich
(2) Department of Mathematics, ETH Zurich
https://arxiv.org/abs/1609.09302v2 —
“This presentation of the symmetry groups G_{i} is
particularly welladapted for the symmetry surfing
philosophy. In particular it is straightforward to
combine them into an overarching symmetry group G
by combining all the generators. The resulting group is
the socalled octad group
G = (Z_{2})^{4}^{ }⋊ A_{8}_{ }.
It can be described as a maximal subgroup of M_{24}
obtained by the setwise stabilizer of a particular
‘reference octad’ in the Golay code, which we take
to be O_{9 }= {3,5,6,9,15,19,23,24} ∈ 𝒢_{24}. The octad
subgroup is of order 322560, and its index in M_{24}
is 759, which is precisely the number of
different reference octads one can choose.”
This “octad group” is in fact the symmetry group of the affine 4space over GF(2),
so described in 1979 in connection not with the Golay code but with the geometry
of the 4×4 square.* Its nature as an affine group acting on the Golay code was
known long before 1979, but its description as an affine group acting on
the 4×4 square may first have been published in connection with the
Cullinane diamond theorem and Abstract 79TA37, “Symmetry invariance in a
diamond ring,” by Steven H. Cullinane in Notices of the American Mathematical
Society , February 1979, pages A193, 194.
* The Galois tesseract .
Update of March 15, 2020 —
Conway and Sloane on the “octad group” in 1993 —
Wednesday, January 22, 2020
Gap Dance
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
Thursday, September 12, 2019
Pattern and Structure
From the previous post —
" . . . Only by the form, the structure,
Can words or music reach
The stillness . . . ."
— Adapted from T. S. Eliot's Four Quartets
by replacing "pattern" with "structure."
Tuesday, September 10, 2019
Congruent Subarrays
A search for "congruent subarrays" yields few results. Hence this post.
Some relevant mathematics: the Cullinane diamond theorem, which
deals with permutations of congruent subarrays.
A related topic: Square Triangles (December 15, 2015).
Tuesday, March 5, 2019
A Block Design 3(16,4,1) as a Steiner Quadruple System:
A Midrash for Wikipedia
Midrash —
Related material —
________________________________________________________________________________
Friday, March 1, 2019
Wikipedia Scholarship (Continued)
This post continues a post from yesterday on the square model of
PG(3,2) that apparently first appeared (presented as such*) in . . .
Cullinane, "Symmetry invariance in a diamond ring,"
Notices of the AMS , pp. A193194, Feb. 1979.
Yesterday's Wikipedia presentation of the square model was today
revised by yet another anonymous author —
Revision history accounting for the above change from yesterday —
The jargon "rm OR" means "remove original research."
The added verbiage about block designs is a smokescreen having
nothing to do with the subject, which is square representation
of the 35 points and lines.
* The 35 squares, each consisting of four 4element subsets, appeared earlier
in the Miracle Octad Generator (MOG) of R. T. Curtis (published in 1976).
They were not at that time presented as constituting a finite geometry,
either affine (AG(4,2)) or projective (PG(3,2)).
Friday, February 22, 2019
Back Issues of AMS Notices
From the online home page of the new March issue —
For instance . . .
Related material now at Wikipedia —
Saturday, September 15, 2018
Eidetic Reduction in Geometry
"Husserl is not the greatest philosopher of all times. — Kurt Gödel as quoted by GianCarlo Rota Some results from a Google search — Eidetic reduction  philosophy  Britannica.com Eidetic reduction, in phenomenology, a method by which the philosopher moves from the consciousness of individual and concrete objects to the transempirical realm of pure essences and thus achieves an intuition of the eidos (Greek: “shape”) of a thing—i.e., of what it is in its invariable and essential structure, apart … Phenomenology Online » Eidetic Reduction
The eidetic reduction: eidos. Method: Bracket all incidental meaning and ask: what are some of the possible invariate aspects of this experience? The research Eidetic reduction – New World Encyclopedia Sep 19, 2017 – Eidetic reduction is a technique in Husserlian phenomenology, used to identify the essential components of the given phenomenon or experience. 
For example —
The reduction of twocolorings and fourcolorings of a square or cubic
array of subsquares or subcubes to lines, sets of lines, cuts, or sets of
cuts between* the subsquares or subcubes.
See the diamond theorem and the eightfold cube.
* Cf. posts tagged Interality and Interstice.
