Thursday, June 21, 2018
"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.
Comments Off on Dirac and Geometry (continued)
Sunday, December 10, 2017
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) —
Comments Off on Geometry
Friday, December 23, 2016
(Continued)
Code Blue
Update of 7:04 PM ET —
The source of the 404 message in the browsing history above
was the footnote below:
Comments Off on Memory, History, Geometry
Friday, December 16, 2016
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 —
Comments Off on Memory, History, Geometry
Monday, December 14, 2015
(Continued)
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
(first published by Simon and Schuster in 1983) —
"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 —
Comments Off on Dirac and Geometry
Friday, November 27, 2015
(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 —
"We acknowledge a theorem's beauty
when we see how the theorem 'fits' in its place,
how it sheds light around itself, like a Lichtung ,
a clearing in the woods."
— GianCarlo Rota, Indiscrete Thoughts ,
Birkhäuser Boston, 1997, page 132
Rota fails to cite the source of his metaphor.
It is Heidegger's 1964 essay, "The End of Philosophy
and the Task of Thinking" —
"The forest clearing [ Lichtung ] is experienced
in contrast to dense forest, called Dickung
in our older language."
— Heidegger's Basic Writings ,
edited by David Farrell Krell,
Harper Collins paperback, 1993, page 441

Comments Off on Einstein and Geometry
Monday, November 23, 2015
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.
Comments Off on Dirac and Line Geometry
Friday, April 25, 2014
Comments Off on Quilt Geometry
Saturday, November 10, 2012
Comments Off on Battlefield Geometry
Friday, November 9, 2012
Comments Off on Battlefield Geometry
Wednesday, April 28, 2010
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.
Comments Off on Eightfold Geometry
Thursday, April 22, 2010
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.
Comments Off on Mere Geometry
Sunday, January 26, 2020
Comments Off on HarmonicAnalysis Building Blocks
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 …
Comments Off on Duke Blocks
Tuesday, January 21, 2020
"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.
Precisely.
Comments Off on EyeinthePyramid Points
Monday, January 20, 2020
The Fourfold Square and Eightfold Cube
Related material: A Google image search for "field dream" + log24.
Comments Off on Dyadic Harmonic Analysis:
Monday, October 21, 2019
Related entertainment —
Detail:
George Steiner —
"Perhaps an insane conceit."
Perhaps.
See Quantum Tesseract Theorem .
Perhaps Not.
See Dirac and Geometry .
Comments Off on Algebra and Space… Illustrated.
Wednesday, October 9, 2019
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."
Comments Off on The Joy of Six
Sunday, July 14, 2019
The Quantum Tesseract Theorem Revisited
From page 274 —
"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.
Comments Off on Old Pathways in Science:
Friday, May 10, 2019
For fans of Resonance Science —
"When the men on the chessboard
get up and tell you where to go …."
Comments Off on I Ching g6
Continues.
Also from Fall Equinox 2018 — Looney Tune for Physicists —
Comments Off on Desperately Seeking Resonance
Thursday, May 9, 2019
(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.
Comments Off on Blade and Chalice at the Museum
Comments Off on Defense Against the Dark Arts
Wednesday, May 8, 2019
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."
Sure it does.
Comments Off on When the Men
Saturday, December 22, 2018
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 —
Comments Off on CremonaRichmond
Wednesday, December 12, 2018
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."
Comments Off on An Inscape for Douthat
Sunday, November 18, 2018
Update of Nov. 19 —
"Design is how it works." — Steve Jobs
See also www.cullinane.design.
Comments Off on Space Music
Sunday, October 21, 2018
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.
Comments Off on For Connoisseurs of Bad Art
Saturday, October 20, 2018
Related Log24 posts — See Blade + Chalice.
Comments Off on Study in Blue and Pink
Tuesday, September 25, 2018
See some posts related to three names
associated with Trinity College, Cambridge —
Atiyah + Shaw + Eddington .
Comments Off on Trinity
Saturday, September 1, 2018
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 —
Comments Off on Ron Shaw — D. 21 June 2016
Monday, March 12, 2018
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."
Comments Off on “Quantum Tesseract Theorem?”
Saturday, February 17, 2018
Michael Atiyah on the late Ron Shaw —
Phrases by Atiyah related to the importance in mathematics
of the twoelement Galois field GF(2) —

