The first 12 pages of my 1976 preprint "Diamond Theory" are
now scanned and uploaded. See a slideshow.
For downloading, all 12 pages are combined in a PDF.
The first 12 pages of my 1976 preprint "Diamond Theory" are
now scanned and uploaded. See a slideshow.
For downloading, all 12 pages are combined in a PDF.
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.
A pdf of a 1977 threepage article with this title
has been added at finitegeometry.org/sc.
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 
"A world of made
is not a world of born— pity poor flesh
and trees, poor stars and stones, but never this
fine specimen of hypermagical
ultraomnipotence."
— e. e. cummings, 1944
For one such specimen, see The Matrix of Abraham—
a 5×5 square that is hypermagical… indeed, diabolical.
Related material on the algebra and geometry underlying some smaller structures
that have also, unfortunately, become associated with the word "magic"—
" … listen: there's a hell
of a good universe next door; let's go"
— e. e. cummings
Happy birthday, e. e.
A post of May 26, 2005, displays, if not the sword,
a place for it —
“The beautiful in mathematics resides in contradiction.
Incommensurability, logoi alogoi, was the first splendor
in mathematics.” — Simone Weil, Oeuvres Choisies,
éd. Quarto , Gallimard, 1999, p. 100
Logos Alogos by S. H. Cullinane
“To a mathematician, mathematical entities have their own existence,
they habitate spaces created by their intention. They do things,
things happen to them, they relate to one another. We can imagine
on their behalf all sorts of stories, providing they don’t contradict
what we know of them. The drama of the diagonal, of the square…”
— Dennis Guedj, abstract of “The Drama of Mathematics,” a talk
to be given this July at the Mykonos conference on mathematics and
narrative. For the drama of the diagonal of the square, see
Poet Wallace Stevens was born 140 years ago today.
For another 140, see Diamond Theory in 1937.
For some notes related to a Stevens poem from 1937,
see "arrowy, still strings" in this journal.
(From his “Structure and Form: Reflections on a Work by Vladimir Propp.”
Translated from a 1960 work in French. It appeared in English as
Chapter VIII of Structural Anthropology, Volume 2 (U. of Chicago Press, 1976).
Chapter VIII was originally published in Cahiers de l’Institut de Science
Économique Appliquée , No. 9 (Series M, No. 7) (Paris: ISEA, March 1960).)
The structure of the matrix of LéviStrauss —
Illustration from Diamond Theory , by Steven H. Cullinane (1976).
The relevant field of mathematics is not Boolean algebra, but rather
Galois geometry.
According to Wallace Stevens:
From Savage Logic— Sunday, March 15, 2009 5:24 PM The Origin of Change
A note on the figure
"Two things of opposite natures seem to depend
— Wallace Stevens, 
This post was suggested by the following passage —
" … the Fano plane ,
a set of seven points
grouped into seven lines
that has been called
'the combinatorialist’s coat of arms.' "
— Blake Stacey in a post with tomorrow's date:
… and by Stacey at another weblog, in a post dated Jan. 29, 2019, …
"(Yes, Bohr was the kind of guy who would choose
the yinyang symbol as his coat of arms.)"
Yes, Stacey is the kind of guy who would casually dismiss
Bohr's coat of arms.
(See also Faust in Copenhagen in this journal)—
» more
Earlier posts have discussed the "story theory of truth"
versus the "diamond theory of truth," as defined by
Richard Trudeau in his 1987 book The NonEuclidean Revolution.
In a New York Times opinion piece for tomorrow's print edition,*
novelist Dara Horn touched on what might be called
"the space theory of truth."
When they return to synagogue, mourners will be greeted
with more ancient words: “May God comfort you
among the mourners of Zion and Jerusalem.”
In that verse, the word used for God is hamakom —
literally, “the place.” May the place comfort you.
[Link added.]
The Source —
See Dara Horn in this journal, as well as Makom.
* "A version of this article appears in print on ,
on Page A23 of the New York edition with the headline:
American Jews Know This Story."
"When times are mysterious
Serious numbers
Will always be heard."
— Paul Simon,
"When Numbers Get Serious"
"There is a pleasantly discursive treatment of
Pontius Pilate's unanswered question 'What is truth?'"
— H. S. M. Coxeter, introduction to Richard J. Trudeau's remarks
on the "story theory" of truth as opposed to the "diamond theory"
of truth in The NonEuclidean Revolution (1987)
The deaths of Roth and Grünbaum on September 14th,
The Feast of the Holy Cross, along with Douthat's column
today titled "Only the Truth Can Save Us Now," suggest a
review of …
<meta name="description"
content="Identidade generativa para o Diamonds Studio
Desenvolvido em conjunto com
http://quadradao.com.br/
http://diamondsstudio.com.br/
Baseado na Diamond Theory by Steven H Cullinane, 1977">
Click on the image for a
relevant Wallace Stevens poem.
A new Facebook page will describe
some background for the above image.
From a Log24 post of March 4, 2008 —
SINGER, ISAAC:
"Sets forth his own aims in writing for children and laments
— An Annotated Listing of Criticism
"She returned the smile, then looked across the room to
— A Swiftly Tilting Planet,
For "the dimension of time," see A Fold in Time, Time Fold,
A Swiftly Tilting Planet is a fantasy for children 
Ibid. —
The pen's point:
John Trever, Albuquerque Journal, 2/29/08
Note the figure on the cover of National Review above —
A related figure from Pentagram Design —
See, more generally, Isaac Singer in this journal.
From the American Mathematical Society homepage today —
From concinnitasproject.org —
"Concinnitas is the title of a portfolio of fine art prints. . . .
The portfolio draws its name from a word famously used
by the Renaissance scholar, artist, architect, and philosopher
Leon Battista Alberti (14041472) to connote the balance of
number, outline, and position (in essence, number, geometry,
and topology) that he believed characterize a beautiful work of art."
The favicon of the Concinnitas Project —
The structure of the Concinnitas favicon —
This structure is from page 15 of
"Diamond Theory," a 1976 preprint —
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.
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
The title is that of a presentation by Arnold Emch
at the 1928 International Congress of Mathematicians:
See also yesterday's "Emch as a Forerunner of S(5, 8, 24)."
Related material: Diamond Theory in 1937.
Further remarks: Christmas 2013 and the fact that
759 × 322,560 = the order of the large Mathieu group M_{24} .
Before the monograph "Diamond Theory" was distributed in 1976,
two (at least) notable figures were published that illustrate
symmetry properties of the 4×4 square:
Hudson in 1905 —
Golomb in 1967 —
It is also likely that some figures illustrating Walsh functions as
twocolor square arrays were published prior to 1976.
Update of Dec. 7, 2016 —
The earlier 1950's diagrams of Veitch and Karnaugh used the
1's and 0's of Boole, not those of Galois.
"Sultan" was a pseudonym of Peter Lindbergh, now a
wellknown fashion photographer. Click image for the source.
Related art — Diamond Theory Roullete, by Radames Ajna,
2013 (Processing code at ReCode Project based on
"Diamond Theory" by Steven H. Cullinane, 1977).
The "unity" of the title was suggested by this morning's update
at the end of yesterday's post Paz.
For the Plato of the title, see the Sept. 27, 2016, post
For glyphs representing the "unity of opposites" of the title,
see a webpage linked to here on Groundhog Day 2014 —
The above image is related to Jung's remarks on Coincidentia
Oppositorum . (See also coincidentia in this journal.)
A different Jung, in a new video with analogues of the rapidly
flashing images in Ajna's webpage "Diamond Theory Roullete" —
The above video promotes Google's new opensource "Noto" font —
According to McLuhan
Marshall McLuhan writing to Ezra Pound on Dec. 21, 1948—
"The American mind is not even close to being amenable
to the ideogram principle as yet. The reason is simply this.
America is 100% 18th Century. The 18th century had
chucked out the principle of metaphor and analogy—
the basic fact that as A is to B so is C to D. AB:CD.
It can see AB relations. But relations in four terms are still
verboten. This amounts to deep occultation of nearly all
human thought for the U.S.A.
I am trying to devise a way of stating this difficulty as it exists.
Until stated and publicly recognized for what it is, poetry and
the arts can’t exist in America."
For context, see Cameron McEwen,
"Marshall McLuhan, John Pick, and Gerard Manley Hopkins."
(Renascence , Fall 2011, Vol. 64 Issue 1, 5576)
A relation in four terms —
A : B :: C : D as Model : Crutch :: Metaphor : Ornament —
See also Dueling Formulas and Symmetry.
"Studies of spin½ theories in the framework of projective geometry
have been undertaken before." — Y. Jack Ng and H. van Dam,
February 20, 2009
For one such framework,* see posts from that same date
four years earlier — February 20, 2005.
* A 4×4 array. See the 1977, 1978, and 1986 versions by
Steven H. Cullinane, the 1987 version by R. T. Curtis, and
the 1988 ConwaySloane version illustrated below —
Cullinane, 1977
Cullinane, 1978
Cullinane, 1986
Curtis, 1987
Update of 10:42 PM ET on Sunday, June 19, 2016 —
The above images are precursors to …
Conway and Sloane, 1988
Update of 10 AM ET Sept. 16, 2016 — The excerpt from the
1977 "Diamond Theory" article was added above.
From the webpage Diamond Theory Bibliography —
Golomb, Solomon W.
Shift register sequences (Revised edition)
Aegean Park Press, Laguna Hills, CA, 1982
The fifteen "stencils" in Golomb's Fig. VIII8, page 219,
are the same as the fifteen affine hyperplanes that
account for patterns' symmetry in diamond theory.
This figure occurs in a discussion of Rademacher
Walsh functions.
See Fields of Force and recent posts.
From PR Newswire in July 2011 —
Campus Crusade for Christ Adopts New Name: Cru
60yearold Int'l Ministry Aims to Increase
Relevance and Global Effectiveness
Related material:
A December 7th New York Times column:
A current exhibition by Joseph Kosuth in Oslo:
From the two texts by Mondrian at the right hand of Kosuth —
"The positive and negative states of being bring about action."
"Through its pure relationships, purely abstract art
can approach the expression of the universal …."
These texts may be viewed as glosses on the following image —
Click image for related posts.
This post's title was suggested by the previous post
and by today's news of a notable sale of a onecopy
record album, "Once Upon a Time in Shaolin."
See as well posts from Tuesday, March 11, 2014,
the day Emma Watson unveiled a new trailer —
The following slides are from lectures on “Advanced Boolean Algebra” —
The small Boolean spaces above correspond exactly to some small
Galois spaces. These two names indicate approaches to the spaces
via Boolean algebra and via Galois geometry .
A reading from Atiyah that seems relevant to this sort of algebra
and this sort of geometry —
” ‘All you need to do is give me your soul: give up geometry
and you will have this marvellous machine.’ (Nowadays you
can think of it as a computer!) “
Related material — The article “Diamond Theory” in the journal
Computer Graphics and Art , Vol. 2 No. 1, February 1977. That
article, despite the word “computer” in the journal’s title, was
much less about Boolean algebra than about Galois geometry .