Monday, August 27, 2018
Children of the Six Sides
From the former date above —
Saturday, September 17, 2016 
From the latter date above —
Tuesday, October 18, 2016
Parametrization

From March 2018 —
Saturday, June 9, 2018
SASTRA paper
Monday, May 7, 2018
Data
(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, 759faceted) 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
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 194575,
the three artists who will be featured, the three seminal
figures of the era, will be not Pollock, de Kooning, and
Johnsbut 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 —
— and in 20132015 papers by Anne Taormina and Katrin Wendland:
Wednesday, April 11, 2018
Behance Post
I just came across the 2010 web page
https://pantone.ccnsite.com/gallery/HELDERvisualidentity/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/TheDiamondTheorem.
Thursday, March 29, 2018
“Before Creation Itself . . .”
From the Diamond Theorem Facebook page —
A question three hours ago at that page —
“Is this Time Cube?”
Notes toward an answer —
And from SixSet Geometry in this journal . . .
Wednesday, March 28, 2018
On Unfairly Excluding Asymmetry
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
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 highenergy 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 . . .
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 2subsets of a 6set, but
did not discuss the 4×4 square as an affine space.
For the connection of the 15 Kummer modular 2subsets with the 16
element affine space over the twoelement Galois field GF(2), see my note
of May 26, 1986, "The 2subsets of a 6set 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 (19801981), Talk no. 577, pp. 258276.
Monday, December 18, 2017
Wheelwright and the Dance
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" —
"The Fire, Air, Earth, and Water" —
Thursday, November 30, 2017
The Matrix for Quantum Mystics
Scholia on the title — See Quantum + Mystic in this journal.
"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 LéviStrauss, 1976
I prefer the earlier, betterknown, 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 2627, 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
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
Continued from the previous post and from posts
now tagged Dueling Formulas —
The fourdiamond formula of Jung and
the fourdot "as" of Claude LéviStrauss:
Simplified versions of the diamonds and the dots —
I prefer Jung. For those who prefer LéviStrauss —
First edition, Cornell University Press, 1970.
A related tale — "A Meaning, Like."
Monday, June 26, 2017
Upgrading to Six
This post was suggested by the previous post — Four Dots —
and by the phrase "smallest perfect" in this journal.
Related material (click to enlarge) —
Detail —
From the work of Eddington cited in 1974 by von Franz —
See also Dirac and Geometry and Kummer in this journal.
Updates from the morning of June 27 —
Ron Shaw on Eddington's triads "associated in conjugate pairs" —
For more about hyperbolic and isotropic lines in PG(3,2),
see posts tagged Diamond Theorem Correlation.
For Shaw, in memoriam — See Contrapuntal Interweaving and The Fugue.
Monday, June 19, 2017
Dead End
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
Tuesday, May 2, 2017
Image Albums
Pinterest boards uploaded to the new m759.net/piwigo —
Update of May 2 —
Update of May 3 —
Update of May 8 —
Art Space board created at Pinterest
Friday, April 7, 2017
Personal Identity
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.
Backstory —
The previous post
and The Crimson Abyss.
Saturday, March 4, 2017
Hidden Figure
Thursday, March 2, 2017
Raiders of the Lost Crucible Continues
See, too, this evening's A Common Space
and earlier posts on Raiders of the Lost Crucible.
Also not without relevance —

The death of the photographer who took
the above cover photo  A Hate Speech for Harvard
 A recent University of Bradford thesis —
Sunday, January 8, 2017
A Theory of Everything
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 —
Tuesday, October 18, 2016
Parametrization
The term "parametrization," as discussed in Wikipedia,
seems useful for describing labelings that are not, at least
at first glance, of a vectorspace nature.
Examples: The labelings of a 4×4 array by a blank space
plus the 15 twosubsets of a sixset (Hudson, 1905) or by a
blank plus the 5 elements and the 10 twosubsets of a fiveset
(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 vectorspace coordinates. He describes it
as simply a mapping to a set of reproducible symbols .
(But Weyl does imply that these symbols should, like vectorspace
coordinates, admit a group of transformations among themselves
that can be used to describe transformations of the pointspace
being coordinatized.)
Tuesday, September 27, 2016
Chomsky and LéviStrauss in China
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 notparticularlydeep 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 LéviStrauss 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 goingbeyond 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 —
Tuesday, September 20, 2016
Savage Logic
From "The Cerebral Savage," by Clifford Geertz —
(Encounter, Vol. 28 No. 4 (April 1967), pp. 2532.)
Sunday, August 21, 2016
Review
Fugue No. 21
BFlat Major
WellTempered Clavier Book II
Johann Sebastian Bach
by Timothy A. Smith
Theme and Variations
by Steven H. Cullinane
The beginning of each —
Some context —
Tuesday, June 21, 2016
The Central Structure
"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
Sunday, May 22, 2016
Sunday School
A less metaphysical approach to a "preform" —
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
Note the echo of Jung's formula in the diamond theorem.