"The digital revolution based on the 2 symbols (0,1)"

"The algebra of George Boole"

"Binary codes"

"Dirac's spinors, with their up/down dichotomy"
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) —

The 2×2 square

The 2x2x2 cube

The 4×4 square

The 4x4x4 cube
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:
Comments Off on The Binary Revolution
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 —
Comments Off on God’s Dice
Tuesday, October 10, 2017
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.
The LéviStrauss formula —
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.
Comments Off on Another 35Year Wait
Tuesday, September 12, 2017
"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.
Comments Off on Goals
Monday, September 11, 2017
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 …
Comments Off on New Depth
Monday, June 26, 2017
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.
Comments Off on Upgrading to Six
Sunday, March 26, 2017
From a search in this journal for Seagram + Tradition —
Related art: Saturday afternoon's Twin Pillars of Symmetry.
Comments Off on Seagram Studies
Saturday, March 25, 2017
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.
Comments Off on Twin Pillars of Symmetry
… 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.
Comments Off on Like Decorations in a Cartoon Graveyard
Friday, February 3, 2017
Personally, I prefer
the religious symbolism
of Hudson Hawk .
Comments Off on Raiders of the Lost Chalice
Sunday, December 25, 2016
See also Robert M. Pirsig in this journal on Dec. 26, 2012.
Comments Off on Credit Where Due
Saturday, December 24, 2016
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:
Karl Ioganson's Cold Structures,"
by Maria Gough, OCTOBER Magazine, MIT,
Issue 84, Spring 1998, pp. 91117

