Various posts here on the geometry underlying the Mathieu group M_{24}
are now tagged with the phrase “Geometry of Even Subsets.”
For example, a post with this diagram . . .
Various posts here on the geometry underlying the Mathieu group M_{24}
are now tagged with the phrase “Geometry of Even Subsets.”
For example, a post with this diagram . . .
Just as
the finite space PG(3,2) is
the geometry of the 6set, so is
the finite space PG(5,2)
the geometry of the 8set.*
Selah.
* Consider, for the 6set, the 32
(16, modulo complementation)
0, 2, 4, and 6subsets,
and, for the 8set, the 128
(64, modulo complementation)
0, 2, 4, 6, and 8subsets.
Update of 11:02 AM ET the same day:
See also Eightfold Geometry, a note from 2010.
"Just fancy a scale model of Being
made out of string and cardboard."
— Nanavira Thera, 1 October 1957,
on a model of Kummer's Quartic Surface
mentioned by Eddington
"… a treatise on Kummer's quartic surface."
The "supermathematician" Eddington did not see fit to mention
the title or the author of the treatise he discussed.
See Hudson + Kummer in this journal.
See also posts tagged Dirac and Geometry.
See also Symplectic in this journal.
From Gotay and Isenberg, “The Symplectization of Science,”
Gazette des Mathématiciens 54, 5979 (1992):
“… what is the origin of the unusual name ‘symplectic’? ….
Its mathematical usage is due to Hermann Weyl who,
in an effort to avoid a certain semantic confusion, renamed
the then obscure ‘line complex group’ the ‘symplectic group.’
… the adjective ‘symplectic’ means ‘plaited together’ or ‘woven.’
This is wonderfully apt….”
The above symplectic figure appears in remarks on
the diamondtheorem correlation in the webpage
Rosenhain and Göpel Tetrads in PG(3,2). See also
related remarks on the notion of linear (or line ) complex
in the finite projective space PG(3,2) —
Code Blue
Update of 7:04 PM ET —
The source of the 404 message in the browsing history above
was the footnote below:
These are Rothko's Swamps .
See a Log24 search for related meditations.
For all three topics combined, see Coxeter —
" There is a pleasantly discursive treatment
of Pontius Pilate’s unanswered question
‘What is truth?’ "
— Coxeter, 1987, introduction to Trudeau’s
The NonEuclidean Revolution
Update of 10 AM ET — Related material, with an elementary example:
Posts tagged "Defining Form." The example —
See a post by Peter Woit from Sept. 24, 2005 — Dirac's Hidden Geometry.
The connection, if any, with recent Log24 posts on Dirac and Geometry
is not immediately apparent. Some related remarks from a novel —
From Broken Symmetries by Paul Preuss "He pondered the source of her fascination with the occult, which sooner or later seemed to entangle a lot of thoughtful people who were not already mired in establishmentarian science or religion. It was the religious impulse, at base. Even reason itself could function as a religion, he supposed— but only for those of severely limited imagination. He’d toyed with 'psi' himself, written a couple of papers now much quoted by crackpots, to his chagrin. The reason he and so many other theoretical physicists were suckers for the stuff was easy to understand— for twothirds of a century an enigma had rested at the heart of theoretical physics, a contradiction, a hard kernel of paradox. Quantum theory was inextricable from the uncertainty relations. The classical fox knows many things, but the quantummechanical hedgehog knows only one big thing— at a time. 'Complementarity,' Bohr had called it, a rubbery notion the great professor had stretched to include numerous pairs of opposites. Peter Slater was willing to call it absurdity, and unlike some of his older colleagues who, following in Einstein’s footsteps, demanded causal explanations for everything (at least in principle), Peter had never thirsted after 'hidden variables' to explain what could not be pictured. Mathematical relationships were enough to satisfy him, mere formal relationships which existed at all times, everywhere, at once. It was a thin nectar, but he was convinced it was the nectar of the gods. The psychic investigators, on the other hand, demanded to know how the mind and the psychical world were related. Through ectoplasm, perhaps? Some fifth force of nature? Extra dimensions of spacetime? All these naive explanations were on a par with the assumption that psi is propagated by a species of nonlocal hidden variables, the favored explanation of sophisticates; ignotum per ignotius . 'In this connection one should particularly remember that the human language permits the construction of sentences which do not involve any consequences and which therefore have no content at all…' The words were Heisenberg’s, lecturing in 1929 on the irreducible ambiguity of the uncertainty relations. They reminded Peter of Evan Harris Walker’s ingenious theory of the psi force, a theory that assigned psi both positive and negative values in such a way that the mere presence of a skeptic in the near vicinity of a sensitive psychic investigation could force null results. Neat, Dr. Walker, thought Peter Slater— neat, and totally without content. One had to be willing to tolerate ambiguity; one had to be willing to be crazy. Heisenberg himself was only human— he’d persuasively woven ambiguity into the fabric of the universe itself, but in that same set of 1929 lectures he’d rejected Dirac’s thennew wave equations with the remark, 'Here spontaneous transitions may occur to the states of negative energy; as these have never been observed, the theory is certainly wrong.' It was a reasonable conclusion, and that was its fault, for Dirac’s equations suggested the existence of antimatter: the first antiparticles, whose existence might never have been suspected without Dirac’s crazy results, were found less than three years later. Those socalled crazy psychics were too sane, that was their problem— they were too stubborn to admit that the universe was already more bizarre than anything they could imagine in their wildest dreams of wizardry." 
Particularly relevant …
"Mathematical relationships were enough to satisfy him,
mere formal relationships which existed at all times,
everywhere, at once."
Some related pure mathematics —
(A Prequel to Dirac and Geometry)
"So Einstein went back to the blackboard.
And on Nov. 25, 1915, he set down
the equation that rules the universe.
As compact and mysterious as a Viking rune,
it describes spacetime as a kind of sagging mattress…."
— Dennis Overbye in The New York Times online,
November 24, 2015
Some pure mathematics I prefer to the sagging Viking mattress —
Readings closely related to the above passage —
Thomas Hawkins, "From General Relativity to Group Representations:
the Background to Weyl's Papers of 192526," in Matériaux pour
l'histoire des mathématiques au XXe siècle: Actes du colloque
à la mémoire de Jean Dieudonné, Nice, 1996 (Soc. Math.
de France, Paris, 1998), pp. 69100.
The 19thcentury algebraic theory of invariants is discussed
as what Weitzenböck called a guide "through the thicket
of formulas of general relativity."
Wallace Givens, "Tensor Coordinates of Linear Spaces," in
Annals of Mathematics Second Series, Vol. 38, No. 2, April 1937,
pp. 355385.