For later remarks on diamond theory, see finitegeometry.org/sc.
Happy birthday to the late Michael Crichton (Harvard ’64).
See also Diamond Theory Roulette —
Part of the ReCode Project (http://recodeproject.com). Based on "Diamond Theory" by Steven H. Cullinane, originally published in "Computer Graphics and Art" Vol. 2 No. 1, February 1977. Copyright (c) 2013 Radames Ajna — OSI/MIT license (http://recodeproject/license).
Related remarks on Plato for Harvard’s
Graduate School of Design —
See also posts from the above publication date, March 31,
2006, among posts now tagged “The Church in Philadelphia.”
Click to enlarge:
For the hypercube as a vector space over the twoelement field GF(2),
see a search in this journal for Hypercube + Vector + Space .
For connections with the related symplectic geometry, see Symplectic
in this journal and Notes on Groups and Geometry, 19781986.
For the above 1976 hypercube (or tesseract ), see "Diamond Theory,"
by Steven H. Cullinane, Computer Graphics and Art , Vol. 2, No. 1,
Feb. 1977, pp. 57.
"Little emblems of eternity"
— Phrase by Oliver Sacks in today's
New York Times Sunday Review
Some other emblems —
Note the colorinterchange
symmetry of each emblem
under 180degree rotation.
Click an emblem for
some background.
(A sequel to Expanding the Spielraum (Feb. 3, 2015))
"Knowledge, wisdom even, lies in depth, not extension."
— Tim Parks in The New York Review of Books ,
5 PM ET on June 26, 2015
See also Log24 posts on the following figure —
Or: Concepts of Space
1976 according to Cullinane:
1976 according to Plotnick:
“Irony and ridicule are entertaining and effective, and . . .
at the same time they are the agents of a great despair
and stasis in U.S. culture.” — David Foster Wallace,
as quoted by Adam Kirsch today at Salon
This journal Tuesday, Oct. 28, 2014, at 5 PM ET:
“What is a tai chi master, and what is it that he unfolds?”
From an earlier post, Hamlet’s father’s ghost
on “the fretful porpentine”:
Hamlet , Act 1, Scene 5 —
Ghost:
“I could a tale unfold whose lightest word
Would harrow up thy soul, freeze thy young blood,
Make thy two eyes, like stars, start from their spheres,
Thy knotted and combinèd locks to part
And each particular hair to stand on end,
Like quills upon the fretful porpentine:
But this eternal blazon must not be
To ears of flesh and blood.”
Galway Kinnell:
“I roll
this way and that in the great bed, under
the quilt
that mimics this country of broken farms and woods”
— “The Porcupine”
For quiltblock designs that do not mimic farms or woods,
see the cover of Diamond Theory . See also the quotations
from Wallace Stevens linked to in the last line of yesterday’s
post in memory of Kinnell.
“… a bee for the remembering of happiness” — Wallace Stevens
A print copy of next Sunday’s New York Times Book Review
arrived in today’s mail. From the frontpage review:
Marcel Theroux on The Book of Strange New Things ,
a novel by Michel Faber —
“… taking a standard science fiction premise and
unfolding it with the patience and focus of a
tai chi master, until it reveals unexpected
connections, ironies and emotions.”
What is a tai chi master, and what is it that he unfolds?
Perhaps the taijitu symbol and related material will help.
The Origin of Change
“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”
(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.
"There is such a thing as a tesseract." — Madeleine L'Engle
An approach via the Omega Matrix:
See, too, Rosenhain and Göpel as The Shadow Guests .
"I need a photo opportunity, I want a shot at redemption.
Don't want to end up a cartoon in a cartoon graveyard."
– Paul Simon
"The theory of poetry, that is to say, the total of the theories of poetry, often seems to become in time a mystical theology or, more simply, a mystique. The reason for this must by now be clear. The reason is the same reason why the pictures in a museum of modern art often seem to become in time a mystical aesthetic, a prodigious search of appearance, as if to find a way of saying and of establishing that all things, whether below or above appearance, are one and that it is only through reality, in which they are reflected or, it may be, joined together, that we can reach them. Under such stress, reality changes from substance to subtlety, a subtlety in which it was natural for Cézanne to say: 'I see planes bestriding each other and sometimes straight lines seem to me to fall' or 'Planes in color…. The colored area where shimmer the souls of the planes, in the blaze of the kindled prism, the meeting of planes in the sunlight.' The conversion of our Lumpenwelt went far beyond this. It was from the point of view of another subtlety that Klee could write: 'But he is one chosen that today comes near to the secret places where original law fosters all evolution. And what artist would not establish himself there where the organic center of all movement in time and space– which he calls the mind or heart of creation– determines every function.' Conceding that this sounds a bit like sacerdotal jargon, that is not too much to allow to those that have helped to create a new reality, a modern reality, since what has been created is nothing less.
— Wallace Stevens, Harvard College Class of 1901, "The Relations between Poetry and Painting" in The Necessary Angel (Knopf, 1951) 
For background on the planes illustrated above,
see Diamond theory in 1937.
The authors of the following offer an introduction to symmetry
in quilt blocks. They assume, perhaps rightly, that their audience
is intellectually impaired:
“A quilt block is made of 16 smaller squares.
Each small square consists of two triangles.”
Study this example of definition.
(It applies quite precisely to the sorts of square patterns
discussed in the 1976 monograph Diamond Theory , but
has little relevance for quilt blocks in general.)
Some background for those who are not intellectually impaired:
Robinson’s book Definition , in this journal and at Amazon.
Or: Death Edit
Log24 on the reported date of Sturtevant’s death:
Conceptual ArtFiled under: Uncategorized — m759 @ 2:01 AM
Yesterday’s online New York Times has the following quote: “The idea becomes a machine that makes the art.” For instance, some conceptual art not by LeWitt: Diamond Theory Roulette (Feb. 2, 2014). 
See the above weblog post honoring a Swiss artist‘s
“wit, his perception, his genius, his horizon,
his determination, his humour, his friendship,
and his immeasurable kindness.”
Not a bad sendoff. Contrast with events at Harvard
on the date of the artist’s death.
Related material: An album cover, and …
See also this journal in September 2008.
As far as Swiss art goes, I personally prefer the work of, say,
Karl Gerstner and Paul Talman.
Raven’s Progressive Matrices intelligence test—
Wechsler Adult Intelligence Scale test—
Related art — (Click images for further details.)
Patterns suggesting those of the Raven test:
Patterns suggesting those of the Wechsler test:
The latter patterns were derived from the former.
Yesterday’s online New York Times has the following quote:
“The idea becomes a machine that makes the art.”
— Sol LeWitt
For instance, some conceptual art not by LeWitt:
Diamond Theory Roulette (Feb. 2, 2014).
“For every kind of vampire, there is a kind of cross.” — Gravity’s Rainbow
“I don’t write exclusively on Jewish themes or about Jewish characters.
My collection of short stories, Strange Attractors , contained nine pieces,
five of which were, to some degree, Jewish, and this ratio has provided me
with a precise mathematical answer (for me, still the best kind of answer)
to the question of whether I am a Jewish writer. I am fiveninths a Jewish writer.”
— Rebecca Goldstein, “Against Logic”
Midrashim for Rebecca:
The Diamond Theory vs. the Story Theory (of truth)
Story Theory and the Number of the Beast
The Palm Sunday post “Gray Space”
For those who prefer the diamond theory of truth,
a “precise mathematical” view of a Gray code —
For those who prefer the story theory of truth,
Thursday with the Nashes —
The actors who portrayed Mr. and Mrs. John Nash in
‘A Beautiful Mind’ now portray Mr. and Mrs. Noah…
"… 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.
From Northrop Frye's The Great Code: The Bible and Literature , Ch. 3: Metaphor I — "In the preceding chapter we considered words in sequence, where they form narratives and provide the basis for a literary theory of myth. Reading words in sequence, however, is the first of two critical operations. Once a verbal structure is read, and reread often enough to be possessed, it 'freezes.' It turns into a unity in which all parts exist at once, without regard to the specific movement of the narrative. We may compare it to the study of a music score, where we can turn to any part without regard to sequential performance. The term 'structure,' which we have used so often, is a metaphor from architecture, and may be misleading when we are speaking of narrative, which is not a simultaneous structure but a movement in time. The term 'structure' comes into its proper context in the second stage, which is where all discussion of 'spatial form' and kindred critical topics take their origin." 
Related material:
"The Great Code does not end with a triumphant conclusion or the apocalypse that readers may feel is owed them or even with a clear summary of Frye’s position, but instead trails off with a series of verbal winks and nudges. This is not so great a fault as it would be in another book, because long before this it has been obvious that the forward motion of Frye’s exposition was illusory, and that in fact the book was devoted to a constant reexamination of the same basic data from various closely related perspectives: in short, the method of the kaleidoscope. Each shake of the machine produces a new symmetry, each symmetry as beautiful as the last, and none of them in any sense exclusive of the others. And there is always room for one more shake."
— Charles Wheeler, "Professor Frye and the Bible," South Atlantic Quarterly 82 (Spring 1983), pp. 154164, reprinted in a collection of reviews of the book. 
For code in a different sense, but related to the first passage above,
see Diamond Theory Roullete, a webpage by Radamés Ajna.
For "the method of the kaleidoscope" mentioned in the second
passage above, see both the Ajna page and a webpage of my own,
Kaleidoscope Puzzle.
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 —
“… her mind rotated the facts….”
Related material— hypercube rotation,* in the context
of rotational symmetries of the Platonic solids:
“I’ve heard of affairs that are strictly Platonic”
* Footnote added on Dec. 26, 2013 —
See Arnold Emch, “Triple and Multiple Systems, Their Geometric
Configurations and Groups,” Trans. Amer. Math. Soc. 31 (1929),
No. 1, 25–42.
On page 42, Emch describes the above method of rotating a
hypercube’s 8 facets (i.e., threedimensional cubes) to count
rotational symmetries —
See also Diamond Theory in 1937.
Also on p. 42, Emch mentions work of Carmichael on a
Steiner system with the Mathieu group M_{11} as automorphism
group, and poses the problem of finding such systems and
groups that are larger. This may have inspired the 1931
discovery by Carmichael of the Steiner system S(5, 8, 24),
which has as automorphisms the Mathieu group M_{24} .
See also last night's "Pink Champagne on Ice" post.
The "ice" in that post's title refers to the white lines
forming a tesseract in the book cover's background—
"icy white and crystalline," as Johnny Mercer put it.
(A Tune for Josefine, Nov. 25.)
See also the tag Diamond Theory tesseract in this journal.
New! Improved!
"Euclid's edifice loomed in my consciousness
as a marvel among sciences, unique in its
clarity and unquestionable validity."