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.
Comments Off on Early X Piece
Thursday, December 22, 2016
See also, from the above publication date, Hudson's Inscape.
The inscape is illustrated in posts now tagged Laughing Academy.
Comments Off on The LaughHospital
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.
Comments Off on Constructivist Witness
Monday, December 19, 2016
See also all posts now tagged Memory, History, Geometry.
Comments Off on ART WARS
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.
Comments Off on Tetrahedral CayleySalmon Model
Sunday, December 18, 2016
Click image to enlarge.
See also the large Desargues configuration in this journal.
Comments Off on Two Models of the Small Desargues Configuration
Saturday, December 17, 2016
Continuing the "Memory, History, Geometry" theme
from yesterday …
See Tetrahedral, Oblivion, and Tetrahedral Oblivion.
"Welcome home, Jack."
Comments Off on Tetrahedral Death Star
Friday, December 16, 2016
Comments Off on Read Something That Means Something
"… 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
Comments Off on Rothko’s Swamps
Tuesday, December 13, 2016
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
Comments Off on The Thirteenth Novel
Saturday, December 10, 2016
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.
Comments Off on Folk Etymology
Tuesday, September 13, 2016
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.
Comments Off on Parametrizing the 4×4 Array
Monday, September 12, 2016
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.
Comments Off on The Kummer Lattice
Tuesday, July 12, 2016
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 —
Comments Off on Group Elements and Skew Lines
Friday, June 3, 2016
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:
Comments Off on Bruins and van Dam
Wednesday, May 25, 2016
"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.
Comments Off on Framework
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 that structure in this journal, for instance —
See as well yesterday morning's post.
Comments Off on Kummer and Dirac
Tuesday, May 24, 2016
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
Comments Off on Rosenhain and Göpel Revisited
Monday, February 8, 2016
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 ).
Comments Off on A Game with Four Letters
Saturday, November 21, 2015
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.
Comments Off on The Zero System
Friday, November 20, 2015
(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.
Comments Off on Anticommuting Dirac Matrices as Skew Lines
Thursday, November 19, 2015
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 —
Comments Off on Highlights of the DiracMathieu Connection
Friday, November 13, 2015
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)—
Comments Off on A Connection between the 16 Dirac Matrices and the Large Mathieu Group
Thursday, March 26, 2015
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).
Comments Off on The Möbius Hypercube
Monday, March 23, 2015
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."
Comments Off on Gallucci’s Möbius Configuration
Wednesday, August 13, 2014
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) —
Comments Off on Symplectic Structure continued
Sunday, August 3, 2014
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:
(Click to enlarge.)
Comments Off on The Omega Matrix
Thursday, July 31, 2014
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.
Comments Off on Zero System
Friday, March 21, 2014
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 —
Comments Off on Three Constructions of the Miracle Octad Generator
Thursday, March 20, 2014
Click image for more details.
To enlarge image, click here.
Comments Off on Classical Galois
Thursday, February 6, 2014
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).
Comments Off on The Representation of Minus One
Friday, January 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 —
Comments Off on The 4×4 Relativity Problem
Friday, December 20, 2013
(On His Dies Natalis )…
An Exceptional Isomorphism Between Geometric and
Combinatorial Steiner Triple Systems Underlies
the Octads of the M_{24} Steiner System S(5, 8, 24).
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.
Comments Off on For Emil Artin
Monday, October 14, 2013
Heraclitus, Fragment 60 (Diels number):
See also Blade and Chalice and, for a less Faustian
approach, Universe of Discourse.
Further context: Not Theology.
Comments Off on Up and Down
Saturday, September 21, 2013
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 .
Comments Off on Geometric Incarnation
Thursday, September 5, 2013
(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 3space." The affine 4space
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
nonisotropic 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)."
Comments Off on Moonshine II
Friday, July 5, 2013
Short Story — (Click image for some details.)
Parts of a longer story —
The Galois Tesseract and Priority.
Comments Off on Mathematics and Narrative (continued)
Monday, June 10, 2013
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.
Comments Off on Galois Coordinates
Saturday, June 1, 2013
"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—
Comments Off on Permanence
Tuesday, May 28, 2013
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):
Comments Off on Codes
Sunday, May 19, 2013
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
(for instance) Peter J. Cameron in a 1976
Cambridge University Press book —
Parallelisms of Complete Designs .
See the proof of Theorem 3A.13 on pages 59 and 60.
The vector space structure as it occurs in a 4×4 array
of the sort that appears in the Curtis Miracle Octad
Generator may first have been pointed out by me,
Steven H. Cullinane, in an AMS abstract submitted in
October 1978, some nine years before the Curtis article.

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).
Comments Off on Priority Claim
Sunday, April 28, 2013
… 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.
Comments Off on The Octad Generator
Thursday, April 25, 2013
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 .
Comments Off on Rosenhain and Göpel Revisited
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:
Comments Off on Note on the MOG Correspondence
Saturday, April 6, 2013
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).
Comments Off on Pascal via Curtis
Thursday, February 28, 2013
A different dodecahedral space (Log24 on Oct. 3, 2011)—
Comments Off on Paperweights
Wednesday, February 13, 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
and mathematical motivation. In the first case one tries
to convince with a telling conceptual story; in the second
one relies more on the elegance of some emergent
mathematical structure. If there is a tradition in logic
it favours the former, but I have a sneaking affection for
the latter. Of course the distinction is not so clear cut.
Elegant mathematics will of itself tell a tale, and one with
the merit of simplicity. This may carry philosophical
weight. But that cannot be guaranteed: in the end one
cannot escape the need to form a judgement of significance."
– J. M. E. Hyland. "Proof Theory in the Abstract." (pdf)
Annals of Pure and Applied Logic 114, 2002, 4378.