An attempt by LéviStrauss 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
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 .
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 allmeaningful, 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 LéviStrauss 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
"… 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): 165181,
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
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
twentyone. 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
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 —
The Post 's remarks are of particular interest:
Philip Kennicott in The Washington Post , Dec. 28, 2015, “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 protodigital 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 selfreplicating 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 —
Lyche's mirrorsonthewall 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
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 213214
Cullinane, Steven H., Notes on Groups and Geometry, 19781986
Related material:
The 6set 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 19941995)—
Thursday, October 15, 2015
Contrapuntal Interweaving
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:
.
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
Norwegian Sculpture Biennial 2015 catalog, p. 70 —
" 'Ambassadørene' er fysiske former som presenterer
ikkfysiske 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”
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. xiiixiv):
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
(A review)
Galois space:
Counting symmetries of Galois space:
The reason for these graphic symmetries in affine Galois space —
symmetries of the underlying projective Galois space:
Monday, June 15, 2015
Omega Matrix
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 fourdimensional
affine space over the twoelement Galois
field, the appropriate Ω is the 4×4 grid above.
Sunday, May 17, 2015
Thursday, May 7, 2015
Paradigm for Pedagogues
Illustrations from a post of Feb. 17, 2011:
Plato’s paradigm in the Meno —
Changed paradigm in the diamond theorem (2×2 case) —
Saturday, April 4, 2015
Harrowing of Hell (continued)
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 3031). 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
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 —

An invariance of symmetry The diamond theorem on a 4x4x4 cube, and a sketch of the proof. 
831001  Portrait of O A table of the octahedral group O using the 24 patterns from the 2×2 case of the diamond theorem. 
831016  Study of O A different way of looking at the octahedral group, using cubes that illustrate the 2x2x2 case of the diamond theorem. 
840915  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
From "How the Guggenheim Got Its Visual Identity,"
by Caitlin Dover, November 4, 2013 —
For the square and halfsquare in the above logo
as independent design elements, see
the Cullinane diamond theorem.
For the circle and halfcircle 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
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 highconcept 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/140824DiamondTheoremCorrelation1202w.jpg
http://www.log24.com/log/pix14B/140731DiamondTheoremCorrelation747w.jpg
http://www.log24.com/log/pix14B/140824Picturing_the_Smallest1986.gif
http://www.log24.com/log/pix14B/140806ProjPoints.gif
For some backstory, see ProjPoints.gif and "Symplectic Polarity" in this journal.
Wednesday, November 19, 2014
The Eye/Mind Conflict
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
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 194575,
the three artists who will be featured, the three seminal
figures of the era, will be not Pollock, de Kooning, and
Johnsbut 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, 5979 (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….”
— Steven H. Cullinane,
diamond theorem illustration
Sunday, November 2, 2014
A Singular Place
"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:
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 …
Friday, October 31, 2014
Structure
Introducing a group of 322,560 affine transformations of Dürer’s ‘Magic’ Square
The four vectorspace 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
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
Structured gray matter:
Graphic symmetries of Galois space:
The reason for these graphic symmetries in affine Galois space —
symmetries of the underlying projective Galois space:
Tuesday, September 9, 2014
Smoke and Mirrors
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 nineteenthcentury
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
The Folding
Cynthia Zarin in The New Yorker , issue dated April 12, 2004—
“Time, for L’Engle, is accordionpleated. 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
3dimensional 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:
Continuous Euclidean space to discrete Galois space*
Euclidean space:
Counting symmetries in Euclidean space:
Galois space:
Counting symmetries of Galois space:
The reason for these graphic symmetries in affine Galois space —
symmetries of the underlying projective Galois space:
* For related remarks, see posts of May 2628, 2012.
Sunday, June 8, 2014
Vide
“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.”
— GianCarlo 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 (rowbased) lines with 10 corresponding
vertically oriented (columnbased) 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:
Wednesday, May 14, 2014
Two Year College
See last night’s pentagram photo and a post from May 13, 2012.
That post links to a littleknown video of a 1972 film.
A speech from the film was used by Oslo artist Josefine Lyche as a
voiceover in her 2011 goldenratio video (with pentagrams) that she
exhibited along with a large, wallfilling 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
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 TurynCurtis construction
from the University of Cambridge —
— Slide by “Dr. Parker” — Apparently Richard A. Parker —
Lecture 4, “Discovering M_{24},” in slides for lectures 18 from lectures
at Cambridge in 20102011 on “Sporadic and Related Groups.”
See also the Parker lectures of 20122013 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+4partitions of an 8set with the 35 lines of the projective 3space
over the 2element field, PG(3, 2), is essentially the same correspondence
as that constituting Curtis’s 1976 MOG.