—Richard J. Trudeau in
The NonEuclidean Revolution (First published in 1986)
Readers of this journal will be aware that Springer's new page
advertising Trudeau's book, pictured above, is a baitandswitch
operation. In the chapter advertised, Trudeau promotes what he
calls "the Diamond Theory of Truth" as a setup for his real goal,
which he calls "the Story Theory of Truth."
For an earlier use of the phrase "Diamond Theory" in
connection with geometry, see a publication from 1977.
Mathematics:
A review of posts from earlier this month —
Wednesday, September 4, 2013

Narrative:
Aooo.
Happy birthday to Stephen King.
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.
"How do you get young people excited
about space? How do you get them interested
not just in watching movies about space,
or in playing video games set in space …
but in space itself?"
— Megan Garber in The Atlantic , Aug. 16, 2012
One approach:
"There is such a thing as a tesseract" and
Diamond Theory in 1937.
See, too, Baez in this journal.
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 .
"What on earth is
— Said to be an annotation 
In the spirit of the late Thomas Guinzburg…
See also "Concrete Universal" in this journal.
Related material— From a Bloomsday reply
to a Diamond Theory reader's comment, an excerpt—
The reader's comment suggests the following passages from
the book by Stirling quoted above—
Here Stirling plays a role analogous to that of Professor Irwin Corey
accepting the National Book Award for Gravity's Rainbow in 1974.
For those who prefer Trudeau's
"Story Theory" of truth to his "Diamond Theory"
Related material: Click images below for the original posts.
See as well the novel "Lexicon" at Amazon.com
and the word "lexicon" in this journal.
From a weblog post on June 11, 2013, by one Pete Trbovich:
Here again, I don't think Steven Cullinane is really unhinged per se. At the very least, his geometric study is fun to play with, particularly when you find this toy. And I'm not really sure that anything he says is wrong per se. But you might find yourself asking "So what?" or more to the point, "Why is this supposed to be the central theory to explaining life, the universe, and everything?" 
It isn't supposed to be such a theory.
I do not know why Trbovich thinks it is .
— Steven H. Cullinane
Update of 11 PM June 16:
For one such central theory of everything, see
the I Ching . Diamond theory is, unlike that
Chinese classic, pure mathematics, but the larger
of the binarycoordinate structures it is based on
are clearly isomorphic, simply as structures , to
the I Ching 's 64 hexagrams.
Make of this what you will.
Today's previous post on coordinate systems
suggests a look at the phrase "Galois coordinates."
A search shows that the phrase, though natural,
has apparently not been used before 2011* for solutions
to what Hermann Weyl called "the relativity problem."
A thorough historical essay on Galois coordinatization
in this sense would require more academic resources
than I have available. It would likely describe a number
of applications of Galoisfield coordinates to square
(and perhaps to cubical) arrays that were studied before
1976, the date of my Diamond Theory monograph.
But such a survey might not find any such pre1976
coordinatization of a 4×4 array by the 16 elements
of the vector 4space over the Galois field with two
elements, GF(2).
Such coordinatizations are important because of their
close relationship to the Mathieu group M _{24 }.
See a preprint by Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of Kummer
surfaces in the Mathieu group M _{24} ," with its remark
denying knowledge of any such coordinatization
prior to a 1989 paper by R. T. Curtis.
Related material:
Some images related to Galois coordinates, excerpted
from a Google search today (click to enlarge)—
* A rather abstract 2011 paper that uses the phrase
"Galois coordinates" may have some implications
for the naive form of the relativity problem
related to square and cubical arrays.
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):
Found this morning in a search:
A logline is a onesentence summary of your script.
www.scriptologist.com/Magazine/Tips/Logline/logline.html
It’s the short blurb in TV guides that tells you what a movie
is about and helps you decide if you’re interested …
The search was suggested by a screenwriting weblog post,
“Loglines: WHAT are you doing?“.
What is your story about?
No, seriously, WHAT are you writing about?
Who are the characters? What happens to them?
Where does it take place? What’s the theme?
What’s the style? There are nearly a million
little questions to answer when you set out
to tell a story. But it all starts with one
super, overarching question.
What are you writing about? This is the first
big idea that we pull out of the ether, sometimes
before we even have any characters.
What is your story about?
The screenwriting post was found in an earlier search for
the highlighted phrase.
The screenwriting post was dated December 15, 2009.
What I am doing now is checking for synchronicity.
This weblog on December 15, 2009, had a post
titled A Christmas Carol. That post referred to my 1976
monograph titled Diamond Theory .
I guess the script I’m summarizing right now is about
the heart of that theory, a group of 322,560 permutations
that preserve the symmetry of a family of graphic designs.
For that group in action, see the Diamond 16 Puzzle.
The “super overarching” phrase was used to describe
this same group in a different context:
This is from “Mathieu Moonshine,” a webpage by Anne Taormina.
A logline summarizing my approach to that group:
Finite projective geometry explains
the surprising symmetry properties
of some simple graphic designs—
found, for instance, in quilts.
The story thus summarized is perhaps not destined for movie greatness.
Story, Structure, and the Galois Tesseract
Recent Log24 posts have referred to the
"Penrose diamond" and Minkowski space.
The Penrose diamond has nothing whatever
to do with my 1976 monograph "Diamond Theory,"
except for the diamond shape and the connection
of the Penrose diamond to the Klein quadric—
The Klein quadric occurs in the fivedimensional projective space
over a field. If the field is the twoelement Galois field GF(2), the
quadric helps explain certain remarkable symmetry properties
of the R. T. Curtis Miracle Octad Generator (MOG), hence of
the large Mathieu group M_{24}. These properties are also
relevant to the 1976 "Diamond Theory" monograph.
For some background on the quadric, see (for instance)…
See also The Klein Correspondence,
Penrose SpaceTime, and a Finite Model.
Related material:
"… one might crudely distinguish between philosophical – J. M. E. Hyland. "Proof Theory in the Abstract." (pdf) 
Those who prefer story to structure may consult
Yesterday's post Treasure Hunt, on a Brazilian weblog,
suggests a review of Brazil in this journal. The post
most relevant to yesterday's remarks is from
August 15, 2003, with a link, now broken, to the work
of Brazilian artist Nicole Sigaud* that also uses the
four halfsquare tiles used in 1704 by Sebastien Truchet
and somewhat later by myself in Diamond Theory
(see a 1977 version).
A more recent link that works:
http://vismath9.tripod.com/sigaud/eindex.html ANACOM PROJECT
APPLICATIONS
© 1997 – 2002 Nicole Sigaud 
* Sigaud shares the interests of her fellow Brazilian
whose weblog was the subject of yesterday's
Treasure Hunt.—
"For many years I have dedicated myself to the study
of medieval magic, demonology, Kabbalah, Astrology,
Alchemy, Tarot and divination in general."
— Nicole Sigaud (translated by Google) in a selfprofile:
http://www.recantodasletras.com.br/autor.php?id=78359.
I do not share the interest of these authors in such matters,
except as they are reflected in the works of authors like
Charles Williams and Umberto Eco.
The Mathematical Association of America (MAA)
newsmagazine Focus for December 2012/January 2013:
The Babylonian tablet on the cover illustrates the
"Mathematical Treasures" article.
A search for related material yields a Babylonian tablet
reproduced in a Brazilian weblog on July 4, 2012:
In that weblog on the same day, July 4, 2012,
another post quotes at length my Diamond Theory page,
starting with the following image from that page—
That Brazilian post recommends use of geometry together
with Tarot and astrology. I do not concur with this
recommendation, but still appreciate the mention.
The finite (i.e., Galois) field GF(16),
according to J. J. Seidel in 1974—
The same field according to Steven H. Cullinane in 1986,
in its guise as the affine 4space over GF(2)—
The same field, again disguised as an affine 4space,
according to John H. Conway and N.J.A. Sloane in
Sphere Packings, Lattices, and Groups , first published in 1988—
The above figure by Conway and Sloane summarizes, using
a 4×4 array, the additive vectorspace structure of the finite
field GF(16).
This structure embodies what in Euclidean space is called
the parallelogram rule for vector addition—
(Thanks to June Lester for the 3D (uvw) part of the above figure.)
For the transition from this colored Euclidean hypercube
(used above to illustrate the parallelogram rule) to the
4×4 Galois space (illustrated by Cullinane in 1979 and
Conway and Sloane in 1988— or later… I do not have
their book’s first edition), see Diamond Theory in 1937,
Vertex Adjacency in a Tesseract and in a 4×4 Array,
Spaces as Hypercubes, and The Galois Tesseract.
For some related narrative, see tesseract in this journal.
(This post has been added to finitegeometry.org.)
Update of August 9, 2013—
Coordinates for hypercube vertices derived from the
parallelogram rule in four dimensions were better
illustrated by Jürgen Köller in a web page archived in 2002.
Update of August 13, 2013—
The four basis vectors in the 2002 Köller hypercube figure
are also visible at the bottom of the hypercube figure on
page 7 of “Diamond Theory,” excerpts from a 1976 preprint
in Computer Graphics and Art , Vol. 2, No. 1, February 1977.
A predecessor: Coxeter’s 1950 hypercube figure from
“SelfDual Configurations and Regular Graphs.”
Memories, Dreams, Reflections
by C. G. Jung
Recorded and edited By Aniela Jaffé, translated from the German
by Richard and Clara Winston, Vintage Books edition of April 1989
From pages 195196:
“Only gradually did I discover what the mandala really is:
‘Formation, Transformation, Eternal Mind’s eternal recreation.’*
And that is the self, the wholeness of the personality, which if all
goes well is harmonious, but which cannot tolerate selfdeceptions.”
* Faust , Part Two, trans. by Philip Wayne (Harmondsworth,
England, Penguin Books Ltd., 1959), p. 79. The original:
… Gestaltung, Umgestaltung,
Des ewigen Sinnes ewige Unterhaltung….
Jung’s “Formation, Transformation” quote is from the realm of
the Mothers (Faust , Part Two, Act 1, Scene 5: A Dark Gallery).
The speaker is Mephistopheles.
See also Prof. Bruce J. MacLennan on this realm
in a Web page from his Spring 2005 seminar on Faust:
“In alchemical terms, F is descending into the dark, formless
primary matter from which all things are born. Psychologically
he is descending into the deepest regions of the
collective unconscious, to the source of life and all creation.
Mater (mother), matrix (womb, generative substance), and matter
all come from the same root. This is Faust’s next encounter with
the feminine, but it’s obviously of a very different kind than his
relationship with Gretchen.”
The phrase “Gestaltung, Umgestaltung ” suggests a more mathematical
approach to the Unterhaltung . Hence…
Part I: Mothers
“The ultimate, deep symbol of motherhood raised to
the universal and the cosmic, of the birth, sending forth,
death, and return of all things in an eternal cycle,
is expressed in the Mothers, the matrices of all forms,
at the timeless, placeless originating womb or hearth
where chaos is transmuted into cosmos and whence
the forms of creation issue forth into the world of
place and time.”
— Harold Stein Jantz, The Mothers in Faust:
The Myth of Time and Creativity ,
Johns Hopkins Press, 1969, page 37
Part II: Matrices
Part III: Spaces and Hypercubes
Click image for some background.