Tensors (also used by Einstein in 1915) are related to
the theory of line complexes in threedimensional
projective space and to the matrices used by Dirac
in his 1928 work on quantum mechanics.
For those who prefer metaphors to mathematics —
Rota fails to cite the source of his metaphor.

Some background for my post of Nov. 20,
"Anticommuting Dirac Matrices as Skew Lines" —
His earlier paper that Bruins refers to, "Line Geometry
and Quantum Mechanics," is available in a free PDF.
For a biography of Bruins translated by Google, click here.
For some additional historical background going back to
Eddington, see Gary W. Gibbons, "The Kummer
Configuration and the Geometry of Majorana Spinors,"
pages 3952 in Oziewicz et al., eds., Spinors, Twistors,
Clifford Algebras, and Quantum Deformations:
Proceedings of the Second Max Born Symposium held
near Wrocław, Poland, September 1992 . (Springer, 2012,
originally published by Kluwer in 1993.)
For morerecent remarks on quantum geometry, see a
paper by Saniga cited in today's update to my Nov. 20 post.
Related material from Wikipedia— Baseball metaphors for sex.
Related web pages:
Miracle Octad Generator,
Generating the Octad Generator,
Geometry of the 4×4 Square
Related folklore:
“It is commonly known that there is a bijection between the 35 unordered triples of a 7set [i.e., the 35 partitions of an 8set into two 4sets] and the 35 lines of PG(3,2) such that lines intersect if and only if the corresponding triples have exactly one element in common.” –“Generalized Polygons and Semipartial Geometries,” by F. De Clerck, J. A. Thas, and H. Van Maldeghem, April 1996 minicourse, example 5 on page 6
The Miracle Octad Generator may be regarded as illustrating the folklore.
Update of August 20, 2010–
For facts rather than folklore about the above bijection, see The Moore Correspondence.
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.
The 15 2subsets of a 6set correspond to the 15 points of PG(3,2).
(Cullinane, 1986*)
The 35 3subsets of a 7set correspond to the 35 lines of PG(3,2).
(Conwell, 1910)
The 56 3subsets of an 8set correspond to the 56 spreads of PG(3,2).
(Seidel, 1970)
Each correspondence above may have been investigated earlier than
indicated by the above dates , which are the earliest I know of.
See also Correspondences in this journal.
* The above 1986 construction of PG(3,2) from a 6set also appeared
in the work of other authors in 1994 and 2002 . . .
Addendum at 5:09 PM suggested by an obituary today for Stephen Joyce:
See as well the word correspondences in
“James Joyce and the Hermetic Tradition,” by William York Tindall
(Journal of the History of Ideas , Jan. 1954).
The Wall Street Journal Jan. 24 on a Duke University professor —
"Dr. Daubechies is best known for her work on mathematical structures
called wavelets; her discoveries have been so influential, in fact, that
these are referred to in the field as Daubechies wavelets. She describes
them as 'mathematical building blocks' that can be used to extract the
essential elements of images or signals without losing their quality—
in effect, a new universal language for scientists and researchers."
See also this journal on January 2021, and …
"it remains only to choose a pleasing arrangement of {1, 2, … 7}
to label the eyeinthepyramid points.
there are, as it’ll turn out, 168 of ’em that’ll work."
— Comment at a weblog on November 27, 2010.
See also Log24 on that date.
The 11/27/2010 comment was on a post dated November 23, 2010.
See also Log24 on that date.
The Fourfold Square and Eightfold Cube
Related material: A Google image search for "field dream" + log24.
“The key is the cocktail that begins the proceedings.”
– Brian Harley, Mate in Two Moves
“Just as these lines that merge to form a key
Are as chess squares . . . .” — Katherine Neville, The Eight
“The complete projective group of collineations and dualities of the
[projective] 3space is shown to be of order [in modern notation] 8! ….
To every transformation of the 3space there corresponds
a transformation of the [projective] 5space. In the 5space, there are
determined 8 sets of 7 points each, ‘heptads’ ….”
— George M. Conwell, “The 3space PG (3, 2) and Its Group,”
The Annals of Mathematics , Second Series, Vol. 11, No. 2 (Jan., 1910),
pp. 6076.
“It must be remarked that these 8 heptads are the key to an elegant proof….”
— Philippe Cara, “RWPRI Geometries for the Alternating Group A_{8},” in
Finite Geometries: Proceedings of the Fourth Isle of Thorns Conference
(July 1621, 2000), Kluwer Academic Publishers, 2001, ed. Aart Blokhuis,
James W. P. Hirschfeld, Dieter Jungnickel, and Joseph A. Thas, pp. 6197.
Exercise: Use the Guitart 7cycles below to relate the 56 triples
in an 8set (such as the eightfold cube) to the 56 triangles in
a wellknown Kleinquartic hyperbolicplane tiling. Then use
the correspondence of the triples with the 56 spreads of PG(3,2)
to construct M_{24}.
Click image below to download a Guitart PowerPoint presentation.
See as well earlier posts also tagged Triangles, Spreads, Mathieu.
(Adapted from Eightfold Geometry, a note of April 28, 2010.
See also the recent post Geometry of 6 and 8.)
The post “Triangles, Spreads, Mathieu” of October 29 has been
updated with an illustration from the Curtis Miracle Octad Generator.
Related material — A search in this journal for “56 Triangles.”
There are many approaches to constructing the Mathieu
group M_{24}. The exercise below sketches an approach that
may or may not be new.
Exercise:
It is wellknown that …
There are 56 triangles in an 8set.
There are 56 spreads in PG(3,2).
The alternating group A_{n }is generated by 3cycles.
The alternating group A_{8 }is isomorphic to GL(4,2).
Use the above facts, along with the correspondence
described below, to construct M_{24}.
Some background —
A Log24 post of May 19, 2013, cites …
Peter J. Cameron in a 1976 Cambridge U. Press
book — Parallelisms of Complete Designs .
See the proof of Theorem 3A.13 on pp. 59 and 60.
See also a Google search for “56 triangles” “56 spreads” Mathieu.
Update of October 31, 2019 — A related illustration —
Update of November 2, 2019 —
See also p. 284 of Geometry and Combinatorics:
Selected Works of J. J. Seidel (Academic Press, 1991).
That page is from a paper published in 1970.
Update of December 20, 2019 —
Related entertainment —
Detail:
George Steiner —
"Perhaps an insane conceit."
Perhaps.
See Quantum Tesseract Theorem .
Perhaps Not.
See Dirac and Geometry .
