The previous post suggests a flashback to June 8, 2014 —
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In the simplest case of a projective space Vide Classical Geometry in Light of Galois Geometry.
* The two types of lines named are derived from
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Friday, April 3, 2026
Diagon Alley
Sunday, December 17, 2023
Scholastic Stochastic
"Let G be the group generatied by … arbitrarily mixing random
permutations of rows and of columns with random permutations
of the four 2×2 quadrants…. G is isomorphic to the affine group
on V4 (GF(2))." — Steven H. Cullinane, October 1978.
Compare and contrast with another discussion of randomness and
affine groups, from American Mathematical Monthly, November 1995 —
Related material not cited by Poole in 1995 —
J. E. Johnson, "Markov-type Lie groups in GL(n, R),"
J. Math. Phys. 26, 252-257 (1985).
The above Johnson citation is from an article that also discusses
the work of Poole —
Irene Paniello, "On Actions on Cubic Stochastic Matrices,"
arXiv preprint dated April 22, 2017. Published in
Markov Processes and Related Fields, 2017, v.23, Issue 2, 325-348.
Saturday, September 23, 2023
The Cullinane Diamond Theorem at Wikipedia
This post was prompted by the recent removal of a reference to
the theorem on the Wikipedia "Diamond theorem" disambiguation
page. The reference, which has been there since 2015, was removed
because it linked to an external source (Encyclopedia of Mathematics)
instead of to a Wikipedia article.
For anyone who might be interested in creating a Wikipedia article on
my work, here are some facts that might be reformatted for that website . . .
https://en.wikipedia.org/wiki/
User:Cullinane/sandbox —
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Cullinane diamond theorem The theorem uses finite geometry to explain some symmetry properties of some simple graphic designs, like those found in quilts, that are constructed from chevrons or diamonds. The theorem was first discovered by Steven H. Cullinane in 1975 and was published in 1977 in Computer Graphics and Art. The theorem was also published as an abstract in 1979 in Notices of the American Mathematical Society. The symmetry properties described by the theorem are related to those of the Miracle Octad Generator of R. T. Curtis. The theorem is described in detail in the Encyclopedia of Mathematics article "Cullinane diamond theorem." References Steven H. Cullinane, "Diamond theory," Computer Graphics and Art, Vol. 2, No. 1, February 1977, pages 5-7. _________, Abstract 79T-A37, "Symmetry invariance in a diamond ring," Notices of the American Mathematical Society, February 1979, pages A-193, 194. _________, "Cullinane diamond theorem," Encyclopedia of Mathematics. R. T. Curtis, A new combinatorial approach to M24, Mathematical Proceedings of the Cambridge Philosophical Society, 1976, Vol. 79, Issue 1, pages 24-42. |
Thursday, March 4, 2021
Teaching the Academy to See
“Art bears the same relationship to society
that the dream bears to mental life. . . .
Like art, the dream mediates between order
and chaos. So, it is half chaos. That is why
it is not comprehensible. It is a vision, not
a fully fledged articulated production.
Those who actualize those half-born visions
into artistic productions are those who begin
to transform what we do not understand into
what we can at least start to see.”
— A book published on March 2, 2021:
Beyond Order , by Jordan Peterson
The inarticulate, in this case, is Rosalind Krauss:
A “raid on the inarticulate” published in Notices of the
American Mathematical Society in the February 1979 issue —
Friday, February 26, 2021
Non-Chaos Non-Magic
For fans of “WandaVision” —
“1978 was perhaps the seminal year in the origin of chaos magic. . . .”
— Wikipedia article on Chaos Magic
Non-Chaos Non-Magic from Halloween 1978 —
Related material —
A doctoral student of a different Peter Cameron —
( Not to be confused with The Tin Man’s Hat. )
Monday, August 31, 2020
Seals: Compare and Contrast
Saturday, March 7, 2020
The “Octad Group” as Symmetries of the 4×4 Square
From "Mathieu Moonshine and Symmetry Surfing" —
(Submitted on 29 Sep 2016, last revised 22 Jan 2018)
by Matthias R. Gaberdiel (1), Christoph A. Keller (2),
and Hynek Paul (1)
(1) Institute for Theoretical Physics, ETH Zurich
(2) Department of Mathematics, ETH Zurich
https://arxiv.org/abs/1609.09302v2 —
"This presentation of the symmetry groups Gi is
particularly well-adapted for the symmetry surfing
philosophy. In particular it is straightforward to
combine them into an overarching symmetry group G
by combining all the generators. The resulting group is
the so-called octad group
G = (Z2)4 ⋊ A8 .