Part IV: Forms
Forms from the I Ching :
Click image for some background.
Forms from Diamond Theory :
Click image for some background.
The concept of "deep structure," once a popular meme,
has long been abandoned by Chomskians.
It still applies, however, to the 1976 mathematics, diamond theory ,
underlying the formal patterns discussed in a Royal Society paper
this year.
A review of deep structure, from the Wikipedia article Cartesian linguistics—
[Numbers in parentheses refer to pages in the original 1966 Harper edition of Chomsky's book Cartesian Linguistics .] Deep structure vs. surface structure "Pursuing the fundamental distinction between body and mind, Cartesian linguistics characteristically assumes that language has two aspects" (32). These are namely the sound/character of a linguistic sign and its significance (32). Semantic interpretation or phonetic interpretation may not be identical in Cartesian linguistics (32). Deep structures are often only represented in the mind (a mirror of thought), as opposed to surface structures, which are not. Deep structures vary less between languages than surface structures. For instance, the transformational operations to derive surface forms of Latin and French may obscure common features of their deep structures (39). Chomsky proposes, "In many respects, it seems to me quite accurate, then, to regard the theory of transformational generative grammar, as it is developing in current work, as essentially a modern and more explicit version of the PortRoyal theory" (39). Summary of Port Royal Grammar The Port Royal Grammar is an often cited reference in Cartesian Linguistics and is considered by Chomsky to be a more than suitable example of Cartesian linguistic philosophy. "A sentence has an inner mental aspect (a deep structure that conveys its meaning) and an outer, physical aspect as a sound sequence"***** This theory of deep and surface structures, developed in Port Royal linguistics, meets the formal requirements of language theory. Chomsky describes it in modern terms as "a base system that generates deep structures and a transformational system that maps these into surface structures", essentially a form of transformational grammar akin to modern studies (42). 
The corresponding concepts from diamond theory are…
"Deep structure"— The line diagrams indicating the underlying
structure of varying patterns
"A base system that generates deep structures"—
Group actions on square arrays… for instance, on the 4×4 square
"A transformational system"— The decomposition theorem
that maps deep structure into surface structure (and viceversa)
Do you know where the mushrooms are?
Above: Image from Log24 on Dec. 4th, 2012, at 4:23 PM ET.
See also… on that date at that time …
The American College of Neuropsychopharmacology… (click to enlarge)—
… 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
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 —
"First steps"? The mathematics underlying such patterns
was presented 35 years earlier, in Diamond Theory.
* See GombrichDouat in this journal.
The 1976 monograph "Diamond Theory" was an example
of "programmed art" in the sense established by, for
instance, Karl Gerstner. The images were produced
according to strict rules, and were in this sense
"programmed," but were drawn by hand.
Now an actual computer program has been written,
based on the Diamond Theory excerpts published
in the Feb. 1977 issue of Computer Graphics and Art
(Vol. 2, No. 1, pp. 57), that produces copies of some of
these images (and a few malformed images not in
Diamond Theory).
See Isaac Gierard's program at GitHub—
https://github.com/matthewepler/ReCode_Project/
blob/dda7b23c5ad505340b468d9bd707fd284e6c48bf/
isaac_gierard/StevenHCullinane_DiamondTheory/
StevenHCullinane_DiamondTheory.pde
As the suffix indicates, this program is in the
Processing Development Environment language.
It produces the following sketch:
The rationale for selecting and arranging these particular images is not clear,
and some of the images suffer from defects (exercise: which ones?), but the
overall effect of the sketch is pleasing.
For some background for the program, see The ReCode Project.
It is good to learn that the Processing language is welladapted to making the
images in such sketches. The overall structure of the sketch gives, however,
no clue to the underlying theory in "Diamond Theory."
For some related remarks, see Theory (Sept. 30, 2012).
* For the title, see Darko Fritz, "Notions of the Program in 1960s Art."
A 1976 monograph:
A 2012 mixtape cover:
A new "Diamond Theory" image found on the Web
today links my work to the "Stoned Ape Theory"
of human evolution due to Terence McKenna.
This link is via a picture, apparently copied from deviantart.com,
of two apes contemplating some psychedelic mushrooms.
The picture is titled "Stoned Ape Theory." The mushrooms in
the picture are apparently taken from an image at DrugNet.net:
Actually, the mathematical work called "diamond theory"
has nothing whatever to do with psychedelic experiences,
although some of the illustrations may appeal to McKenna fans.
Quotes from the Bremen site
http://dada.compartbremen.de/ —
" '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.
Happy birthday to…
Today's sermon, by MarieLouise 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…
The circle above is marked with a version For the square, see the diamond theorem. "Two things of opposite natures seem to depend — Wallace Stevens, 
A review of two theories of truth described
by a clergyman, Richard J. Trudeau, in
The NonEuclidean Revolution—
"But, I asked, is there a difference
between fiction and nonfiction?
'Not much,' she said, shrugging."
— New Yorker profile of tesseract
author Madeleine L'Engle
(Click image for some background.)
See also the links on a webpage at finitegeometry.org.
(An episode of Mathematics and Narrative )
A report on the August 9th opening of Sondheim's Into the Woods—
Amy Adams… explained why she decided to take on the role of the Baker’s Wife.
“It’s the ‘Be careful what you wish’ part,” she said. “Since having a child, I’m really aware that we’re all under a social responsibility to understand the consequences of our actions.” —Amanda Gordon at businessweek.com
Related material—
Amy Adams in Sunshine Cleaning "quickly learns the rules and ropes of her unlikely new market. (For instance, there are products out there specially formulated for cleaning up a 'decomp.')" —David Savage at Cinema Retro
Compare and contrast…
1. The following item from Walpurgisnacht 2012—
2. The six partitions of a tesseract's 16 vertices
into four parallel faces in Diamond Theory in 1937—
Title of a treatise by Dominique Douat—
"Méthode pour faire une infinité de desseins différens avec des carreaux mipartis de deux couleurs par une ligne diagonale : ou observations du Père Dominique Doüat Religieux Carmes de la Province de Toulouse sur un mémoire inséré dans l'Histoire de l'Académie Royale des Sciences de Paris l'année 1704, présenté par le Révérend Père Sébastien Truchet religieux du même ordre, Académicien honoraire " (Paris, 1722)
"The earliest (and perhaps the rarest) treatise on the theory of design"
— E. H. Gombrich, 1979, in The Sense of Order
A facsimile version (excerpts, 108 pp., Feb. 5, 2010) of this treatise is available from
http://jacquesandre.fr/ed/ in a 23.1 MB pdf.
Sample page—
For a treatise on the finite geometry underlying such designs (based on a monograph I wrote in 1976, before I had heard of Douat or his predecessor Truchet), see Diamond Theory.
From the current Wikipedia article "Symmetry (physics)"—
"In physics, symmetry includes all features of a physical system that exhibit the property of symmetry—that is, under certain transformations, aspects of these systems are 'unchanged', according to a particular observation. A symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is 'preserved' under some change.
A family of particular transformations may be continuous (such as rotation of a circle) or discrete (e.g., reflection of a bilaterally symmetric figure, or rotation of a regular polygon). Continuous and discrete transformations give rise to corresponding types of symmetries. Continuous symmetries can be described by Lie groups while discrete symmetries are described by finite groups (see Symmetry group)."….
"A discrete symmetry is a symmetry that describes noncontinuous changes in a system. For example, a square possesses discrete rotational symmetry, as only rotations by multiples of right angles will preserve the square's original appearance."
Note the confusion here between continuous (or discontinuous) transformations and "continuous" (or "discontinuous," i.e. "discrete") groups .
This confusion may impede efforts to think clearly about some pure mathematics related to current physics— in particular, about the geometry of spaces made up of individual units ("points") that are not joined together in a continuous manifold.
For an attempt to forestall such confusion, see Noncontinuous Groups.
For related material, see Erlanger and Galois as well as the opening paragraphs of Diamond Theory—
Symmetry is often described as invariance under a group of transformations. An unspoken assumption about symmetry in Euclidean 3space is that the transformations involved are continuous.
Diamond theory rejects this assumption, and in so doing reveals that Euclidean symmetry may itself be invariant under rather interesting groups of noncontinuous (and asymmetric) transformations. (These might be called noncontinuous groups, as opposed to socalled discontinuous (or discrete ) symmetry groups. See Weyl's Symmetry .)
For example, the affine group A on the 4space over the 2element field has a natural noncontinuous and asymmetric but symmetrypreserving action on the elements of a 4×4 array. (Details)
(Version first archived on March 27, 2002)
Update of Sunday, February 19, 2012—
The abuse of language by the anonymous authors
of the above Wikipedia article occurs also in more
reputable sources. For instance—
Some transformations referred to by Brading and Castellani
and their editees as "discrete symmetries" are, in fact, as
linear transformations of continuous spaces, themselves
continuous transformations.
This unfortunate abuse of language is at least made explicit
in a 2003 text, Mathematical Perspectives on Theoretical
Physics (Nirmala Prakash, Imperial College Press)—
"… associated[*] with any given symmetry there always exists
a continuous or a discrete group of transformations….
A symmetry whose associated group is continuous (discrete)
is called a continuous (discrete ) symmetry ." — Pp. 235, 236
[* Associated how?]
From the Crafoord Prize website—
Related meta mathematical image from Diamond Theory—
Mathematical image related to combinatorics—
See also permutahedron in this journal.
In memory of artist Ronald Searle—
Searle reportedly died at 91 on December 30th.
From Log24 on that date—
Click the above image for some context.
Update of 9:29 PM EST Jan. 3, 2012—
Theorum
Theorum (rhymes with decorum, apparently) is a neologism proposed by Richard Dawkins in The Greatest Show on Earth to distinguish the scientific meaning of theory from the colloquial meaning. In most of the opening introduction to the show, he substitutes "theorum" for "theory" when referring to the major scientific theories such as evolution. Problems with "theory" Dawkins notes two general meanings for theory; the scientific one and the general sense that means a wild conjecture made up by someone as an explanation. The point of Dawkins inventing a new word is to get around the fact that the lay audience may not thoroughly understand what scientists mean when they say "theory of evolution". As many people see the phrase "I have a theory" as practically synonymous with "I have a wild guess I pulled out of my backside", there is often confusion about how thoroughly understood certain scientific ideas are. Hence the well known creationist argument that evolution is "just a theory" – and the often cited response of "but gravity is also just a theory". To convey the special sense of thoroughness implied by the word theory in science, Dawkins borrowed the mathematical word "theorem". This is used to describe a well understood mathematical concept, for instance Pythagoras' Theorem regarding right angled triangles. However, Dawkins also wanted to avoid the absolute meaning of proof associated with that word, as used and understood by mathematicians. So he came up with something that looks like a spelling error. This would remove any person's emotional attachment or preconceptions of what the word "theory" means if it cropped up in the text of The Greatest Show on Earth , and so people would (in "theory ") have no other choice but to associate it with only the definition Dawkins gives. This phrase has completely failed to catch on, that is, if Dawkins intended it to catch on rather than just be a device for use in The Greatest Show on Earth . When googled, Google will automatically correct the spelling to theorem instead, depriving this very page its rightful spot at the top of the results.