Those who prefer story to structure may consult
 today's previous post on the Penrose diamond
 the remarks of Scott Aaronson on August 17, 2012
 the remarks in this journal on that same date
 the geometry of the 4×4 array in the context of M_{24}.
Comments Off on Form:
Friday, February 8, 2013
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:
Symplectic Geometry Lies at the Very
Foundations of Physics and Mathematics
Mark J. Gotay
Department of Mathematics
University of Hawai‘i
James A. Isenberg
Institute of Theoretical Science and Department of Mathematics
University of Oregon
February 18, 1992
Acknowledgments:
We would like to thank Jerry Marsden and Alan Weinstein
for their comments on previous drafts.
Published in: Gazette des Mathématiciens 54, 5979 (1992).
Opening:
"Physics is geometry . This dictum is one of the guiding
principles of modern physics. It largely originated with
Albert Einstein…."

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.
Comments Off on Dictum
Tuesday, February 5, 2013
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.
Comments Off on Grail
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.
Comments Off on Arsenal
Monday, January 21, 2013
(Continued from March 15, 2001)
For one sort of regimentation, see Elements of Geometry.
Comments Off on Shining Forth
Saturday, January 5, 2013
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.”
Comments Off on Vector Addition in a Finite Field
Saturday, December 8, 2012
… 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.
Comments Off on Defining the Contest…
Monday, November 19, 2012
From today's noon post—
"In all his poems with all their enchantments
for the poet himself, there is the final enchantment
that they are true. The significance of the poetic act
then is that it is evidence. It is instance and illustration.
It is an illumination of a surface,
the movement of a self in the rock.
Above all it is a new engagement with life.
It is that miracle to which the true faith of the poet
attaches itself."
— Wallace Stevens at Bard College, March 30, 1951
Stevens also said at Bard that
"When Joan of Arc said:
Have no fear: what I do, I do by command.
My brothers of Paradise tell me what I have to do.
these words were the words of an hallucination.
No matter what her brothers of Paradise drove her to do,
what she did was never a poetic act of faith in reality
because it could not be."
There are those who would dispute this.
Some related material:
"Ageometretos me eisito."—
"Let no one ignorant of geometry enter."—
Said to be a saying of Plato, part of the
seal of the American Mathematical Society—
A poetic approach to geometry—
"A surface" and "the rock," from All Saints' Day, 2012—
— and from 1981—
Some mathematical background for poets in Purgatory—
"… the Klein correspondence underlies Conwell's discussion
of eight heptads. These play an important role in another
correspondence, illustrated in the Miracle Octad Generator
of R. T. Curtis, that may be used to picture actions
of the large Mathieu group M_{24}."
Comments Off on Poetry and Truth
Monday, October 8, 2012
Related entertainment—
The song being performed in the above trailer
for Air America is "A Horse with No Name."
See "Instantia Crucis" and "Winning."
Comments Off on Air America
Thursday, July 12, 2012
An example of lines in a Galois space * —
The 35 lines in the 3dimensional Galois projective space PG(3,2)—
(Click to enlarge.)
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.)
Comments Off on Galois Space
Monday, June 18, 2012
"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—
Comments Off on Surface
Sunday, June 17, 2012
A Google search today yielded no results
for the phrase "congruent group actions."
Places where this phrase might prove useful include—
Comments Off on Congruent Group Actions
Saturday, April 14, 2012
Two papers suggested by Google searches tonight—
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)—
citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.104…
You +1'd this publicly. Undo
File Format: PDF/Adobe Acrobat – Quick View
by PJ CAMERON – 1975 – Cited by 14 – Related articles
PETER J. CAMERON. It is known that, if G is a triply transitive permutation group
on a finite set X with a regular … S3 the symmetric group on 3 letters, and PGL (2, 5)
the 2dimensional projective general linear … Received 24 October, 1973