Note that in the pictures below of the 15 twosubsets of a sixset,
the symbols 1 through 6 in Hudson's square array of 1905 occupy the
same positions as the anticommuting Dirac matrices in Arfken's 1985
square array. Similarly occupying these positions are the skew lines
within a generalized quadrangle (a line complex) inside PG(3,2).
Related narrative — The "Quantum Tesseract Theorem."
The Quantum Tesseract Theorem Revisited
"The secret is that the supermathematician expresses by the anticommutation
of his operators the property which the geometer conceives as perpendicularity
of displacements. That is why on p. 269 we singled out a pentad of anticommuting
operators, foreseeing that they would have an immediate application in describing
the property of perpendicular directions without using the traditional picture of space.
They express the property of perpendicularity without the picture of perpendicularity.
Thus far we have touched only the fringe of the structure of our set of sixteen Eoperators.
Only by entering deeply into the theory of electrons could I show the whole structure
coming into evidence."
A related illustration, from posts tagged Dirac and Geometry —
Compare and contrast Eddington's use of the word "perpendicular"
with a later use of the word by Saniga and Planat.
(For other posts on the continuing triumph of entertainment
over truth, see a Log24 search for "Night at the Museum.")
See also yesterday's post When the Men and today's previous post.
In Memoriam . . .
"When the men on the chessboard
get up and tell you where to go …."
"The I Ching encodes the geometry of the fabric of spacetime."
The following are some notes on the history of Clifford algebras
and finite geometry suggested by the "Clifford Modules" link in a
Log24 post of March 12, 2005 —
A more recent appearance of the configuration —
Some images, and a definition, suggested by my remarks here last night
on Apollo and Ross Douthat's remarks today on "The Return of Paganism" —
In finite geometry and combinatorics,
an inscape is a 4×4 array of square figures,
each figure picturing a subset of the overall 4×4 array:
Related material — the phrase
"Quantum Tesseract Theorem" and …
A. An image from the recent
film "A Wrinkle in Time" —
B. A quote from the 1962 book —
"There's something phoney
in the whole setup, Meg thought.
There is definitely something rotten
in the state of Camazotz."
This journal ten years ago today —
Surprise Package
From a talk by a Melbourne mathematician on March 9, 2018 —
The source — Talk II below —
Search Resultspdf of talk I (March 8, 2018)www.math.sci.hiroshimau.ac.jp/branched/…/AitchisonHiroshima2018Talk12.pdf Iain Aitchison. Hiroshima University March 2018 … Immediate: Talk given last year at Hiroshima (originally Caltech 2010). pdf of talk II (March 9, 2018) (with model for M24)www.math.sci.hiroshimau.ac.jp/branched/files/…/AitchisonHiroshima22018.pdf Iain Aitchison. Hiroshima University March 2018. (IRA: Hiroshima 032018). Highly symmetric objects II. Abstractwww.math.sci.hiroshimau.ac.jp/branched/files/2018/abstract/Aitchison.txt Iain AITCHISON Title: Construction of highly symmetric Riemann surfaces , related manifolds, and some exceptional objects, I, II Abstract: Since antiquity, some … 
Related material —
The 56 triangles of the eightfold cube . . .
Image from Christmas Day 2005.
Yesterday afternoon's post "Study in Blue and Pink" featured
an image related to the "Blade and Chalice" of Dan Brown …
Requiem for a comics character known as "The Blue Blade" —
"We all float down here."
About the corresponding "Pink Chalice," the less said the better.
See some posts related to three names
associated with Trinity College, Cambridge —
The date of Ron Shaw's 2016 death appears to be June 21:
All other Internet sources I have seen omit the June 21 date.
This journal on that date —
The title is from a phrase spoken, notably, by Yul Brynner
to Christopher Plummer in the 1966 film “Triple Cross.”
Related structures —
Greg Egan’s animated image of the Klein quartic —
For a smaller tetrahedral arrangement, within the Steiner quadruple
system of order 8 modeled by the eightfold cube, see a book chapter
by Michael Huber of Tübingen —
For further details, see the June 29 post Triangles in the Eightfold Cube.
See also, from an April 2013 philosophical conference:
Abstract for a talk at the City University of New York:
The Experience of Meaning Once the question of truth is settled, and often prior to it, what we value in a mathematical proof or conjecture is what we value in a work of lyric art: potency of meaning. An absence of clutter is a feature of such artifacts: they possess a resonant clarity that allows their meaning to break on our inner eye like light. But this absence of clutter is not tantamount to ‘being simple’: consider Eliot’s Four Quartets or Mozart’s late symphonies. Some truths are complex, and they are simplified at the cost of distortion, at the cost of ceasing to be truths. Nonetheless, it’s often possible to express a complex truth in a way that precipitates a powerful experience of meaning. It is that experience we seek — not simplicity per se , but the flash of insight, the sense we’ve seen into the heart of things. I’ll first try to say something about what is involved in such recognitions; and then something about why an absence of clutter matters to them. 
For the talk itself, see a YouTube video.
The conference talks also appear in a book.
The book begins with an epigraph by Hilbert —
From a post of July 25, 2008, “56 Triangles,” on the Klein quartic
and the eightfold cube —
“Baez’s discussion says that the Klein quartic’s 56 triangles
can be partitioned into 7 eighttriangle Egan ‘cubes’ that
correspond to the 7 points of the Fano plane in such a way
that automorphisms of the Klein quartic correspond to
automorphisms of the Fano plane. Show that the
56 triangles within the eightfold cube can also be partitioned
into 7 eighttriangle sets that correspond to the 7 points of the
Fano plane in such a way that (affine) transformations of the
eightfold cube induce (projective) automorphisms of the Fano plane.”
Related material from 1975 —
More recently …
Remarks related to a recent film and a notsorecent film.
For some historical background, see Dirac and Geometry in this journal.
Also (as Thas mentions) after Saniga and Planat —
The SanigaPlanat paper was submitted on December 21, 2006.
Excerpts from this journal on that date —
"Open the pod bay doors, HAL."
Michael Atiyah on the late Ron Shaw —
Phrases by Atiyah related to the importance in mathematics
of the twoelement Galois field GF(2) —
These phrases are from the yearend review of Trinity College,
Cambridge, Trinity Annual Record 2017 .
I prefer other, purely geometric, reasons for the importance of GF(2) —
See Finite Geometry of the Square and Cube.
See also today’s earlier post God’s Dice and Atiyah on the theology of
(Boolean) algebra vs. (Galois) geometry:
On a Trinity classmate of Ian Macdonald (see previous post)—
Atiyah's eulogy of Shaw in Trinity Annual Record 2017
is on pages 137 through 146. The conclusion —
The title refers to today's earlier post "The 35Year Wait."