Some backgound— In this journal, "Diamond Theory of Truth."
James Hillman reportedly died on Thursday, October 27, 2011.
For some commentary, see Wednesday's link to 779—
Lurching Toward Decision
"Suskind… nails, I think, Obama's intellectual blind spot. Indeed, Obama himself nails it, telling Suskind that he was too inclined to search for 'the perfect technical answer' to the myriad of complex issues coming at him."
— Frank Rich on Ron Suskind's new book about the White House, Confidence Men
Very distantly related material—
From "Confidence Game," an Oct. 12, 2008, post in this journal, a quasiEuropean perspective—
Kaleidoscope turning… – Roger Zelazny, Eye of Cat 
See also …
Gravity’s Rainbow , Penguin Classics, 1995, page 742:
"… knowing his Tarot, we would expect to look among the Humility, among the gray and preterite souls, to look for him adrift in the hostile light of the sky, the darkness of the sea….
Now there’s only a long cat’seye of bleak sunset left over the plain tonight, bright gray against a purple ceiling of clouds, with an iris of
742"
From math16.com—
Quotations on Realism

The story of the diamond mine continues
(see Coordinated Steps and Organizing the Mine Workers)—
From The Search for Invariants (June 20, 2011):
The conclusion of Maja Lovrenov's
"The Role of Invariance in Cassirer’s Interpretation of the Theory of Relativity"—
"… physical theories prove to be theories of invariants
with regard to certain groups of transformations and
it is exactly the invariance that secures the objectivity
of a physical theory."
— SYNTHESIS PHILOSOPHICA 42 (2/2006), pp. 233–241
Related material from Sunday's New York Times travel section—
Yesterday's post Ad Meld featured Harry Potter (succeeding in business),
a 4×6 array from a video of the song "Abracadabra," and a link to a post
with some background on the 4×6 Miracle Octad Generator of R.T. Curtis.
A search tonight for related material on the Web yielded…
Weblog post by Steve Richards titled "The Search for Invariants:
The Diamond Theory of Truth, the Miracle Octad Generator
and Metalibrarianship." The artwork is by Steven H. Cullinane.
Richards has omitted Cullinane's name and retitled the artwork.
The author of the post is an artist who seems to be interested in the occult.
His post continues with photos of pages, some from my own work (as above), some not.
My own work does not deal with the occult, but some enthusiasts of "sacred geometry" may imagine otherwise.
The artist's post concludes with the following (note also the beginning of the preceding post)—
"The Struggle of the Magicians" is a 1914 ballet by Gurdjieff. Perhaps it would interest Harry.
From Savage Logic— Sunday, March 15, 2009 5:24 PM The Origin of Change A note on the figure "Two things of opposite natures seem to depend — Wallace Stevens, 
For the title, see Palm Sunday.
"There is a pleasantly discursive treatment of
Pontius Pilate's unanswered question 'What is truth?'" — H. S. M. Coxeter, 1987
From this date (April 22) last year—
Richard J. Trudeau in The NonEuclidean Revolution , chapter on "Geometry and the Diamond Theory of Truth"– "… Plato and Kant, and most of the philosophers and scientists in the 2200year interval between them, did share the following general presumptions: (1) Diamonds– informative, certain truths about the world– exist. Presumption (1) is what I referred to earlier as the 'Diamond Theory' of truth. It is far, far older than deductive geometry." Trudeau's book was published in 1987. The nonEuclidean* figures above illustrate concepts from a 1976 monograph, also called "Diamond Theory." Although nonEuclidean,* the theorems of the 1976 "Diamond Theory" are also, in Trudeau's terminology, diamonds. * "NonEuclidean" here means merely "other than Euclidean." No violation of Euclid's parallel postulate is implied. 
Trudeau comes to reject what he calls the "Diamond Theory" of truth. The trouble with his argument is the phrase "about the world."
Geometry, a part of pure mathematics, is not about the world. See G. H. Hardy, A Mathematician's Apology .
An Ordinary Evening in New Haven  

line 540 (xxx.18):  In which hundreds of eyes, in one mind, see at once. 
The cover art of a 1976 monograph, "Diamond Theory," was described in this morning's post.
As Madeleine L'Engle noted in 1976, the cover art resembles the character Proginoskes in her novel A Wind in the Door.
A search today for Proginoskes yields a description by Brendan Kidwell…
A link at Kidwell's site leads to a weblog by Jeff Atwood, a founder of Stack Overflow, a programmers' questionandanswer site.
(Stack Overflow is said to have inspired the similar site for mathematicians, Math Overflow.)
Yesterday Atwood discussed technical writing.
This suggests a look at Robert M. Pirsig on that subject in his 1974 philosophical novel Zen and the Art of Motorcycle Maintenance.
(See also a document on Pirsig's technicalwriting background.)
Pirsig describes his novel as "a sort of Chautauqua."
This, together with the Stevens and Proginoskes quotes above, leads back to the Log24 Feb. 1 post The Search.
An image from that post (click to enlarge)—
Here the apparently fragmented nature of the set of
images imagined as rising above the podium of the
Hall of Philosophy at Chautauqua rather naturally
echoes Stevens's "hundreds of eyes" remark.
The following is from the weblog of a high school mathematics teacher—
This is related to the structure of the figure on the cover of the 1976 monograph Diamond Theory—
Each small square pattern on the cover is a Latin square,
with elements that are geometric figures rather than letters or numerals.
All orderfour Latin squares are represented.
For a deeper look at the structure of such squares, let the highschool
chart above be labeled with the letters A through X, and apply the
fourcolor decomposition theorem. The result is 24 structural diagrams—
Some of the squares are structurally congruent under the group of 8 symmetries of the square.
This can be seen in the following regrouping—
(Image corrected on Jan. 25, 2011– "seven" replaced "eight.")
* Retitled "The Order4 (i.e., 4×4) Latin Squares" in the copy at finitegeometry.org/sc.
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.
and the New York Lottery
A search in this journal for yesterday's evening number in the New York Lottery, 359, leads to…
The Cerebral Savage:
On the Work of Claude LéviStrauss
by Clifford Geertz
Shown below is 359, the final page of Chapter 13 in
The Interpretation of Cultures: Selected Essays by Clifford Geertz,
New York, 1973: Basic Books, pp. 345359 —
This page number 359 also appears in this journal in an excerpt from Dan Brown's novel Angels & Demons—
See this journal's entries for March 115, 2009, especially…
Sunday, March 15, 2009 5:24 PM
Philosophy and Poetry: The Origin of Change A note on the figure "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" Sunday, March 15, 2009 11:00 AM Ides of March Sermon: Angels, Demons,
"Symbology" "On Monday morning, 9 March, after visiting the Mayor of Rome and the Municipal Council on the Capitoline Hill, the Holy Father spoke to the Romans who gathered in the square outside the Senatorial Palace…
'… a verse by Ovid, the great Latin poet, springs to mind. In one of his elegies he encouraged the Romans of his time with these words: "Perfer et obdura: multo graviora tulisti." "Hold out and persist: (Tristia, Liber V, Elegia XI, verse 7).'" This journal
on 9 March: Note the colorinterchange Related material:

The symmetry of the yinyang symbol, of the diamondtheorem symbol, and of Brown's Illuminati Diamond is also apparent in yesterday's midday New York lottery number (see above).
"Savage logic works like a kaleidoscope…." — Clifford Geertz on LéviStrauss
Notes on Mathematics and Narrative
Background—
Commentary on The Wicker Man—
Originally The Wicker Man was not wellreceived by critics in the UK. It was considered
to be bizarre, disturbing, and uncomfortable, with the hasty editing making the story confusing
and out of order…. Today this movie is considered a cult classic and has been called
the “Citizen Kane of horror films” by some reviewers. How did this film become a cult classic?
Real estate motto— Location, Location, Location.
Illustration— The fire leap scene from Wicker Man, filmed at Castle Kennedy—
In today's New York Times, Michiko Kakutani reviews a summer thriller
by Kevin Guilfoile. The Thousand is in the manner of Dan Brown's
2003 The Da Vinci Code or of Katherine Neville's 1988 The Eight .
From the review—
What connects these disparate events, it turns out, is a sinister organization
called the Thousand, made up of followers of the ancient Greek mathematician
and philosopher Pythagoras (yes, the same Pythagoras associated with
the triangle theorem that we learned in school).
As Mr. Guilfoile describes it, this organization is part Skull and Bones,
part Masonic lodge, part something much more twisted and nefarious….
The plot involves, in part,
… an eccentric artist’s mysterious masterwork, made up of thousands of
individually painted tiles that may cohere into an important message….
Not unlike the tiles in the Diamond Theory cover (see yesterday's post)
or, more aptly, the entries in this journal.
A brief prequel to the above dialogue—
In lieu of songs, here is a passage by Patrick Blackburn
more relevant to the art of The Thousand—
See also the pagan fire leaping in Dancing at Lughnasa.
Narrative Sequence
In today's New York Times, Michiko Kakutani reviews a summer thriller by Kevin Guilfoile. The Thousand is in the manner of Dan Brown's 2003 The Da Vinci Code or of Katherine Neville's 1988 The Eight .
From the review—
What connects these disparate events, it turns out, is a sinister organization called the Thousand, made up of followers of the ancient Greek mathematician and philosopher Pythagoras (yes, the same Pythagoras associated with the triangle theorem that we learned in school).
As Mr. Guilfoile describes it, this organization is part Skull and Bones, part Masonic lodge, part something much more twisted and nefarious….
The plot involves, in part,
… an eccentric artist’s mysterious masterwork, made up of thousands of individually painted tiles that may cohere into an important message….
Not unlike the tiles in the Diamond Theory cover (see yesterday's post) or, more aptly, the entries in this journal.
Or— Childhood's Rear End
This post was suggested by…
Related material:
The Zeppelin album cover, featuring rear views of nude children, was shot at the Giant's Causeway.
From a page at ledzeppelin.org—
See also Richard Rorty on Heidegger—
Safranski, the author of ''Schopenhauer and the Wild Years of Philosophy,'' never steps back and pronounces judgment on Heidegger, but something can be inferred from the German title of his book: ''Ein Meister aus Deutschland'' (''A Master From Germany''). Heidegger was, undeniably, a master, and was very German indeed. But Safranski's spinechilling allusion is to Paul Celan's bestknown poem, ''Death Fugue.'' In Michael Hamburger's translation, its last lines are:
death is a master from Germany his eyes are blue
he strikes you with leaden bullets his aim is true
a man lives in the house your golden hair Margarete
he sets his pack on us he grants us a grave in the air
he plays with the serpents and daydreams death is a master from Germany
your golden hair Margarete
your ashen hair Shulamith.