Illustration from Cameron (1973)—
Comments Off on Scottish Algebra
Monday, January 23, 2012
(Continued)
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.
Comments Off on How It Works
Monday, January 16, 2012
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—
(Click for larger version.)
Exercise— Devise a test for "naturality" of
such mappings and apply it to the above.
Comments Off on Mapping Problem
Thursday, January 12, 2012
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.
Comments Off on Triangles Are Square
Thursday, January 5, 2012
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 …
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….
— Four Quartets
Comments Off on Precisely
Wednesday, January 4, 2012
I revised the cubes image and added a new link to
an explanatory image in posts of Dec. 30 and Jan. 3
(and at finitegeometry.org). (The cubes now have
quaternion "i , j , k " labels and the cubes now
labeled "k " and "k " were switched.)
I found some relevant remarks here and here.
Comments Off on Revision
Saturday, December 31, 2011
(Continued)
"Design is how it works." — Steve Jobs
From a commercial testprep firm in New York City—
From the date of the above uploading—
From a New Year's Day, 2012, weblog post in New Zealand—
From Arthur C. Clarke, an early version of his 2001 monolith—
"So they left a sentinel, one of millions they have scattered
throughout the Universe, watching over all worlds with the
promise of life. It was a beacon that down the ages has been
patiently signaling the fact that no one had discovered it.
Perhaps you understand now why that crystal pyramid was set…."
The numerical (not crystal) pyramid above is related to a sort of
mathematical block design known as a Steiner system.
For its relationship to the graphic block design shown above,
see the webpages Block Designs and The Diamond Theorem
as well as The Galois Tesseract and R. T. Curtis's classic paper
"A New Combinatorial Approach to M_{24}," which contains the following
version of the above numerical pyramid—
For graphic block designs, I prefer the blocks (and the parents)
of Grand Rapids to those of New York City.
For the barbed tail of Clarke's "Angel" story, see the New Zealand post
of New Year's Day mentioned above.
Comments Off on The Uploading
Friday, December 30, 2011
The following picture provides a new visual approach to
the order8 quaternion group's automorphisms.
Click the above image for some context.
Here the cube is called "eightfold" because the eight vertices,
like the eight subcubes of a 2×2×2 cube,* are thought of as
independently movable. See The Eightfold Cube.
See also…
Related material: Robin Chapman and Karen E. Smith
on the quaternion group's automorphisms.
* See Margaret Wertheim's Christmas Eve remarks on mathematics
and the following eightfold cube from an institute she cofounded—
© 2005 The Institute for Figuring
Photo by Norman Brosterman
fom the Inventing Kindergarten
exhibit at The Institute for Figuring
(cofounded by Margaret Wertheim)
Comments Off on Quaternions on a Cube
Wednesday, November 2, 2011
A search today, All Souls Day, for relevant learning
at All Souls College, Oxford, yields the person of
Sir Michael Dummett and the following scholarly page—
(Click to enlarge.)
My own background is in mathematics rather than philosophy.
From a mathematical point of view, the cells discussed above
seem related to some "universals" in an example of Quine.
In Quine's example,* universals are certain equivalence classes
(those with the "same shape") of a family of figures
(33 convex regions) selected from the 2^{8 }= 256 subsets
of an eightelement set of plane regions.
A smaller structure, closer to Wright's concerns above,
is a universe of 2^{4 }= 16 subsets of a 4element set.
The number of elements in this universe of Concepts coincides,
as it happens, with the number obtained by multiplying out
the title of T. S. Eliot's Four Quartets .
For a discussion of functions that map "cells" of the sort Wright
discusses— in the quartets example, four equivalence classes,
each with four elements, that partition the 16element universe—
onto a fourelement set, see Poetry's Bones.
For some philosophical background to the Wright passage
above, see "The Concept Horse," by Harold W. Noonan—
Chapter 9, pages 155176, in Universals, Concepts, and Qualities ,
edited by P. F. Strawson and Arindam Chakrabarti,
Ashgate Publishing, 2006.
For a different approach to that concept, see Devil's Night, 2011.