A check of my activities 35 years ago this fall, in the autumn
of 1982, yields a formula I prefer to the nonsensical, but famous,
"canonical formula" of Claude LéviStrauss.
My "inscape" formula, from a note of Sept. 22, 1982 —
S = f ( f ( X ) ) .
Some mathematics from last year related to the 1982 formula —
See also Inscape in this journal and posts tagged Dirac and Geometry.
"Truth and clarity remained his paramount goals…"
— Benedict Nightingale in today's online New York TImes on an
English theatre director, founder of the Royal Shakespeare Company,
who reportedly died yesterday at 86.
See also Paramount in this journal.
A sentence from the New York Times Wire discussed in the previous post —
"Through characters like Wolverine and Swamp Thing,
he helped bring a new depth to his art form."
For Wolverine and Swamp Thing in posts related to a different
art form — geometry — see …
This post was suggested by the previous post — Four Dots —
and by the phrase "smallest perfect" in this journal.
Related material (click to enlarge) —
Detail —
From the work of Eddington cited in 1974 by von Franz —
See also Dirac and Geometry and Kummer in this journal.
Updates from the morning of June 27 —
Ron Shaw on Eddington's triads "associated in conjugate pairs" —
For more about hyperbolic and isotropic lines in PG(3,2),
see posts tagged Diamond Theorem Correlation.
For Shaw, in memoriam — See Contrapuntal Interweaving and The Fugue.
From a search in this journal for Seagram + Tradition —
Related art: Saturday afternoon's Twin Pillars of Symmetry.
The phrase "twin pillars" in a New York Times Fashion & Style
article today suggests a look at another pair of pillars —
This pair, from the realm of memory, history, and geometry disparaged
by the late painter Mark Rothko, might be viewed by Rothko
as "parodies of ideas (which are ghosts)." (See the previous post.)
For a relationship between a 3dimensional simplex and the {4, 3, 3},
see my note from May 21, 2014, on the tetrahedron and the tesseract.
… Continued from April 11, 2016, and from …
A tribute to Rothko suggested by the previous post —
For the idea of Rothko's obstacles, see Hexagram 39 in this journal.
Personally, I prefer
the religious symbolism
of Hudson Hawk .
In memory of an American artist whose work resembles that of
the Soviet constructivist Karl Ioganson (c. 18901929).
The American artist reportedly died on Thursday, Dec. 22, 2016.
"In fact, the (re)discovery of this novel structural principle was made in 194849 by a young American artist whom Koleichuk also mentions, Kenneth Snelson. In the summer of 1948, Snelson had gone to study with Joseph Albers who was then teaching at Black Mountain College. . . . One of the first works he made upon his return home was Early X Piece which he dates to December 1948 . . . . "
— "In the Laboratory of Constructivism: 
The word "constructivism" also refers to a philosophy of mathematics.
See a Log24 post, "Constructivist Witness," of 1 AM ET on the above
date of death.
See also, from the above publication date, Hudson's Inscape.
The inscape is illustrated in posts now tagged Laughing Academy.
The title refers to a philosophy of mathematics.
For those who prefer metaphor… Folk Etymology.
See also Stages of Math at Princeton's
Institute for Advanced Study in March 2013 —
— and in this journal starting in August 2014.
The figure below is one approach to the exercise
posted here on December 10, 2016.
Some background from earlier posts —
Click the image below to enlarge it.
Continuing the "Memory, History, Geometry" theme
from yesterday …
See Tetrahedral, Oblivion, and Tetrahedral Oblivion.
"Welcome home, Jack."
"… you don’t write off an aging loved one
just because he or she becomes cranky."
— Peter Schjeldahl on Rothko in The New Yorker ,
issue dated December 19 & 26, 2016, page 27
He was cranky in his forties too —
See Rothko + Swamp in this journal.
Related attitude —
From Subway Art for Times Square Church , Nov. 7
John Updike on Don DeLillo's thirteenth novel, Cosmopolis —
" DeLillo’s postChristian search for 'an order at some deep level'
has brought him to global computerization:
'the zerooneness of the world, the digital imperative . . . . ' "
— The New Yorker , issue dated March 31, 2003
On that date ….
Related remark —
" There is a pleasantly discursive treatment
of Pontius Pilate’s unanswered question
‘What is truth?’ "
— Coxeter, 1987, introduction to Trudeau’s
The NonEuclidean Revolution
Images from Burkard Polster's Geometrical Picture Book —
See as well in this journal the large Desargues configuration, with
15 points and 20 lines instead of 10 points and 10 lines as above.
Exercise: Can the large Desargues configuration be formed
by adding 5 points and 10 lines to the above Polster model
of the small configuration in such a way as to preserve
the smallconfiguration model's striking symmetry?
(Note: The related figure below from May 21, 2014, is not
necessarily very helpful. Try the Wolfram Demonstrations
model, which requires a free player download.)
Labeling the Tetrahedral Model (Click to enlarge) —
Related folk etymology (see point a above) —
Related literature —
The concept of "fire in the center" at The New Yorker ,
issue dated December 12, 2016, on pages 3839 in the
poem by Marsha de la O titled "A Natural History of Light."
Cézanne's Greetings.
For a concise historical summary of the interplay between
the geometry of an 8set and that of a 16set that is
involved in the the Miracle Octad Generator approach
to the large Mathieu group M_{24}, see Section 2 of …
Alan R. Prince
A near projective plane of order 6 (pp. 97105)
Innovations in Incidence Geometry
Volume 13 (Spring/Fall 2013).
This interplay, notably discussed by Conwell and
by Edge, involves spreads and Conwell’s heptads .
Update, morning of the following day (7:07 ET) — related material:
See also “56 spreads” in this journal.
The previous post discussed the parametrization of
the 4×4 array as a vector 4space over the 2element
Galois field GF(2).
The 4×4 array may also be parametrized by the symbol
0 along with the fifteen 2subsets of a 6set, as in Hudson's
1905 classic Kummer's Quartic Surface —
Hudson in 1905:
These two ways of parametrizing the 4×4 array — as a finite space
and as an array of 2element sets — were related to one another
by Cullinane in 1986 in describing, in connection with the Curtis
"Miracle Octad Generator," what turned out to be 15 of Hudson's
1905 "Göpel tetrads":
A recap by Cullinane in 2013:
Click images for further details.
The previous post quoted Tom Wolfe on Chomsky's use of
the word "array."
An example of particular interest is the 4×4 array
(whether of dots or of unit squares) —
.
Some context for the 4×4 array —
The following definition indicates that the 4×4 array, when
suitably coordinatized, underlies the Kummer lattice .