No one familiar with Heidegger's work can read Celan's poem without recalling Heidegger's famous dictum: ''Language is the house of Being. In its home man dwells.'' Nobody who makes this association can reread the poem without having the images of Hitler and Heidegger — two men who played with serpents and daydreamed — blend into each other. Heidegger's books will be read for centuries to come, but the smell of smoke from the crematories — the ''grave in the air'' — will linger on their pages.
Heidegger is the antithesis of the sort of philosopher (John Stuart Mill, William James, Isaiah Berlin) who assumes that nothing ultimately matters except human happiness. For him, human suffering is irrelevant: philosophy is far above such banalities. He saw the history of the West not in terms of increasing freedom or of decreasing misery, but as a poem. ''Being's poem,'' he once wrote, ''just begun, is man.''
For Heidegger, history is a sequence of ''words of Being'' — the words of the great philosophers who gave successive historical epochs their selfimage, and thereby built successive ''houses of Being.'' The history of the West, which Heidegger also called the history of Being, is a narrative of the changes in human beings' image of themselves, their sense of what ultimately matters. The philosopher's task, he said, is to ''preserve the force of the most elementary words'' — to prevent the words of the great, housesofBeingbuilding thinkers of the past from being banalized.
Related musical meditations—
Shine On (Saturday, April 21, 2007), Shine On, Part II, and Built (Sunday, April 22, 2007).
Related pictorial meditations—
The Giant's Causeway at Peter J. Cameron's weblog
and the cover illustration for Diamond Theory (1976)—
The connection between these two images is the following from Cameron's weblog today—
… as we saw, there are two different Latin squares of order 4;
one, but not the other, can be extended to a complete set
of 3 MOLS [mutually orthogonal Latin squares].
The underlying structures of the square pictures in the Diamond Theory cover are those of the two different Latin squares of order 4 mentioned by Cameron.
Connection with childhood—
The children's book A Wind in the Door, by Madeleine L'Engle. See math16.com. L'Engle's fantasies about children differ from those of Arthur C. Clarke and Led Zeppelin.
Hoax and Hype
Four Years Ago Today—
There is Plato's diamond—
and there is diamond theory—
… but there is no "Plato's Diamond Theory."
See, however, today's noon entry, "Plato's Code."
"You gotta be true to your code…" —Sinatra
Romancing the
NonEuclidean Hyperspace
Backstory —
Mere Geometry, Types of Ambiguity,
Dream Time, and Diamond Theory, 1937
For the 1937 grid, see Diamond Theory, 1937.
The grid is, as Mere Geometry points out, a nonEuclidean hyperspace.
For the diamonds of 2010, see Galois Geometry and Solomon’s Cube.
“…geometrically organized, with the parts labeled”
— Ursula K. Le Guin on what she calls “the Euclidean utopia”
“There is such a thing as a tesseract.”
Related material– Diamond Theory, 1937
Stanford Encyclopedia of Philosophy —
“Mereology (from the Greek μερος, ‘part’) is the theory of parthood relations: of the relations of part to whole and the relations of part to part within a whole. Its roots can be traced back to the early days of philosophy, beginning with the Presocratics….”
A nonEuclidean* approach to parts–
Corresponding nonEuclidean*
projective points —
Richard J. Trudeau in The NonEuclidean Revolution, chapter on “Geometry and the Diamond Theory of Truth”–
“… Plato and Kant, and most of the philosophers and scientists in the 2200year interval between them, did share the following general presumptions:
(1) Diamonds– informative, certain truths about the world– exist.
(2) The theorems of Euclidean geometry are diamonds.
Presumption (1) is what I referred to earlier as the ‘Diamond Theory’ of truth. It is far, far older than deductive geometry.”
Trudeau’s book was published in 1987. The nonEuclidean* figures above illustrate concepts from a 1976 monograph, also called “Diamond Theory.”
Although nonEuclidean,* the theorems of the 1976 “Diamond Theory” are also, in Trudeau’s terminology, diamonds.
* “NonEuclidean” here means merely “other than Euclidean.” No violation of Euclid’s parallel postulate is implied.
Today's previous entry was "Gameplayers of the Academy."
David Corfield in the March 2010
European Mathematical Society newsletter—
"Staying on the theme of games, the mathematician
Alexandre Borovik* once told me he thinks of mathematics
as a MassivelyMultiplayer Online RolePlaying Game. If
so, it would show up very clearly the difference between
internal and external viewpoints. Inside the game people
are asking each other whether they were right about
something they encountered in it– 'When you entered
the dungeon did you see that dragon in the fireplace or
did I imagine it?' But someone observing them from the
outside wants to shout: 'You’re not dealing with anything
real. You’ve just got a silly virtual reality helmet on.' External
nominalists say the same thing, if more politely, to
mathematical practitioners. But in an important way the
analogy breaks down. Even if the players interact with
the game to change its functioning in unforeseen ways,
there were the original programmers who set the bounds
for what is possible by the choices they made. When they
release the next version of the game they will have made
changes to allow new things to happen. In the case of
mathematics, it’s the players themselves who make these
choices. There’s no further layer outside.
What can we do then instead to pin down internal reality?"
*See previous references to Borovik in this journal.
Related material:
The Diamond Theory vs. the Story Theory of Truth,
Infantilizing the Audience, and
It's Still the Same Old Story…God of War III
Truth, Geometry, Algebra
The following notes are related to A Simple Reflection Group of Order 168.
1. According to H.S.M. Coxeter and Richard J. Trudeau
“There is a pleasantly discursive treatment of Pontius Pilate’s unanswered question ‘What is truth?’.”
— Coxeter, 1987, introduction to Trudeau’s The NonEuclidean Revolution
1.1 Trudeau’s Diamond Theory of Truth
1.2 Trudeau’s Story Theory of Truth
2. According to Alexandre Borovik and Steven H. Cullinane
2.1 Coxeter Theory according to Borovik
2.1.1 The Geometry–
Mirror Systems in Coxeter Theory
2.1.2 The Algebra–
Coxeter Languages in Coxeter Theory
2.2 Diamond Theory according to Cullinane
2.2.1 The Geometry–
Examples: Eightfold Cube and Solomon’s Cube
2.2.2 The Algebra–
Examples: Cullinane and (rather indirectly related) Gerhard Grams
Summary of the story thus far:
Diamond theory and Coxeter theory are to some extent analogous– both deal with reflection groups and both have a visual (i.e., geometric) side and a verbal (i.e., algebraic) side. Coxeter theory is of course highly developed on both sides. Diamond theory is, on the geometric side, currently restricted to examples in at most three Euclidean (and six binary) dimensions. On the algebraic side, it is woefully underdeveloped. For material related to the algebraic side, search the Web for generators+relations+”characteristic two” (or “2“) and for generators+relations+”GF(2)”. (This last search is the source of the Grams reference in 2.2.2 above.)
From the September 1953 Bulletin of the American Mathematical Society—
Emil Artin, in a review of Éléments de mathématique, by N. Bourbaki, Book II, Algebra, Chaps. IVII–
"We all believe that mathematics is an art. The author of a book, the lecturer in a classroom tries to convey the structural beauty of mathematics to his readers, to his listeners. In this attempt he must always fail. Mathematics is logical to be sure; each conclusion is drawn from previously derived statements. Yet the whole of it, the real piece of art, is not linear; worse than that its perception should be instantaneous. We all have experienced on some rare occasions the feeling of elation in realizing that we have enabled our listeners to see at a moment's glance the whole architecture and all its ramifications. How can this be achieved? Clinging stubbornly to the logical sequence inhibits the visualization of the whole, and yet this logical structure must predominate or chaos would result."
Art Versus Chaos
From an exhibit,
"Reimagining Space"
The above tesseract (4D hypercube)
sculpted in 1967 by Peter Forakis
provides an example of what Artin
called "the visualization of the whole."
For related mathematical details see
Diamond Theory in 1937.
"'The test?' I faltered, staring at the thing.
'Yes, to determine whether you can live
in the fourth dimension or only die in it.'"
— Fritz Leiber, 1959
See also the Log24 entry for
Nov. 26, 2009, the date that
Forakis died.
"There is such a thing
as a tesseract."
— Madeleine L'Engle, 1962
Related material:
"Harrowing cuteness,"* The Eden Express, and a search on "harrowing" in this journal
* Perhaps a typo, but still a memorable phrase.
A Sequel to Koestler's
The Call Girls
Gilles Deleuze, Negotiations 19721990,
Columbia University Press paperback, 1997, p. 137–
"Academics' lives are seldom interesting."
But then there is Matt Lee of the University of Greenwich.
See his weblog subtitled "notes and thoughts on philosophy"… particularly his post "Diamond time, daimon time," of August 20, 2009.
See also my own post of August 20, 2009– "Sophists"– and my earlier post "Daimon Theory" of March 12, 2003:
More about Lee:
"Chaos majik is a form of modern witchcraft."
More about magick:
Noetic Symbology
(Log24 on October 25, 2009)
Unitarian Universalist Origins: Our Historic Faith—
“In sixteenthcentury Transylvania, Unitarian congregations were established for the first time in history.”
Gravity’s Rainbow–
“For every kind of vampire, there is a kind of cross.”
Unitarian minister Richard Trudeau—
“… I called the belief that
(1) Diamonds– informative, certain truths about the world– exist
the ‘Diamond Theory’ of truth. I said that for 2200 years the strongest evidence for the Diamond Theory was the widespread perception that
(2) The theorems of Euclidean geometry are diamonds….
As the news about nonEuclidean geometry spread– first among mathematicians, then among scientists and philosophers– the Diamond Theory began a long decline that continues today.
Factors outside mathematics have contributed to this decline. Euclidean geometry had never been the Diamond Theory’s only ally. In the eighteenth century other fields had seemed to possess diamonds, too; when many of these turned out to be manmade, the Diamond Theory was undercut. And unlike earlier periods in history, when intellectual shocks came only occasionally, received truths have, since the eighteenth century, been found wanting at a dizzying rate, creating an impression that perhaps no knowledge is stable.
Other factors notwithstanding, nonEuclidean geometry remains, I think, for those who have heard of it, the single most powerful argument against the Diamond Theory*– first, because it overthrows what had always been the strongest argument in favor of the Diamond Theory, the objective truth of Euclidean geometry; and second, because it does so not by showing Euclidean geometry to be false, but by showing it to be merely uncertain.” —The NonEuclidean Revolution, p. 255
H. S. M. Coxeter, 1987, introduction to Trudeau’s book—
“There is a pleasantly discursive treatment of Pontius Pilate’s unanswered question ‘What is truth?’.”
As noted here on Oct. 8, 2008 (A Yom Kippur Meditation), Coxeter was aware in 1987 of a more technical use of the phrase “diamond theory” that is closely related to…
At right below, an image from the opening of Fox Studios Australia in Sydney on November 7, 1999. The Fox ceremonies included, notably, Kylie Minogue singing “Diamonds are a Girl’s Best Friend.”