See The Diamond Theorem, Finite Relativity, Galois Space,
Generating the Octad Generator, and The Klein Correspondence.
Update of March 22March 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
S_{3} 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 “byhand” 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 —
Saturday, March 15, 2014
Tuesday, March 11, 2014
Depth
"… 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
Part II: An Inch Wide
See a search for "square inch space" in this journal.
See also recent posts with the tag depth.
Thursday, February 6, 2014
The Representation of Minus One
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 —
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 …
… 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
The sixteendot 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—
The 1977 matrix Q is echoed in the following from 2002—
A different representation of Cullinane’s 1977 square model of the
16point affine geometry over the twoelement Galois field GF(2)
is supplied by Conway and Sloane in Sphere Packings, Lattices and Groups
(first published in 1988) :
Here a, b, c, d are basis vectors in the vector 4space over GF(2).
(For a 1979 version of this vector space, see AMS Abstract 79TA37.)
See also a 2011 publication of the Mathematical Association of America —
Wednesday, January 8, 2014
Occupy Space
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—
Thursday, December 5, 2013
Wednesday, October 16, 2013
Theme and Variations
Josefine Lyche’s large wall version of the twentyfour 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
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…
The Philosopher's Gaze , by David Michael Levin, The postmetaphysical question—question for a postmetaphysical 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 comingtopass of the ontological difference that is constitutive of all the local figureground 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 responseability, our capacity as beings gifted with vision, to measure up to the responsibility for perceptual responsiveness laid down for us in the "primordial decision" (Entscheid ) of the ontological difference (ibid.). To recognize the operation of the ontological difference taking place in the figureground difference of the perceptual Gestalt is to recognize the ontological difference as the primordial Riß , the primordial Urteil 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 socalled 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
The Galois tesseract appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977—
The Galois tesseract is the basis for a representation of the smallest
projective 3space, 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—
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
I added links today in the following Wikipedia articles:
 Diamond theorem (disambiguation page)
 Miracle Octad Generator
 Binary Golay code
The links will probably soon be deleted,
but it seemed worth a try.
Tuesday, July 9, 2013
Vril Chick
Profile picture of "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
Keywords (to help place my artwork in the (See also the original catalog page.) 
Clearly most of this (the nonhighlighted 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)
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:
NonBlarney from a rural outpost—
Illustration for the generalized diamond theorem,
by Steven H. Cullinane:
Sunday, June 9, 2013
Sicilian Reflections
(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:
Thursday, June 6, 2013
Review Comment Submitted
The Mathematical Association of America has a
submitareview 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 106108, "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
The Daily Princetonian today:
A different cover act, discussed here Saturday:
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
"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—
Tuesday, May 28, 2013
Codes
The hypercube model of the 4space over the 2element Galois field GF(2):
The phrase Galois tesseract may be used to denote a different model
of the above 4space: 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 Galoistesseract model of the 4space over GF(2).
The thirtyfive 4×4 structures within the MOG:
Curtis himself first described these 35 square MOG patterns
combinatorially, (as his title indicated) rather than
algebraically or geometrically:
A later book coauthored by Sloane, first published in 1988,
did recognize the 4×4 MOG patterns as based on the 4×4
Galoistesseract 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 MacWilliamsSloane book was first published):
Tuesday, April 2, 2013
Baker on Configurations
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. 212218)—
On Certain Elementary Configurations, and
on the Complete Figure for Pappus's Theorem
and Vol. II, Note II (pp. 219236)—
On the Hexagrammum Mysticum of Pascal.
Monday's elucidation of Baker's Desarguestheorem figure
treats the figure as a 15_{4}20_{3 }configuration (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."
A related nocciolo :
Click on the nocciolo for some
geometric background.
Tuesday, March 19, 2013
Mathematics and Narrative (continued)
Angels & Demons meet Hudson Hawk
Dan Brown's fourelements diamond in Angels & Demons :
The Leonardo Crystal from Hudson Hawk :
Mathematics may be used to relate (very loosely)
Dan Brown's fanciful diamond figure to the fanciful
Leonardo Crystal from Hudson Hawk …

Compare Brown's fictional Illuminati Diamond to the
nonfictional figures in The Diamond Theorem and
Theme and Variations. 
Compare the fictional Leonardo Crystal to Hudson's
nonfictional desmic system of tetrahedra (above), and
see, in Rosenhain and Göpel Tetrads in PG(3,2), how
the diamond theorem is related to Hudson's work.
For the tetrads ' relationship to tetrahedra , see
Hudson's own book.
"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
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 4dimensional 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 15point 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.