Further background on the Kummer lattice:
Alice Garbagnati and Alessandra Sarti,
"Kummer Surfaces and K3 surfaces
with $(Z/2Z)^4$ symplectic action."
To appear in Rocky Mountain J. Math. —
The above article is written from the viewpoint of traditional
algebraic geometry. For a less traditional view of the underlying
affine 4space from finite geometry, see the website
Finite Geometry of the Square and Cube.
Some further context …
"To our knowledge, the relation of the Golay code
to the Kummer lattice … is a new observation."
— Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of
Kummer surfaces in the Mathieu group M_{24 }"
As noted earlier, Taormina and Wendland seem not to be aware of
R. W. H. T. Hudson's use of the (uncoordinatized*) 4×4 array in his
1905 book Kummer's Quartic Surface. The array was coordinatized,
i.e. given a "vector space structure," by Cullinane eight years prior to
the cited remarks of Curtis.
* Update of Sept. 14: "Uncoordinatized," but parametrized by 0 and
the 15 twosubsets of a sixset. See the post of Sept. 13.
The following passage by Igor Dolgachev (Good Friday, 2003)
seems somewhat relevant (via its connection to Kummer's 16_{6} )
to previous remarks here on Dirac matrices and geometry —
Note related remarks from E. M. Bruins in 1959 —
A review of some recent posts on Dirac and geometry,
each of which mentions the late physicist Hendrik van Dam:
The first of these posts mentions the work of E. M. Bruins.
Some earlier posts that cite Bruins:
"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 "Projective Geometry and PTSymmetric Dirac Hamiltonian,"
Y. Jack Ng and H. van Dam,
Physics Letters B , Volume 673, Issue 3,
23 March 2009, Pages 237–239
(http://arxiv.org/abs/0901.2579v2, last revised Feb. 20, 2009)
" Studies of spin½ theories in the framework of projective geometry
have been undertaken before. See, e.g., Ref. [4]. ^{1 }"
" ^{1} These papers are rather mathematical and technical.
The authors of the first two papers discuss the Dirac equation
in terms of the PluckerKlein correspondence between lines of
a threedimensional projective space and points of a quadric
in a fivedimensional projective space. The last paper shows
that the Dirac equation bears a certain relation to Kummer’s
surface, viz., the structure of the Dirac ring of matrices is
related to that of Kummer’s 16_{6} configuration . . . ."
[4]
O. Veblen
Proc. Natl. Acad. Sci. USA , 19 (1933), p. 503
Full Text via CrossRef
E.M. Bruins
Proc. Nederl. Akad. Wetensch. , 52 (1949), p. 1135
F.C. Taylor Jr., Master thesis, University of North Carolina
at Chapel Hill (1968), unpublished
A remark of my own on the structure of Kummer’s 16_{6} configuration . . . .
See as well yesterday morning's post.
The authors Taormina and Wendland in the previous post
discussed some mathematics they apparently did not know was
related to a classic 1905 book by R. W. H. T. Hudson, Kummer's
Quartic Surface .
"This famous book is a prototype for the possibility
of explaining and exploring a manyfaceted topic of
research, without focussing on general definitions,
formal techniques, or even fancy machinery. In this
regard, the book still stands as a highly recommendable,
unparalleled introduction to Kummer surfaces, as a
permanent source of inspiration and, last but not least,
as an everlasting symbol of mathematical culture."
— Werner Kleinert, Mathematical Reviews ,
as quoted at Amazon.com
Some 4×4 diagrams from that book are highly relevant to the
discussion by Taormina and Wendland of the 4×4 squares within
the 1974 Miracle Octad Generator of R. T. Curtis that were later,
in 1987, described by Curtis as pictures of the vector 4space over
the twoelement Galois field GF(2).
Hudson did not think of his 4×4 diagrams as illustrating a vector space,
but he did use them to picture certain subsets of the 16 cells in each
diagram that he called Rosenhain and Göpel tetrads .
Some related work of my own (click images for related posts)—
Rosenhain tetrads as 20 of the 35 projective lines in PG(3,2)
Göpel tetrads as 15 of the 35 projective lines in PG(3,2)
Related terminology describing the Göpel tetrads above
Related material — Posts tagged Dirac and Geometry.
For an example of what Eddington calls "an open mind,"
see the 1958 letters of Nanavira Thera.
(Among the "Early Letters" in Seeking the Path ).
For the title phrase, see Encyclopedia of Mathematics .
The zero system illustrated in the previous post*
should not be confused with the cinematic Zero Theorem .
* More precisely, in the part showing the 15 lines fixed under
a zerosystem polarity in PG(3,2). For the zero system
itself, see diamondtheorem correlation.
(Continued from November 13)
The work of Ron Shaw in this area, ca. 19941995, does not
display explicitly the correspondence between anticommutativity
in the set of Dirac matrices and skewness in a line complex of
PG(3,2), the projective 3space over the 2element Galois field.
Here is an explicit picture —
References:
Arfken, George B., Mathematical Methods for Physicists , Third Edition,
Academic Press, 1985, pages 213214
Cullinane, Steven H., Notes on Groups and Geometry, 19781986
Shaw, Ron, "Finite Geometry, Dirac Groups, and the Table of
Real Clifford Algebras," undated article at ResearchGate.net
Update of November 23:
See my post of Nov. 23 on publications by E. M. Bruins
in 1949 and 1959 on Dirac matrices and line geometry,
and on another author who gives some historical background
going back to Eddington.
Some morerecent related material from the Slovak school of
finite geometry and quantum theory —
The matrices underlying the Saniga paper are those of Pauli, not
those of Dirac, but these two sorts of matrices are closely related.
For the connection of the title, see the post of Friday, November 13th, 2015.
For the essentials of this connection, see the following two documents —
Note that the six anticommuting sets of Dirac matrices listed by Arfken
correspond exactly to the six spreads in the above complex of 15 projective
lines of PG(3,2) fixed under a symplectic polarity (the diamond theorem
correlation ). As I noted in 1986, this correlation underlies the Miracle
Octad Generator of R. T. Curtis, hence also the large Mathieu group.