For the mathematical properties of the red windmill (moulin rouge) figure at left, see Diamond Theory. 
First you will need to
prepare your sacred space….
Calling the Corners (or Quarters)
is something you will always do.”
Pilate Goes
to Kindergarten
“There is a pleasantly discursive
treatment of Pontius Pilate’s
unanswered question
‘What is truth?’.”
— H. S. M. Coxeter, 1987,
introduction to Trudeau’s
remarks on the “Story Theory“
of truth as opposed to the
“Diamond Theory” of truth in
The NonEuclidean Revolution
Consider the following question in a paper cited by V. S. Varadarajan:
E. G. Beltrametti, “Can a finite geometry describe physical spacetime?” Universita degli studi di Perugia, Atti del convegno di geometria combinatoria e sue applicazioni, Perugia 1971, 57–62.
Simplifying:
“Can a finite geometry describe physical space?”
Simplifying further:
“Yes. Vide ‘The Eightfold Cube.'”
Getting All
the Meaning In
Webpage heading for the
2009 meeting of the
American Comparative
Literature Association:
The mysterious symbols on
the above map suggest the
following reflections:
From A Cure of the Mind: The Poetics of Wallace Stevens, by Theodore Sampson, published by Black Rose Books Ltd., 2000–
Page x:
"… if what he calls 'the spirit's alchemicana' (CP [Collected Poems] 471) addresses itself to the irrational element in poetry, to what extent is such an element dominant in his theory and practice of poetry, and therefore in what way is Stevens' intricate verbal music dependent on his irrational use of language– a 'pure rhetoric of a language without words?' (CP 374)?"
From "'When Novelists Become Cubists:' The Prose Ideograms of Guy Davenport," by Andre Furlani:
Laurence Zachar argues that Davenport's writing is situated "aux frontieres intergeneriques" where manifold modes are brought into concord: "L'etonnant chez Davenport est la facon don't ce materiau qui parait l'incarnation meme du chaos– hermetique, enigmatique, obscur, avec son tropplein de references– se revele en fait etre construit, ordonne, structure. Plus l'on s'y plonge, et plus l'on distingue de cohesion dans le texte." 'What astonishes in Davenport is the way in which material that seems the very incarnation of chaos– hermetic, enigmatic, obscure, with its proliferation of allusions– in fact reveals itself to be constructed, organized, structured. The more one immerses oneself in them the more one discerns the texts' cohesion.' (62). Davenport also works along the intergeneric border between text and graphic, for he illustrates many of his texts. (1) "The prime use of words is for imagery: my writing is drawing," he states in an interview (Hoeppfner 123). Visual imagery is not subordinated to writing in Davenport, who draws on the assemblage practice of superimposing image and writing. "I trust the image; my business is to get it onto the page," he writes in the essay "Ernst Machs Max Ernst." "A page, which I think of as a picture, is essentially a texture of images. […] The text of a story is therefore a continuous graph, kin to the imagist poem, to a collage (Ernst, Willi Baumeister, El Lissitzky), a page of Pound, a Brakhage film" (Geography 37475). Note: (1.) Davenport is an illustrator of books (such as Hugh Kenner's The Stoic Comedians and The Counterfeiters) and journals (such as The Kenyon Review, Parnassus, and Paideuma). His art is the subject of Erik Anderson Reece's monograph, A Balance of Quinces, which reveals the inseparable relationship between Davenport's literary and pictorial work. References: Davenport, Guy. The Geography of the Imagination. San Francisco: North Point Press, 1981. Rpt. New York: Pantheon, 1992. Hoepffner, Bernard. "Pleasant Hill: An Interview with Guy Davenport." Conjunctions 24 (1995): 11824. Reece, Erik Anderson. A Balance of Quinces: The Paintings and Drawings of Guy Davenport. New York: New Directions, 1996. Zachar, Laurence. "Guy Davenport: Une Mosaique du genres." Recherches Anglaises et NordAmericaines 21 (1994): 5163. 
"… when novelists become Cubists; that is, when they see the possibilities of making a hieroglyph, a coherent symbol, an ideogram of the total work. A symbol comes into being when an artist sees that it is the only way to get all the meaning in."
— Guy Davenport, The Geography of the Imagination
The Rest
of the Story
Today's previous entry discussed the hermeneutics of the midday NY and PA lottery numbers.
Lotteries on Reba's birthday, 2009 
Pennsylvania (No revelation) 
New York (Revelation) 
Midday (No belief) 
No belief, no revelation 726 
Revelation without belief 378 
Evening (Belief) 
Belief without revelation 006 
Belief and revelation 091 
Interpretations of the evening numbers–
The PA evening number, 006, may be viewed as a followup to the PA midday 726 (or 7/26, the birthday of Kate Beckinsale and Carl Jung). Here 006 is the prestigious "00" number assigned to Beckinsale.
The NY evening number, 091, may be viewed as a followup to the NY midday 378 (the number of pages in The Innermost Kernel by Suzanne Gieser, published by Springer, 2005)–
Page 91: The entire page is devoted to the title of the book's Part 3– "The Copenhagen School and Psychology"–
The next page begins: "With the crisis of physics, interest in epistemological and psychological questions grew among many theoretical physicists. This interest was particularly marked in the circle around Niels Bohr."
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.
"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"
The square above is marked
with a graphic design
related to the fourdiamond
figure of Jung's Aion.
The Square of Oppositon
at Stanford Encylopedia of Philosophy
The Square of Opposition
in its original form
"The diagram above is from a ninth century manuscript of Apuleius' commentary on Aristotle's Perihermaneias, probably one of the oldest surviving pictures of the square."
— Edward Buckner at The Logic Museum
From the webpage "Semiotics for Beginners: Paradigmatic Analysis," by Daniel Chandler:
The Semiotic Square
"The structuralist semiotician Algirdas Greimas introduced the semiotic square (which he adapted from the 'logical square' of scholastic philosophy) as a means of analysing paired concepts more fully (Greimas 1987,* xiv, 49). The semiotic square is intended to map the logical conjunctions and disjunctions relating key semantic features in a text. Fredric Jameson notes that 'the entire mechanism… is capable of generating at least ten conceivable positions out of a rudimentary binary opposition' (in Greimas 1987,* xiv). Whilst this suggests that the possibilities for signification in a semiotic system are richer than the either/or of binary logic, but that [sic] they are nevertheless subject to 'semiotic constraints' – 'deep structures' providing basic axes of signification."
* Greimas, Algirdas (1987): On Meaning: Selected Writings in Semiotic Theory (trans. Paul J Perron & Frank H Collins). London: Frances Pinter
Another version of the semiotic square:
Here is a more explicit figure representing the Klein group:
There is also the logical
diamond of opposition —
A semiotic (as opposed to logical)
diamond has been used to illustrate
remarks by Fredric Jameson,
a Marxist literary theorist:
"Introduction to Algirdas Greimas, Module on the Semiotic Square," by Dino Felluga at Purdue University–
The semiotic square has proven to be an influential concept not only in narrative theory but in the ideological criticism of Fredric Jameson, who uses the square as "a virtual map of conceptual closure, or better still, of the closure of ideology itself" ("Foreword"* xv). (For more on Jameson, see the [Purdue University] Jameson module on ideology.) Greimas' schema is useful since it illustrates the full complexity of any given semantic term (seme). Greimas points out that any given seme entails its opposite or "contrary." "Life" (s_{1}) for example is understood in relation to its contrary, "death" (s_{2}). Rather than rest at this simple binary opposition (S), however, Greimas points out that the opposition, "life" and "death," suggests what Greimas terms a contradictory pair (S), i.e., "notlife" (s_{1}) and "notdeath" (s_{2}). We would therefore be left with the following semiotic square (Fig. 1): As Jameson explains in the Foreword to Greimas' On Meaning, "s_{1} and s_{2}"—which in this example are taken up by "notdeath" and "notlife"—"are the simple negatives of the two dominant terms, but include far more than either: thus 'nonwhite' includes more than 'black,' 'nonmale' more than 'female'" (xiv); in our example, notlife would include more than merely death and notdeath more than life.
* Jameson, Fredric. "Foreword." On Meaning: Selected Writings in Semiotic Theory. By Algirdas Greimas. Trans. Paul J. Perron and Frank H. Collins. Minneapolis: U of Minnesota P, 1976. 
— The Gameplayers of Zan, by M.A. Foster
"For every kind of vampire,
there is a kind of cross."
— Thomas Pynchon,
Gravity's Rainbow
Crosses used by semioticians
to baffle their opponents
are illustrated above.
Some other kinds of crosses,
and another kind of opponent:
Monday, July 11, 2005
Logos
for St. Benedict's Day Click on either of the logos below for religious meditations– on the left, a Jewish meditation from the Conference of Catholic Bishops; on the right, an Aryan meditation from Stormfront.org. Both logos represent different embodiments of the "story theory" of truth, as opposed to the "diamond theory" of truth. Both logos claim, in their own ways, to represent the eternal Logos of the Christian religion. I personally prefer the "diamond theory" of truth, represented by the logo below. See also the previous entry Sunday, July 10, 2005 Mathematics
and Narrative Click on the title for a narrative about Nikolaos K. Artemiadis,
"First of all, I'd like to
— Remark attributed to Plato

The Origin of Change
A note on the figure
from this morning's sermon:
"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"
Angels, Demons,
"Symbology"
"On Monday morning, 9 March, after visiting the Mayor of Rome and the Municipal Council on the Capitoline Hill, the Holy Father spoke to the Romans who gathered in the square outside the Senatorial Palace…
'… a verse by Ovid, the great Latin poet, springs to mind. In one of his elegies he encouraged the Romans of his time with these words:
"Perfer et obdura: multo graviora tulisti."
"Hold out and persist:
you have got through
far more difficult situations."
(Tristia, Liber V, Elegia XI, verse 7).'"
Note the colorinterchange
symmetry of each symbol
under 180degree rotation.
Related material:
The Illuminati Diamond:
A possible source for Brown's term "symbology" is a 1995 web page, "The Rotation of the Elements," by one "John Opsopaus." (Cf. Art History Club.)
"The four qualities are the key to understanding the rotation of the elements and many other applications of the symbology of the four elements." –John Opsopaus
* "…ambigrams were common in symbology…." —Angels & Demons
Humorism
"Always with a
little humor."
— Dr. Yen Lo
From Temperament: A Brief Survey
For other interpretations
of the above shape, see
The Illuminati Diamond.
from Jung's Aion:
As for rotation, see the ambigrams in Dan Brown's Angels & Demons (to appear as a film May 15) and the following figures:
A related note on
"Angels & Demons"
director Ron Howard:
This journal on October 8, 2008, at noon: “There is a pleasantly discursive treatment of Pontius Pilate’s unanswered question ‘What is truth?'” Trudeau’s 1987 book uses the phrase “diamond theory” to denote the philosophical theory, common since Plato and Euclid, that there exist truths (which Trudeau calls “diamonds”) that are certain and eternal– for instance, the truth in Euclidean geometry that the sum of a triangle’s angles is 180 degrees. Insidehighered.com on “Future readers may consider Updike our era’s Mozart; Mozart was once written off as a tooprolific composer of ‘charming nothings,’ and some speak of Updike that way.” — Comment by BPJ 
Updike died on January 27.