* Admittedly artificial. See From a Logical Point of View , IV, 3
Comments Off on The Poetry of Universals
Monday, October 3, 2011
The following may help show why R.T. Curtis calls his approach
to sporadic groups symmetric generation—
(Click to enlarge.)
Related material— Yesterday's Symmetric Generation Illustrated.
Comments Off on Mathieu Symmetry
Sunday, October 2, 2011
R.T. Curtis in a 1990 paper* discussed his method of "symmetric generation" of groups as applied to the Mathieu groups M_{ 12} and M_{ 24}.
See Finite Relativity and the Log24 posts Relativity Problem Revisited (Sept. 20) and Symmetric Generation (Sept. 21).
Here is some exposition of how this works with M_{ 12 }.
* "Geometric Interpretations of the ‘Natural’ Generators of the Mathieu groups," Mathematical Proceedings of the Cambridge Philosophical Society (1990), Vol. 107, Issue 01, pp. 1926.
Comments Off on Symmetric Generation Illustrated
Wednesday, September 21, 2011
Suggested by yesterday's Relativity Problem Revisited and by Cassirer on Objectivity—
From Symmetric Generation of Groups , by R.T. Curtis (Cambridge U. Press, 2007)—
"… we are saying much more than that G ≅ M_{ 24} is generated by
some set of seven involutions, which would be a very weak
requirement. We are asserting that M_{ 24} is generated by a set
of seven involutions which possesses all the symmetries of L_{3}(2)
acting on the points of the 7point projective plane…."
— Symmetric Generation , p. 41
"It turns out that this approach is particularly revealing and that
many simple groups, both sporadic and classical, have surprisingly
simple definitions of this type."
— Symmetric Generation , p. 42
See also (click to enlarge)—
Cassirer's remarks connect the concept of objectivity with that of object .
The above quotations perhaps indicate how the Mathieu group M_{ 24} may be viewed as an object.
"This is the moment which I call epiphany. First we recognise that the object is one integral thing, then we recognise that it is an organised composite structure, a thing in fact: finally, when the relation of the parts is exquisite, when the parts are adjusted to the special point, we recognise that it is that thing which it is. Its soul, its whatness, leaps to us from the vestment of its appearance. The soul of the commonest object, the structure of which is so adjusted, seems to us radiant. The object achieves its epiphany."
— James Joyce, Stephen Hero
For a simpler object "which possesses all the symmetries of L_{3}(2) acting on the points of the 7point projective plane…." see The Eightfold Cube.
For symmetric generation of L_{3}(2) on that cube, see A Simple Reflection Group of Order 168.
Comments Off on Symmetric Generation
Tuesday, September 20, 2011
A footnote was added to Finite Relativity—
Background:
Weyl on what he calls the relativity problem—
"The relativity problem is one of central significance throughout geometry and algebra and has been recognized as such by the mathematicians at an early time."
– Hermann Weyl, 1949, "Relativity Theory as a Stimulus in Mathematical Research"
"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, 1946, The Classical Groups , Princeton University Press, p. 16
…. A note of Feb. 20, 1986, supplied an example of such coordinatizations in finite geometry. In that note, the group of mediating transformations acted directly on coordinates within a 4×4 array. When the 4×4 array is embedded in a 4×6 array, a larger and more interesting group, M_{ 24} (containing the original group), acts on the larger array. There is no obvious solution to Weyl's relativity problem for M_{ 24}. That is, there is no obvious way* to apply exactly 24 distinct transformable coordinatesets (or symbolstrings ) to the 24 array elements in such a way that the natural group of mediating transformations of the 24 symbolstrings is M_{ 24}. ….
Footnote of Sept. 20, 2011:
* R.T. Curtis has, it seems, a nonobvious way that involves strings of seven symbols. His abstract for a 1990 paper says that in his construction "The generators of M_{ 24} are defined… as permutations of twentyfour 7cycles in the action of PSL_{2}(7) on seven letters…."
See "Geometric Interpretations of the ‘Natural’ Generators of the Mathieu groups," by R.T. Curtis, Mathematical Proceedings of the Cambridge Philosophical Society (1990), Vol. 107, Issue 01, pp. 1926. (Rec. Jan. 3, 1989, revised Feb. 3, 1989.) This paper was published online on Oct. 24, 2008.