References:
Arfken, George B., Mathematical Methods for Physicists , Third Edition,
Academic Press, 1985, pages 213214
Cullinane, Steven H., Notes on Groups and Geometry, 19781986
Related material:
The 6set in my 1986 note above also appears in a 1996 paper on
the sixteen Dirac matrices by David M. Goodmanson —
Background reading:
Ron Shaw on finite geometry, Clifford algebras, and Dirac groups
(undated compilation of publications from roughly 19941995)—
The incidences of points and planes in the
Möbius 8_{4 } configuration (8 points and 8 planes,
with 4 points on each plane and 4 planes on each point),
were described by Coxeter in a 1950 paper.*
A table from Monday's post summarizes Coxeter's
remarks, which described the incidences in
spatial terms, with the points and planes as the vertices
and faceplanes of two mutually inscribed tetrahedra —
Monday's post, "Gallucci's Möbius Configuration,"
may not be completely intelligible unless one notices
that Coxeter has drawn some of the intersections in his
Fig. 24, a schematic representation of the pointplane
incidences, as dotless, and some as hollow dots. The figure,
"Gallucci's version of Möbius's 8_{4}," is shown below.
The hollow dots, representing the 8 points (as opposed
to the 8 planes ) of the configuration, are highlighted in blue.
Here a plane (represented by a dotless intersection) contains
the four points that are represented in the square array as lying
in the same row or same column as the plane.
The above Möbius incidences appear also much earlier in
Coxeter's paper, in figures 6 and 5, where they are shown
as describing the structure of a hypercube.
In figures 6 and 5, the dotless intersections representing
planes have been replaced by solid dots. The hollow dots
have again been highlighted in blue.
Figures 6 and 5 demonstrate the fact that adjacency in the set of
16 vertices of a hypercube is isomorphic to adjacency in the set
of 16 subsquares of a square 4×4 array, provided that opposite
sides of the array are identified, as in Fig. 6. The digits in
Coxeter's labels above may be viewed as naming the positions
of the 1's in (0,1) vectors (x_{4}, x_{3}, x_{2}, x_{1}) over the twoelement
Galois field.^{†} In that context, the 4×4 array may be called, instead
of a Möbius hypercube , a Galois tesseract .
* "SelfDual Configurations and Regular Graphs,"
Bulletin of the American Mathematical Society,
Vol. 56 (1950), pp. 413455
^{†} The subscripts' usual 1234 order is reversed as a reminder
that such a vector may be viewed as labeling a binary number
from 0 through 15, or alternately as labeling a polynomial in
the 16element Galois field GF(2^{4}). See the Log24 post
Vector Addition in a Finite Field (Jan. 5, 2013).
From H. S. M. Coxeter's 1950 paper
"SelfDual Configurations and Regular Graphs,"
a 4×4 array and a more perspicuous rearrangement—
(Click image to enlarge.)
The above rearrangement brings Coxeter's remarks into accord
with the webpage The Galois Tesseract.
Update of Thursday, March 26, 2015 —
For an explanation of Coxeter's Fig. 24, see Thursday's later
post titled "The Möbius Hypercube."
Some background for the part of the 2002 paper by Dolgachev and Keum
quoted here on January 17, 2014 —
Related material in this journal (click image for posts) —
Shown below is the matrix Omega from notes of Richard Evan Schwartz.
See also earlier versions (19761979) by Steven H. Cullinane.
Backstory: The Schwartz Notes (June 1, 2011), and Schwartz on
the American Mathematical Society's current home page:
The title phrase (not to be confused with the film 'The Zero Theorem')
means, according to the Encyclopedia of Mathematics,
a null system , and
"A null system is also called null polarity,
a symplectic polarity or a symplectic correlation….
it is a polarity such that every point lies in its own
polar hyperplane."
See Reinhold Baer, "Null Systems in Projective Space,"
Bulletin of the American Mathematical Society, Vol. 51
(1945), pp. 903906.
An example in PG(3,2), the projective 3space over the
twoelement Galois field GF(2):
See also the 10 AM ET post of Sunday, June 8, 2014, on this topic.
See also a Log24 post on this subject from Dec. 14, 2013,
especially (scroll down) the update of March 9, 2014.
Related material on the TurynCurtis construction
from the University of Cambridge —
— Slide by “Dr. Parker” — Apparently Richard A. Parker —
Lecture 4, “Discovering M_{24},” in slides for lectures 18 from lectures
at Cambridge in 20102011 on “Sporadic and Related Groups.”
See also the Parker lectures of 20122013 on the same topic.
A third construction of Curtis’s 35 4×6 1976 MOG arrays would use
Cullinane’s analysis of the 4×4 subarrays’ affine and projective structure,
and point out the fact that Conwell’s 1910 correspondence of the 35
4+4partitions of an 8set with the 35 lines of the projective 3space
over the 2element field, PG(3, 2), is essentially the same correspondence
as that constituting Curtis’s 1976 MOG.
See The Diamond Theorem, Finite Relativity, Galois Space,
Generating the Octad Generator, and The Klein Correspondence.
Update of March 22March 23 —
Adding together as (0,1)matrices over GF(2) the black parts (black
squares as 1’s, all other squares as 0’s) of the 35 4×6 arrays of the 1976
Curtis MOG would then reveal* the symmetric role played in octads
by what Curtis called the heavy brick , and so reveal also the action of
S_{3} on the three Curtis bricks that leaves invariant the set of all 759
octads of the S(5, 8, 24) constructed from the 35 MOG arrays. For more
details of this “byhand” construction, see Geometry of the 4×4 Square.
For the mathematical properties of the S(5, 8, 24), it is convenient to
have a separate construction, not by hand (such as Turyn’s), of the
extended binary Golay code. See the Brouwer preprint quoted above.
* “Then a miracle occurs,” as in the classic 1977 Sidney Harris cartoon.
Illustration of array addition from March 23 —
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).
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 —
(On His Dies Natalis )…
This is asserted in an excerpt from…
"The smallest nonrank 3 strongly regular graphs
which satisfy the 4vertex condition"
by Mikhail Klin, Mariusz Meszka, Sven Reichard, and Alex Rosa,
BAYREUTHER MATHEMATISCHE SCHRIFTEN 73 (2005), 152212—
(Click for clearer image)
Note that Theorem 46 of Klin et al. describes the role
of the Galois tesseract in the Miracle Octad Generator
of R. T. Curtis (original 1976 version). The tesseract
(a 4×4 array) supplies the geometric part of the above
exceptional geometriccombinatorial isomorphism.
Heraclitus, Fragment 60 (Diels number):
The way up and the way down is one and the same. ὁδὸς ἄνω κάτω μία καὶ ὡυτή hodòs áno káto mía kaì houté 
See also Blade and Chalice and, for a less Faustian
approach, Universe of Discourse.
Further context: Not Theology.
The Kummer 16_{6} configuration is the configuration of sixteen
6sets within a 4×4 square array of points in which each 6set
is determined by one of the 16 points of the array and
consists of the 3 other points in that point's row and the
3 other points in that point's column.
See Configurations and Squares.