On the same date,
Mozart was born.
Requiem
Mr. Best entered, tall, young, mild, light. He bore in his hand with grace a notebook, new, large, clean, bright. — James Joyce, Ulysses, 
The New York Times of Sunday, May 6, 2007, on a writer of pulp fiction:
His early novels, written in two weeks or less, were published in doubledecker Ace paperbacks that included two books in one, with a lurid cover for each. “If the Holy Bible was printed as an Ace Double,” an editor once remarked, “it would be cut down to two 20,000word halves with the Old Testament retitled as ‘Master of Chaos’ and the New Testament as ‘The Thing With Three Souls.'”
Epigraph for Part One:
Epigraph for Part Two:
“Beware lest you believe that you can comprehend the Incomprehensible….“
Death of a Classmate
Michael Crichton,
Harvard College, 1964
Authors Michael Crichton and
David Foster Wallace in today’s
New York Times obituaries
The Times’s remarks above
on the prose styles of
Crichton and Wallace–
“compelling formula” vs.
“intricate complexity”–
suggest the following works
of visual art in memory
of Crichton.
“Crystal”—
Some philosophical
remarks related to
the Harvard background
that Crichton and I share–
Hitler’s Still Point
and
The Crimson Passion.
Related material:
Dec. 16, 2003—
Kaleidoscope turning… 
Serious Numbers
A Yom Kippur
Meditation
"When times are mysterious
Serious numbers
Will always be heard."
— Paul Simon,
"When Numbers Get Serious"
"There is a pleasantly discursive treatment of Pontius Pilate's unanswered question 'What is truth?'"
— H. S. M. Coxeter, introduction to Richard J. Trudeau's remarks on the "story theory" of truth as opposed to the "diamond theory" of truth in The NonEuclidean Revolution
Trudeau's 1987 book uses the phrase "diamond theory" to denote the philosophical theory, common since Plato and Euclid, that there exist truths (which Trudeau calls "diamonds") that are certain and eternal– for instance, the truth in Euclidean geometry that the sum of a triangle's angles is 180 degrees. As the excerpt below shows, Trudeau prefers what he calls the "story theory" of truth–
"There are no diamonds. People make up stories about what they experience. Stories that catch on are called 'true.'"
(By the way, the phrase "diamond theory" was used earlier, in 1976, as the title of a monograph on geometry of which Coxeter was aware.)
What does this have to do with numbers?
Pilate's skeptical tone suggests he may have shared a certain confusion about geometric truth with thinkers like Trudeau and the slave boy in Plato's Meno. Truth in a different part of mathematics– elementary arithmetic– is perhaps more easily understood, although even there, the existence of what might be called "nonEuclidean number theory"– i.e., arithmetic over finite fields, in which 1+1 can equal zero– might prove baffling to thinkers like Trudeau.
Trudeau's book exhibits, though it does not discuss, a less confusing use of numbers– to mark the location of pages. For some philosophical background on this version of numerical truth that may be of interest to devotees of the Semitic religions on this evening's High Holiday, see Zen and Language Games.
For uses of numbers that are more confusing, see– for instance– the new website The Daily Beast and the old website Story Theory and the Number of the Beast.
Preview of a Tom Stoppard play presented at Town Hall in Manhattan on March 14, 2008 (Pi Day and Einstein’s birthday):
The play’s title, “Every Good Boy Deserves Favour,” is a mnemonic for the notes of the treble clef EGBDF.
The place, Town Hall, West 43rd Street. The time, 8 p.m., Friday, March 14. One single performance only, to the tinkle– or the clang?– of a triangle. Echoing perhaps the clangclack of Warsaw Pact tanks muscling into Prague in August 1968.
The “u” in favour is the British way, the Stoppard way, “EGBDF” being “a Play for Actors and Orchestra” by Tom Stoppard (words) and André Previn (music).
And what a play!– as luminescent as always where Stoppard is concerned. The music component of the onenighter at Town Hall– a showcase for the Boston University College of Fine Arts– is by a 47piece live orchestra, the significant instrument being, well, a triangle.
When, in 1974, André Previn, then principal conductor of the London Symphony, invited Stoppard “to write something which had the need of a live fulltime orchestra onstage,” the 36yearold playwright jumped at the chance.
One hitch: Stoppard at the time knew “very little about ‘serious’ music… My qualifications for writing about an orchestra,” he says in his introduction to the 1978 Grove Press edition of “EGBDF,” “amounted to a spell as a triangle player in a kindergarten percussion band.”
Review of the same play as presented at Chautauqua Institution on July 24, 2008:
“Stoppard’s modus operandi– to teasingly introduce numerous clever tidbits designed to challenge the audience.”
— Jane Vranish, Pittsburgh PostGazette, Saturday, August 2, 2008
“The leader of the band is tired
And his eyes are growing old
But his blood runs through
My instrument
And his song is in my soul.”
— Dan Fogelberg
“He’s watching us all the time.”
Finnegans Wake, Book II, Episode 2, pp. 296297: I’ll make you to see figuratleavely the whome of your eternal geomater. And if you flung her headdress on her from under her highlows you’d wheeze whyse Salmonson set his seel on a hexengown.^{1} Hissss!, Arrah, go on! Fin for fun! ^{1} The chape of Doña Speranza of the Nacion. 
“…the principle of taking and giving, of learning and teaching, of listening and storytelling, in a word: of reciprocity…. … E. M. Forster famously advised his readers, ‘Only connect.’ ‘Reciprocity’ would be Michael Kruger’s succinct philosophy, with all that the word implies.” — William Boyd, review of Himmelfarb, a novel by Michael Kruger, in The New York Times Book Review, October 30, 1994 Last year’s entry on this date:
The picture above is of the complete graph K_{6 }… Six points with an edge connecting every pair of points… Fifteen edges in all. Diamond theory describes how the 15 twoelement subsets of a sixelement set (represented by edges in the picture above) may be arranged as 15 of the 16 parts of a 4×4 array, and how such an array relates to grouptheoretic concepts, including Sylvester’s synthematic totals as they relate to constructions of the Mathieu group M_{24}. If diamond theory illustrates any general philosophical principle, it is probably the interplay of opposites…. “Reciprocity” in the sense of Lao Tzu. See Reciprocity and Reversal in Lao Tzu. For a sense of “reciprocity” more closely related to Michael Kruger’s alleged philosophy, see the Confucian concept of Shu (Analects 15:23 or 24) described in Kruger’s novel is in part about a Jew: the quintessential Jewish symbol, the star of David, embedded in the K_{6} graph above, expresses the reciprocity of male and female, as my May 2003 archives illustrate. The star of David also appears as part of a graphic design for cubes that illustrate the concepts of diamond theory: Click on the design for details. Those who prefer a Jewish approach to physics can find the star of David, in the form of K_{6}, applied to the sixteen 4×4 Dirac matrices, in A Graphical Representation The star of David also appears, if only as a heuristic arrangement, in a note that shows generating partitions of the affine group on 64 points arranged in two opposing triplets. Having thus, as the New York Times advises, paid tribute to a Jewish symbol, we may note, in closing, a much more sophisticated and subtle concept of reciprocity due to Euler, Legendre, and Gauss. See 
“Finn MacCool ate the Salmon of Knowledge.”
Wikipedia:
“George Salmon spent his boyhood in Cork City, Ireland. His father was a linen merchant. He graduated from Trinity College Dublin at the age of 19 with exceptionally high honours in mathematics. In 1841 at age 21 he was appointed to a position in the mathematics department at Trinity College Dublin. In 1845 he was appointed concurrently to a position in the theology department at Trinity College Dublin, having been confirmed in that year as an Anglican priest.”
Related material:
Kindergarten Theology,
Arrangements for
56 Triangles.
For more on the
arrangement of
triangles discussed
in Finnegans Wake,
see Log24 on Pi Day,
March 14, 2008.
Happy birthday,
Martin Sheen.
The HSBC Logo Designer — Henry Steiner He is an internationally recognized corporate identity consultant. Based in Hong Kong, his work for clients such as HongkongBank, IBM and Unilever is a major influence in Pacific Rim design. Born in Austria and raised in New York, Steiner was educated at Yale under Paul Rand and attended the Sorbonne as a Fulbright Fellow. He is a past President of Alliance Graphique Internationale. Other professional affiliations include the American Institute of Graphic Arts, Chartered Society of Designers, Design Austria, and the New York Art Directors’ Club. His CrossCultural Design: Communicating in the Global Marketplace was published by Thames and Hudson (1995). 
Charles Taylor, “… the object sets up
See also Talking of Michelangelo.

Related material
from today —
Escape from a
cartoon graveyard:
for lifetime achievement
in arts and philosophy
this year goes to
Charles Taylor,
Montreal philosophy professor.
“The Kyoto Prize has been given in three domains since 1984:
advanced technology, basic sciences, and the arts and philosophy.
It is administered by the Inamori Foundation, whose president,
Kazuo Inamori, is founder and chairman emeritus of Kyocera and
KDDI Corporation, two Japanese telecommunications giants.”
“The Kyocera brand symbol is composed of a corporate mark 
Related material —
Charles Taylor,
“Epiphanies of Modernism,”
Chapter 24 of Sources of the Self
(Cambridge U. Press, 1989, p. 477) —
“… the object sets up
a kind of frame or space or field
within which there can be epiphany.”
See also Talking of Michelangelo.
On April 16, the Pope’s birthday, the evening lottery number in Pennsylvania was 441. The Log24 entries of April 17 and April 18 supplied commentaries based on 441’s incarnation as a page number in an edition of Heidegger’s writings. Here is a related commentary on a different incarnation of 441. (For a context that includes both today’s commentary and those of April 17 and 18, see GianCarlo Rota– a Heidegger scholar as well as a mathematician– on mathematical Lichtung.)
From R. D. Carmichael, Introduction to the Theory of Groups of Finite Order (Boston, Ginn and Co., 1937)– an exercise from the final page, 441, of the final chapter, “Tactical Configurations”–
“23. Let G be a multiply transitive group of degree n whose degree of transitivity is k; and let G have the property that a set S of m elements exists in G such that when k of the elements S are changed by a permutation of G into k of these elements, then all these m elements are permuted among themselves; moreover, let G have the property P, namely, that the identity is the only element in G which leaves fixed the n – m elements not in S. Then show that G permutes the m elements S into
m(m – 1) … (m – k + 1)
This exercise concerns an important mathematical structure said to have been discovered independently by the American Carmichael and by the German Ernst Witt.
For some perhaps more comprehensible material from the preceding page in Carmichael– 440– see Diamond Theory in 1937.
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