Some related articles by Curtis:
R.T. Curtis, "Natural Constructions of the Mathieu groups," Math. Proc. Cambridge Philos. Soc. (1989), Vol. 106, pp. 423429
R.T. Curtis. "Symmetric Presentations I: Introduction, with Particular Reference to the Mathieu groups M_{ 12} and M_{ 24}" In Proceedings of 1990 LMS Durham Conference 'Groups, Combinatorics and Geometry' (eds. M. W. Liebeck and J. Saxl), London Math. Soc. Lecture Note Series 165, Cambridge University Press, 1992, pp. 380–396
R.T. Curtis, "A Survey of Symmetric Generation of Sporadic Simple Groups," in The Atlas of Finite Groups: Ten Years On , (eds. R.T. Curtis and R.A. Wilson), London Math. Soc. Lecture Note Series 249, Cambridge University Press, 1998, pp. 39–57
Comments Off on Relativity Problem Revisited
Sunday, September 18, 2011
A transcription—
"Now suppose that α is an element of order 23 in M _{ 24 }; we number the points of Ω
as the projective line ∞, 0, 1, 2, … , 22 so that α : i → i + 1 (modulo 23) and fixes ∞. In
fact there is a full L_{ 2 }(23) acting on this line and preserving the octads…."
— R. T. Curtis, "A New Combinatorial Approach to M _{24 },"
Mathematical Proceedings of the Cambridge Philosophical Society (1976), 79: 2542
Comments Off on Alpha and Omega
Saturday, September 3, 2011
A post of September 1, The Galois Tesseract, noted that the interplay
of algebraic and geometric properties within the 4×4 array that forms
twothirds of the Curtis Miracle Octad Generator (MOG) may first have
been described by Cullinane (AMS abstract 79TA37, 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 4space over the 2element 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 4point 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
Latinsquare 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 M_{24}, 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 misnamed 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: 345353
* For instance:
Update of Sept. 4— This post is now a page at finitegeometry.org.
Comments Off on The Galois Tesseract (continued)
Thursday, September 1, 2011
"Design is how it works." — Steven Jobs (See Symmetry and Design.)
"By far the most important structure in design theory is the Steiner system S(5, 8, 24)."
— "Block Designs," by Andries E. Brouwer
The name Carmichael is not to be found in Booher's thesis. In a reference he does give for the history of S(5,8,24), Carmichael's construction of this design is dated 1937. It should be dated 1931, as the following quotation shows—
From Log24 on Feb. 20, 2010—
"The linear fractional group modulo 23 of order 24•23•11 is often represented as a doubly transitive group of degree 24 on the symbols ∞, 0, 1, 2,…, 22. This transitive group contains a subgroup of order 8 each element of which transforms into itself the set ∞, 0, 1, 3, 12, 15, 21, 22 of eight elements, while the whole group transforms this set into 3•23•11 sets of eight each. This configuration of octuples has the remarkable property that any given set of five of the 24 symbols occurs in one and just one of these octuples. The largest permutation group Γ on the 24 symbols, each element of which leaves this configuration invariant, is a fivefold transitive group of degree 24 and order 24•23•22•21•20•48. This is the Mathieu group of degree 24."
– R. D. Carmichael, "Tactical Configurations of Rank Two," in American Journal of Mathematics, Vol. 53, No. 1 (Jan., 1931), pp. 217240
Epigraph from Ch. 4 of Design Theory , Vol. I:
"Es is eine alte Geschichte,
doch bleibt sie immer neu "
—Heine (Lyrisches Intermezzo XXXIX)
See also "Do you like apples?"
Comments Off on How It Works
Thursday, August 25, 2011
"Design is how it works." — Steven Jobs (See yesterday's Symmetry.)
Today's American Mathematical Society home page—
Some related material—
The above Rowley paragraph in context (click to enlarge)—
"We employ Curtis's MOG …
both as our main descriptive device and
also as an essential tool in our calculations."
— Peter Rowley in the 2009 paper above, p. 122
And the MOG incorporates the
Geometry of the 4×4 Square.
For this geometry's relation to "design"
in the graphicarts sense, see
Block Designs in Art and Mathematics.
Comments Off on Design
Wednesday, August 10, 2011
From math16.com—
Quotations on Realism
and the Problem of Universals:
"It is said that the students of medieval Paris came to blows in the streets over the question of universals. The stakes are high, for at issue is our whole conception of our ability to describe the world truly or falsely, and the objectivity of any opinions we frame to ourselves. It is arguable that this is always the deepest, most profound problem of philosophy. It structures Plato's (realist) reaction to the sophists (nominalists). What is often called 'postmodernism' is really just nominalism, colourfully presented as the doctrine that there is nothing except texts. It is the variety of nominalism represented in many modern humanities, paralysing appeals to reason and truth."
— Simon Blackburn, Think, Oxford University Press, 1999, page 268
"You will all know that in the Middle Ages there were supposed to be various classes of angels…. these hierarchized celsitudes are but the last traces in a less philosophical age of the ideas which Plato taught his disciples existed in the spiritual world."
— Charles Williams, page 31, Chapter Two, "The Eidola and the Angeli," in The Place of the Lion (1933), reprinted in 1991 by Eerdmans Publishing
For Williams's discussion of Divine Universals (i.e., angels), see Chapter Eight of The Place of the Lion.
"People have always longed for truths about the world — not logical truths, for all their utility; or even probable truths, without which daily life would be impossible; but informative, certain truths, the only 'truths' strictly worthy of the name. Such truths I will call 'diamonds'; they are highly desirable but hard to find….The happy metaphor is Morris Kline's in Mathematics in Western Culture (Oxford, 1953), p. 430."
— Richard J. Trudeau, The NonEuclidean Revolution, Birkhauser Boston, 1987, pages 114 and 117
"A new epistemology is emerging to replace the Diamond Theory of truth. I will call it the 'Story Theory' of truth: There are no diamonds. People make up stories about what they experience. Stories that catch on are called 'true.' The Story Theory of truth is itself a story that is catching on. It is being told and retold, with increasing frequency, by thinkers of many stripes…. My own viewpoint is the Story Theory…. I concluded long ago that each enterprise contains only stories (which the scientists call 'models of reality'). I had started by hunting diamonds; I did find dazzlingly beautiful jewels, but always of human manufacture."
— Richard J. Trudeau, The NonEuclidean Revolution, Birkhauser Boston, 1987, pages 256 and 259
Trudeau's confusion seems to stem from the nominalism of W. V. Quine, which in turn stems from Quine's appalling ignorance of the nature of geometry. Quine thinks that the geometry of Euclid dealt with "an emphatically empirical subject matter" — "surfaces, curves, and points in real space." Quine says that Euclidean geometry lost "its old status of mathematics with a subject matter" when Einstein established that space itself, as defined by the paths of light, is nonEuclidean. Having totally misunderstood the nature of the subject, Quine concludes that after Einstein, geometry has become "uninterpreted mathematics," which is "devoid not only of empirical content but of all question of truth and falsity." (From Stimulus to Science, Harvard University Press, 1995, page 55)
— S. H. Cullinane, December 12, 2000
The correct statement of the relation between geometry and the physical universe is as follows:
"The contrast between pure and applied mathematics stands out most clearly, perhaps, in geometry. There is the science of pure geometry, in which there are many geometries: projective geometry, Euclidean geometry, nonEuclidean geometry, and so forth. Each of these geometries is a model, a pattern of ideas, and is to be judged by the interest and beauty of its particular pattern. It is a map or picture, the joint product of many hands, a partial and imperfect copy (yet exact so far as it extends) of a section of mathematical reality. But the point which is important to us now is this, that there is one thing at any rate of which pure geometries are not pictures, and that is the spatiotemporal reality of the physical world. It is obvious, surely, that they cannot be, since earthquakes and eclipses are not mathematical concepts."
— G. H. Hardy, section 23, A Mathematician's Apology, Cambridge University Press, 1940

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—
"Exhibit A is certainly Ljubljana…."
Comments Off on Objectivity
Monday, August 8, 2011
Some background—
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—
Comments Off on Diamond Theory vs. Story Theory (continued)
Older Posts »