The Wikipedia article Kummer surface uses a rather poetic
phrase* to describe the relationship of the 16_{6} to a number
of other mathematical concepts — "geometric incarnation."
Related material from finitegeometry.org —
* Apparently from David Lehavi on March 18, 2007, at Citizendium .
Short Story — (Click image for some details.)
Parts of a longer story —
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.
"What we do may be small, but it has
a certain character of permanence."
— G. H. Hardy, A Mathematician's Apology
The diamond theorem group, published without acknowledgment
of its source by the Mathematical Association of America in 2011—
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):
From an arXiv preprint submitted July 18, 2011,
and last revised on March 11, 2013 (version 4):
"By our construction, this vector space is the dual
of our hypercube F_{2}^{4} built on I \ O_{9}. The vector space
structure of the latter, to our knowledge, is first
mentioned by Curtis in [Cur89]. Hence altogether
our proposition 2.3.4 gives a novel geometric
meaning in terms of Kummer geometry to the known
vector space structure on I \ O_{9}."
[Cur89] reference:
R. T. Curtis, "Further elementary techniques using
the miracle octad generator," Proc. Edinburgh
Math. Soc. 32 (1989), 345353 (received on
July 20, 1987).
— Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of Kummer
surfaces in the Mathieu group M _{24 },"
arXiv.org > hepth > arXiv:1107.3834
"First mentioned by Curtis…."
No. I claim that to the best of my knowledge, the
vector space structure was first mentioned by me,
Steven H. Cullinane, in an AMS abstract submitted
in October 1978, some nine years before the
Curtis article.
Update of the above paragraph on July 6, 2013—
No. The vector space structure was described by
The vector space structure as it occurs in a 4×4 array 
See Notes on Finite Geometry for some background.
See in particular The Galois Tesseract.
For the relationship of the 1978 abstract to Kummer
geometry, see Rosenhain and Göpel Tetrads in PG(3,2).
… And the history of geometry —
Desargues, Pascal, Brianchon and Galois
in the light of complete npoints in space.
(Rewritten for clarity at about 10 AM ET April 29, with quote from Dowling added.
Updated with a reference to a Veblen and Young exercise (on p. 53) on April 30.)
Veblen and Young, Projective Geometry, Vol. I ,
Ginn and Company, 1910, page 39:
"The Desargues configuration. A very important configuration
is obtained by taking the plane section of a complete space fivepoint."
Each of figures 14 and 15 above has 15 points and 20 lines.
The Desargues configuration within each figure is denoted by
10 white points and 10 solid lines, with 3 points on each line and
3 lines on each point. Black points and dashed lines indicate the
complete space fivepoint and lines connecting it to the plane section
containing the Desargues configuration.
In a 1915 University of Chicago doctoral thesis, Archibald Henderson
used a complete space six point to construct a configuration of
15 points and 20 lines in the context not of Desargues ' theorem, but
rather of Brianchon 's theorem and of the Pascal hexagram.
Henderson's 1915 configuration is, it turns out, isomorphic to that of
the 15 points and 20 lines in the configuration constructed via a
complete space five point five years earlier by Veblen and Young.
(See, in Veblen and Young's 1910 Vol. I, exercise 11, page 53:
"A plane section of a 6point in space can be considered as
3 triangles perspective in pairs from 3 collinear points with
corresponding sides meeting in 3 collinear points." This is the
large Desargues configuration. See Classical Geometry in Light of
Galois Geometry.)
For this large Desargues configuration see April 19.
For Henderson's complete six –point, see The SixSet (April 23).
That post ends with figures relating the large Desargues configuration
to the Galois geometry PG(3,2) that underlies the Curtis
Miracle Octad Generator and the large Mathieu group M_{24} —
See also Note on the MOG Correspondence from April 25, 2013.
That correspondence was also discussed in a note 28 years ago, on this date in 1985.
Some historical background for today's note on the geometry
underlying the Curtis Miracle Octad Generator (MOG):
The above incidence diagram recalls those in today's previous post
on the MOG, which is used to construct the large Mathieu group M_{24}.
For some related material that is more uptodate, search the Web
for Mathieu + Kummer .
In light of the April 23 post "The SixSet,"
the caption at the bottom of a note of April 26, 1986
seems of interest:
"The R. T. Curtis correspondence between the 35 lines and the
2subsets and 3subsets of a 6set. This underlies M_{24}."
A related note from today:
Click image for some background.
Shown above is a rearranged version of the
Miracle Octad Generator (MOG) of R. T. Curtis
("A new combinatorial approach to M_{24},"
Math. Proc. Camb. Phil. Soc., 79 (1976), 2542.)
The 8subcell rectangles in the left part of the figure may be
viewed as illustrating (if the top left subcell is disregarded)
the thirtyfive 3subsets of a 7set.
Such a view relates, as the remarks below show, the
MOG's underlying Galois geometry, that of PG(3,2), to
the hexagrammum mysticum of Pascal.
On Danzer's 35_{4} Configuration:
"Combinatorially, Danzer’s configuration can be interpreted
as defined by all 3sets and all 4sets that can be formed
by the elements of a 7element set; each 'point' is represented
by one of the 3sets, and it is incident with those lines
(represented by 4sets) that contain the 3set."
— Branko Grünbaum, "Musings on an Example of Danzer's,"
European Journal of Combinatorics , 29 (2008),
pp. 1910–1918 (online March 11, 2008)
"Danzer's configuration is deeply rooted in
Pascal's Hexagrammum Mysticum ."
— Marko Boben, Gábor Gévay, and Tomaž Pisanski,
"Danzer's Configuration Revisited," arXiv.org, Jan. 6, 2013
For an approach to such configurations that differs from
those of Grünbaum, Boben, Gévay, and Pisanski, see
Classical Geometry in Light of Galois Geometry.
Grünbaum has written little about Galois geometry.
Pisanski has recently touched on the subject;
see Configurations in this journal (Feb. 19, 2013).
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
A review of the life of physicist Arthur Wightman,
who died at 90 on January 13th, 2013. yields
the following.
Wightman at Wikipedia:
"His graduate students include
Arthur Jaffe, Jerrold Marsden, and Alan Sokal."
"I think of Arthur as the spiritual leader
of mathematical physics and his death
really marks the end of an era."
— Arthur Jaffe in News at Princeton , Jan. 30
Marsden at Wikipedia:
"He [Marsden] has laid much of the foundation for
symplectic topology." (Link redirects to symplectic geometry.)