It can be described as a maximal subgroup of M24
obtained by the setwise stabilizer of a particular
'reference octad' in the Golay code, which we take
to be O9 = {3,5,6,9,15,19,23,24} ∈ 𝒢24. The octad
subgroup is of order 322560, and its index in M24
is 759, which is precisely the number of
different reference octads one can choose."
This "octad group" is in fact the symmetry group of the affine 4-space over GF(2),
so described in 1979 in connection not with the Golay code but with the geometry
of the 4×4 square.* Its nature as an affine group acting on the Golay code was
known long before 1979, but its description as an affine group acting on
the 4×4 square may first have been published in connection with the
Cullinane diamond theorem and Abstract 79T-A37, "Symmetry invariance in a
diamond ring," by Steven H. Cullinane in Notices of the American Mathematical
Society , February 1979, pages A-193, 194.
* The Galois tesseract .
Update of March 15, 2020 —
Conway and Sloane on the "octad group" in 1993 —
Friday, March 1, 2019
Wikipedia Scholarship (Continued)
This post continues a post from yesterday on the square model of
PG(3,2) that apparently first appeared (presented as such*) in . . .
Cullinane, "Symmetry invariance in a diamond ring,"
Notices of the AMS , pp. A193-194, Feb. 1979.
Yesterday's Wikipedia presentation of the square model was today
revised by yet another anonymous author —
Revision history accounting for the above change from yesterday —
The jargon "rm OR" means "remove original research."
The added verbiage about block designs is a smokescreen having
nothing to do with the subject, which is square representation
of the 35 points and lines.
* The 35 squares, each consisting of four 4-element subsets, appeared earlier
in the Miracle Octad Generator (MOG) of R. T. Curtis (published in 1976).
They were not at that time presented as constituting a finite geometry,
either affine (AG(4,2)) or projective (PG(3,2)).
Friday, February 22, 2019
Back Issues of AMS Notices
From the online home page of the new March issue —
For instance . . .
Related material now at Wikipedia —
Monday, December 21, 2015
The Eppstein Edit
A Wikipedia edit today by David Eppstein, a professor
at the University of California, Irvine:
See the Fano-plane page before and after the Eppstein edit.
Eppstein deleted my Dec. 6 Fano 3-space image as well as
today's Fano-plane image. He apparently failed to read the
explanatory notes for both the 3-space model and the
2-space model. The research he refers to was original
(in 1979) but has been published for some time now in the
online Encyclopedia of Mathematics, as he could have
discovered by following a link in the notes for the 3-space
model.
For a related recent display of ignorance, see Hint of Reality.
Happy darkest night.
Wednesday, September 3, 2014
Image and Logic, Part Deux
The title refers to the previous post.
Click image for some context.
For further context, see some
mathematics from Halloween 1978.
See also May 12, 2014.
Sunday, June 8, 2014
Vide
"The relevance of a geometric theorem is determined by what the theorem
tells us about space, and not by the eventual difficulty of the proof."
— Gian-Carlo Rota discussing the theorem of Desargues
What space tells us about the theorem :
In the simplest case of a projective space (as opposed to a plane ),
there are 15 points and 35 lines: 15 Göpel lines and 20 Rosenhain lines.*
The theorem of Desargues in this simplest case is essentially a symmetry
within the set of 20 Rosenhain lines. The symmetry, a reflection
about the main diagonal in the square model of this space, interchanges
10 horizontally oriented (row-based) lines with 10 corresponding
vertically oriented (column-based) lines.
Vide Classical Geometry in Light of Galois Geometry.
* Update of June 9: For a more traditional nomenclature, see (for instance)
R. Shaw, 1995. The "simplest case" link above was added to point out that
the two types of lines named are derived from a natural symplectic polarity
in the space. The square model of the space, apparently first described in
notes written in October and December, 1978, makes this polarity clearly visible:
Tuesday, June 3, 2014
Galois Matrices
The webpage Galois.us, on Galois matrices , has been created as
a starting point for remarks on the algebra (as opposed to the geometry)
underlying the rings of matrices mentioned in AMS abstract 79T-A37,
“Symmetry invariance in a diamond ring.”
See also related historical remarks by Weyl and by Atiyah.
Friday, February 28, 2014
Code
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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:
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"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 re-examination 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. 154-164, 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.
Thursday, February 6, 2014
The Representation of Minus One
For the late mathematics educator Zoltan Dienes.
“There comes a time when the learner has identified
the abstract content of a number of different games
and is practically crying out for some sort of picture
by means of which to represent that which has been
gleaned as the common core of the various activities.”