A Wikipedia reference in the symplectic geometry article leads to…
THE SYMPLECTIZATION OF SCIENCE:
Mark J. Gotay
James A. Isenberg February 18, 1992 Acknowledgments:
We would like to thank Jerry Marsden and Alan Weinstein Published in: Gazette des Mathématiciens 54, 5979 (1992). Opening:
"Physics is geometry . This dictum is one of the guiding 
A different account of the dictum:
The strange term Geometrodynamics
is apparently due to Wheeler.
Physics may or may not be geometry, but
geometry is definitely not physics.
For some pure geometry that has no apparent
connection to physics, see this journal
on the date of Wightman's death.
Today's online Telegraph has an obituary of The Troggs'
lead singer Reg Presley, who died yesterday at 71.
The unusually brilliant style of of the unsigned obituary
suggests a review of the life of a fellow Briton—
F. L. Lucas (18941967), author of Style .
According to Wikipedia, Virginia Woolf described Lucas as
"pure Cambridge: clean as a breadknife, and as sharp."
Lucas's acerbic 1923 review of The Waste Land suggests,
in the context of Woolf's remark and of the Blade and Chalice
link at the end of today's previous post, a search for a grail.
Voilà.
The previous post discussed some fundamentals of logic.
The name “Boole” in that post naturally suggests the
concept of Boolean algebra . This is not the algebra
needed for Galois geometry . See below.
Some, like Dan Brown, prefer to interpret symbols using
religion, not logic. They may consult Diamond Mandorla,
as well as Blade and Chalice, in this journal.
See also yesterday’s Universe of Discourse.
(Continued from March 15, 2001)
For one sort of regimentation, see Elements of Geometry.
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.”
… 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.
Related entertainment—
The song being performed in the above trailer
for Air America is "A Horse with No Name."
See "Instantia Crucis" and "Winning."
An example of lines in a Galois space * —
The 35 lines in the 3dimensional Galois projective space PG(3,2)—
There are 15 different individual linear diagrams in the figure above.
These are the points of the Galois space PG(3,2). Each 3set of linear diagrams
represents the structure of one of the 35 4×4 arrays and also represents a line
of the projective space.
The symmetry of the linear diagrams accounts for the symmetry of the
840 possible images in the kaleidoscope puzzle.
* For further details on the phrase "Galois space," see
Beniamino Segre's "On Galois Geometries," Proceedings of the
International Congress of Mathematicians, 1958 [Edinburgh].
(Cambridge U. Press, 1960, 488499.)
(Update of Jan. 5, 2013— This post has been added to finitegeometry.org.)
"Poetry is an illumination of a surface…."
— Wallace Stevens
Some poetic remarks related to a different surface, Klein's Quartic—
This link between the Klein map κ and the Mathieu group M_{24}
is a source of great delight to the author. Both objects were
found in the 1870s, but no connection between them was
known. Indeed, the class of maximal subgroups of M_{24}
isomorphic to the simple group of order 168 (often known,
especially to geometers, as the Klein group; see Baker [8])
remained undiscovered until the 1960s. That generators for
the group can be read off so easily from the map is
immensely pleasing.
— R. T. Curtis, Symmetric Generation of Groups ,
Cambridge University Press, 2007, page 39
Other poetic remarks related to the simple group of order 168—
A Google search today yielded no results
for the phrase "congruent group actions."
Places where this phrase might prove useful include—
Two papers suggested by Google searches tonight—
[PDF] PAPERS HELD OVER FROM THEME ISSUE ON ALGEBRA AND …
ajse.kfupm.edu.sa/articles/271A_08p.pdf
File Format: PDF/Adobe Acrobat – View as HTML by RT Curtis – 2001 – Related articles This paper is based on a talk given at the Scottish Algebra Day 1998 in Edinburgh. …… 
Curtis discusses the exceptional outer automorphism of S_{6}
as arising from group actions of PGL(2,5).
See also Cameron and Galois on PGL(2,5)—
[PDF] ON GROUPS OF DEGREE n AND n1, AND HIGHLYSYMMETRIC …

Illustration from Cameron (1973)—
J. H. Conway in 1971 discussed the role of an elementary abelian group
of order 16 in the Mathieu group M_{24}. His approach at that time was
purely algebraic, not geometric—
For earlier (and later) discussions of the geometry (not the algebra )
of that order16 group (i.e., the group of translations of the affine space
of 4 dimensions over the 2element field), see The Galois Tesseract.
Thursday's post Triangles Are Square posed the problem of
finding "natural" maps from the 16 subsquares of a 4×4 square
to the 16 equilateral subtriangles of an edge4 equilateral triangle.
Here is a trial solution of the inverse problem—
Exercise— Devise a test for "naturality" of
such mappings and apply it to the above.
Coming across John H. Conway's 1991*
pinwheel triangle decomposition this morning—
— suggested a review of a triangle decomposition result from 1984:
Figure A
(Click the below image to enlarge.)
The above 1985 note immediately suggests a problem—
What mappings of a square with c ^{2} congruent parts
to a triangle with c ^{2} congruent parts are "natural"?**
(In Figure A above, whether the 322,560 natural transformations
of the 16part square map in any natural way to transformations
of the 16part triangle is not immediately apparent.)
* Communicated to Charles Radin in January 1991. The Conway
decomposition may, of course, have been discovered much earlier.
** Update of Jan. 18, 2012— For a trial solution to the inverse
problem, see the "Triangles are Square" page at finitegeometry.org.
From a review of Truth and Other Enigmas , a book by the late Michael Dummett—
"… two issues stand out as central, recurring as they do in many of the
essays. One issue is the set of debates about realism, that is, those debates that ask
whether or not one or another aspect of the world is independent of the way we
represent that aspect to ourselves. For example, is there a realm of mathematical
entities that exists fully formed independently of our mathematical activity? Are
there facts about the past that our use of the past tense aims to capture? The other
issue is the view— which Dummett learns primarily from the later Wittgenstein—
that the meaning of an expression is fully determined by its use, by the way it
is employed by speakers. Much of his work consists in attempts to argue for this
thesis, to clarify its content and to work out its consequences. For Dummett one
of the most important consequences of the thesis concerns the realism debate and
for many other philosophers the prime importance of his work precisely consists
in this perception of a link between these two issues."
— Bernhard Weiss, pp. 104125 in Central Works of Philosophy , Vol. 5,
ed. by John Shand, McGillQueen's University Press, June 12, 2006
The above publication date (June 12, 2006) suggests a review of other
philosophical remarks related to that date. See …
"Every partitioning of the set of sixteen ontic states
into four disjoint pure epistemic states
yields a maximally informative measurement."—
For some morepersonal remarks on Dummett, see yesterday afternoon's
"The Stone" weblog in The New York Times.
I caught the sudden look of some dead master….
Powered by WordPress