— Article by “Melanie” at Zoltan Dienes’s website
Dienes reportedly died at 97 on Jan. 11, 2014.
From this journal on that date —
A star figure and the Galois quaternion.
The square root of the former is the latter.
Update of 5:01 PM ET Feb. 6, 2014 —
An illustration by Dienes related to the diamond theorem —
See also the above 15 images in …
… and versions of the 4×4 coordinatization in The 4×4 Relativity Problem
(Jan. 17, 2014).
Sunday, February 2, 2014
Diamond Theory Roulette
A ReCode Project program from Radamés Ajna of São Paulo —
At the program's webpage, click the image to
generate random permutations of rows, columns,
and quadrants. Note the resulting image's ordinary
or color-interchange symmetry.
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Friday, January 17, 2014
The 4×4 Relativity Problem
The sixteen-dot square array in yesterday’s noon post suggests
the following remarks.
“This is the relativity problem: to fix objectively a class of
equivalent coordinatizations and to ascertain the group of
transformations S mediating between them.”
— Hermann Weyl, The Classical Groups ,
Princeton University Press, 1946, p. 16
The Galois tesseract appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977—
The 1977 matrix Q is echoed in the following from 2002—
A different representation of Cullinane’s 1977 square model of the
16-point affine geometry over the two-element Galois field GF(2)
is supplied by Conway and Sloane in Sphere Packings, Lattices and Groups
(first published in 1988) :
Here a, b, c, d are basis vectors in the vector 4-space over GF(2).
(For a 1979 version of this vector space, see AMS Abstract 79T-A37.)
See also a 2011 publication of the Mathematical Association of America —
Saturday, September 21, 2013
Mathematics and Narrative (continued)
Mathematics:
A review of posts from earlier this month —
Wednesday, September 4, 2013
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Narrative:
Aooo.
Happy birthday to Stephen King.
Thursday, September 5, 2013
Moonshine II
(Continued from yesterday)
The foreword by Wolf Barth in the 1990 Cambridge U. Press
reissue of Hudson's 1905 classic Kummer's Quartic Surface
covers some of the material in yesterday's post Moonshine.
The distinction that Barth described in 1990 was also described, and illustrated,
in my 1986 note "Picturing the smallest projective 3-space." The affine 4-space
over the the finite Galois field GF(2) that Barth describes was earlier described—
within a 4×4 array like that pictured by Hudson in 1905— in a 1979 American
Mathematical Society abstract, "Symmetry invariance in a diamond ring."
"The distinction between Rosenhain and Goepel tetrads
is nothing but the distinction between isotropic and
non-isotropic planes in this affine space over the finite field."
The 1990 paragraph of Barth quoted above may be viewed as a summary
of these facts, and also of my March 17, 2013, note "Rosenhain and Göpel
Tetrads in PG(3,2)."
Saturday, September 3, 2011
The Galois Tesseract (continued)
A post of September 1, The Galois Tesseract, noted that the interplay
of algebraic and geometric properties within the 4×4 array that forms
two-thirds of the Curtis Miracle Octad Generator (MOG) may first have
been described by Cullinane (AMS abstract 79T-A37, Notices , Feb. 1979).
Here is some supporting material—
The passage from Carmichael above emphasizes the importance of
the 4×4 square within the MOG.
The passage from Conway and Sloane, in a book whose first edition
was published in 1988, makes explicit the structure of the MOG's
4×4 square as the affine 4-space over the 2-element Galois field.
The passage from Curtis (1974, published in 1976) describes 35 sets
of four "special tetrads" within the 4×4 square of the MOG. These
correspond to the 35 sets of four parallel 4-point affine planes within
the square. Curtis, however, in 1976 makes no mention of the affine
structure, characterizing his 140 "special tetrads" rather by the parity
of their intersections with the square's rows and columns.
The affine structure appears in the 1979 abstract mentioned above—
The "35 structures" of the abstract were listed, with an application to
Latin-square orthogonality, in a note from December 1978—
See also a 1987 article by R. T. Curtis—
Further elementary techniques using the miracle octad generator, by R. T. Curtis. Abstract:
“In this paper we describe various techniques, some of which are already used by devotees of the art, which relate certain maximal subgroups of the Mathieu group M24, as seen in the MOG, to matrix groups over finite fields. We hope to bring out the wealth of algebraic structure* underlying the device and to enable the reader to move freely between these matrices and permutations. Perhaps the MOG was mis-named as simply an ‘octad generator’; in this paper we intend to show that it is in reality a natural diagram of the binary Golay code.”
(Received July 20 1987)
– Proceedings of the Edinburgh Mathematical Society (Series 2) (1989), 32: 345-353
* For instance:
Update of Sept. 4— This post is now a page at finitegeometry.org.
Saturday, November 20, 2010
Search
An Epic Search for Truth
— Subtitle of Logicomix , a work reviewed in the December 2010 Notices of the American Mathematical Society (see previous post).
Some future historian of mathematics may contrast the lurid cover of the December 2010 Notices
Excerpts from Logicomix
with the 1979 cover found in a somewhat less epic search —
Larger view of Google snippet —
For some purely mathematical background, see Finite Geometry of the Square and Cube.
For some background related to searches for truth, see "Coxeter + Trudeau" in this journal.
Sunday, December 13, 2009
Ein Kampf
(Click on image for video.)
See also Tyger! Tyger! and
The Stars My Destination.
Hitler's Peer Review–
See also Abstract 79T-A37
and Scientific American.
Saturday, July 29, 2006
Saturday July 29, 2006
Big Rock
Thanks to Ars Mathematica, a link to everything2.com:
“In mathematics, a big rock is a result which is vastly more powerful than is needed to solve the problem being considered. Often it has a difficult, technical proof whose methods are not related to those of the field in which it is applied. You say ‘I’m going to hit this problem with a big rock.’ Sard’s theorem is a good example of a big rock.”
Another example:
Properties of the Monster Group of R. L. Griess, Jr., may be investigated with the aid of the Miracle Octad Generator, or MOG, of R. T. Curtis. See the MOG on the cover of a book by Griess about some of the 20 sporadic groups involved in the Monster:
The MOG, in turn, illustrates (via Abstract 79T-A37, Notices of the American Mathematical Society, February 1979) the fact that the group of automorphisms of the affine space of four dimensions over the two-element field is also the natural group of automorphisms of an arbitrary 4×4 array.
This affine group, of order 322,560, is also the natural group of automorphisms of a family of graphic designs similar to those on traditional American quilts. (See the diamond theorem.)
This top-down approach to the diamond theorem may serve as an illustration of the “big rock” in mathematics.
For a somewhat simpler, bottom-up, approach to the theorem, see Theme and Variations.
For related literary material, see Mathematics and Narrative and The Diamond as Big as the Monster.
Saturday, June 4, 2005
Saturday June 4, 2005
The 4×4 Square:
French Perspectives
Earendil_Silmarils:
Les Anamorphoses:
"Pour construire un dessin en perspective,
le peintre trace sur sa toile des repères:
la ligne d'horizon (1),
le point de fuite principal (2)
où se rencontre les lignes de fuite (3)
et le point de fuite des diagonales (4)."
_______________________________
Serge Mehl,
Perspective &
Géométrie Projective:
"… la géométrie projective était souvent
synonyme de géométrie supérieure.
Elle s'opposait à la géométrie
euclidienne: élémentaire…
La géométrie projective, certes supérieure
car assez ardue, permet d'établir
de façon élégante des résultats de
la géométrie élémentaire."
Similarly…
Finite projective geometry
(in particular, Galois geometry)
is certainly superior to
the elementary geometry of
quilt-pattern symmetry
and allows us to establish
de façon élégante
some results of that
elementary geometry.
Other Related Material…
from algebra rather than
geometry, and from a German
rather than from the French:
"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 U. Press, 1946
Evariste Galois
Weyl also says that the profound branch
of mathematics known as Galois theory
relativity theory for the set Sigma,
a set which, by its discrete and
finite character, is conceptually
so much simpler than the
infinite set of points in space
or space-time dealt with
by ordinary relativity theory."
— Weyl, Symmetry,
Princeton U. Press, 1952
Metaphor and Algebra…
"Perhaps every science must
start with metaphor
and end with algebra;
and perhaps without metaphor
there would never have been
any algebra."
Max Black, Models and Metaphors, 1962
For metaphor and
algebra combined, see
in a diamond ring,"
A.M.S. abstract 79T-A37,
Notices of the
American Mathematical Society,
February 1979, pages A-193, 194 —
the original version of the 4×4 case
of the diamond theorem.
More on Max Black…
"When approaching unfamiliar territory, we often, as observed earlier, try to describe or frame the novel situation using metaphors based on relations perceived in a familiar domain, and by using our powers of association, and our ability to exploit the structural similarity, we go on to conjecture new features for consideration, often not noticed at the outset. The metaphor works, according to Max Black, by transferring the associated ideas and implications of the secondary to the primary system, and by selecting, emphasising and suppressing features of the primary in such a way that new slants on it are illuminated."
— Paul Thompson, University College, Oxford,
The Nature and Role of Intuition
in Mathematical Epistemology
That intuition, metaphor (i.e., analogy), and association may lead us astray is well known. The examples of French perspective above show what might happen if someone ignorant of finite geometry were to associate the phrase "4×4 square" with the phrase "projective geometry." The results are ridiculously inappropriate, but at least the second example does, literally, illuminate "new slants"– i.e., diagonals– within the perspective drawing of the 4×4 square.
Similarly, analogy led the ancient Greeks to believe that the diagonal of a square is commensurate with the side… until someone gave them a new slant on the subject.
Monday, January 24, 2005
Monday January 24, 2005
Old School Tie
“We are introduced to John Nash, fuddling flat-footed about the Princeton courtyard, uninterested in his classmates’ yammering about their various accolades. One chap has a rather unfortunate sense of style, but rather than tritely insult him, Nash holds a patterned glass to the sun, [director Ron] Howard shows us refracted patterns of light that take shape in a punch bowl, which Nash then displaces onto the neckwear, replying, ‘There must be a formula for how ugly your tie is.’ ”
have been extremely influential.”– Draft of
Computing with Modal Logics
(pdf), by Carlos Areces
and Maarten de Rijke
“Algebra in general is particularly suited for structuring and abstracting. Here, structure is imposed via symmetries and dualities, for instance in terms of Galois connections……. diamonds and boxes are upper and lower adjoints of Galois connections….”
and Applications: A Survey”
(pdf), by Jules Desharnais,
Bernhard Möller, and
Georg Struth, March 2004
Galois Correspondence
Evariste Galois
“Perhaps every science must
start with metaphor
and end with algebra;
and perhaps without metaphor
there would never have been
any algebra.”
— attributed, in varying forms
(1, 2, 3), to Max Black,
Models and Metaphors, 1962
For metaphor and
algebra combined, see
“Symmetry invariance
in a diamond ring,”
A.M.S. abstract 79T-A37,
Notices of the Amer. Math. Soc.,
February 1979, pages A-193, 194 —
the original version of the 4×4 case
of the diamond theorem.
Thursday, May 20, 2004
Thursday May 20, 2004
Parable
"A comparison or analogy. The word is simply a transliteration of the Greek word: parabolé (literally: 'what is thrown beside' or 'juxtaposed'), a term used to designate the geometric application we call a 'parabola.'…. The basic parables are extended similes or metaphors."
— http://religion.rutgers.edu/nt/
primer/parable.html
"If one style of thought stands out as the most potent explanation of genius, it is the ability to make juxtapositions that elude mere mortals. Call it a facility with metaphor, the ability to connect the unconnected, to see relationships to which others are blind."
— Sharon Begley, "The Puzzle of Genius," Newsweek magazine, June 28, 1993, p. 50
"The poet sets one metaphor against another and hopes that the sparks set off by the juxtaposition will ignite something in the mind as well. Hopkins’ poem 'Pied Beauty' has to do with 'creation.' "
— Speaking in Parables, Ch. 2, by Sallie McFague
"The Act of Creation is, I believe, a more truly creative work than any of Koestler's novels…. According to him, the creative faculty in whatever form is owing to a circumstance which he calls 'bisociation.' And we recognize this intuitively whenever we laugh at a joke, are dazzled by a fine metaphor, are astonished and excited by a unification of styles, or 'see,' for the first time, the possibility of a significant theoretical breakthrough in a scientific inquiry. In short, one touch of genius—or bisociation—makes the whole world kin. Or so Koestler believes."
— Henry David Aiken, The Metaphysics of Arthur Koestler, New York Review of Books, Dec. 17, 1964
For further details, see
Speaking in Parables:
A Study in Metaphor and Theology
by Sallie McFague
Fortress Press, Philadelphia, 1975
Introduction
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
"Perhaps every science must start with metaphor and end with algebra; and perhaps without metaphor there would never have been any algebra."
— attributed, in varying forms (1, 2, 3), to Max Black, Models and Metaphors, 1962
For metaphor and algebra combined, see
"Symmetry invariance in a diamond ring," A.M.S. abstract 79T-A37, Notices of the Amer. Math. Soc., February 1979, pages A-193, 194 — the original version of the 4×4 case of the diamond theorem.
Saturday, July 20, 2002
Saturday July 20, 2002
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Initial Xanga entry. Updated Nov. 18, 2006.