Wednesday, September 23, 2020

Geometry of Even Subsets

Filed under: General — Tags: — m759 @ 12:06 AM

Various posts here on the geometry underlying the Mathieu group M24
are now tagged with the phrase “Geometry of Even Subsets.”

For example, a post with this diagram . . .

Sunday, December 8, 2019

Geometry of 6 and 8

Filed under: General — Tags: , , — m759 @ 4:03 AM

Just as
the finite space PG(3,2) is
the geometry of the 6-set, so is
the finite space PG(5,2)
the geometry of the 8-set.*


* Consider, for the 6-set, the 32
(16, modulo complementation)
0-, 2-, 4-, and 6-subsets,
and, for the 8-set, the 128
(64, modulo complementation)
0-, 2-, 4-, 6-, and 8-subsets.

Update of 11:02 AM ET the same day:

See also Eightfold Geometry, a note from 2010.

Thursday, June 21, 2018

Dirac and Geometry (continued)

"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 "super-mathematician" 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.

Sunday, December 10, 2017


Google search result for Plato + Statesman + interlacing + interweaving

See also Symplectic in this journal.

From Gotay and Isenberg, “The Symplectization of Science,”
Gazette des Mathématiciens  54, 59-79 (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….”

IMAGE- A symplectic structure -- i.e. a structure that is symplectic (meaning plaited or woven)

The above symplectic  figure appears in remarks on
the diamond-theorem 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) —

Anticommuting Dirac matrices as spreads of projective lines

Ron Shaw on the 15 lines of the classical generalized quadrangle W(2), a general linear complex in PG(3,2)

Friday, December 23, 2016

Memory, History, Geometry

Filed under: General,Geometry — Tags: — m759 @ 6:48 PM


Code Blue

Update of 7:04 PM ET —

The source of the 404 message in the browsing history above
was the footnote below:

Friday, December 16, 2016

Memory, History, Geometry

Filed under: General,Geometry — Tags: — m759 @ 9:48 AM

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 Non-Euclidean Revolution

Update of 10 AM ET —  Related material, with an elementary example:

Posts tagged "Defining Form." The example —

IMAGE- Triangular models of the 4-point affine plane A and 7-point projective plane PA

Monday, December 14, 2015

Dirac and Geometry

Filed under: General,Geometry — Tags: , — m759 @ 10:30 AM


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 two-thirds 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 quantum-mechanical 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 then-new 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 so-called 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

Anticommuting Dirac matrices as spreads of projective lines

Friday, November 27, 2015

Einstein and Geometry

Filed under: General,Geometry — Tags: — m759 @ 2:01 PM

(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 space-time 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 1925-26
," 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. 69-100.

The 19th-century 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. 355-385.

Tensors (also used by Einstein in 1915) are related to 
the theory of line complexes in three-dimensional
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." 
— Gian-Carlo 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

Monday, November 23, 2015

Dirac and Line Geometry

Some background for my post of Nov. 20,
"Anticommuting Dirac Matrices as Skew Lines" —

First page of 'Configurations in Quantum Mechanics,' by E.M. Bruins, 1959

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 39-52 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 more-recent remarks on quantum geometry, see a
paper by Saniga cited in today's update to my Nov. 20 post

Friday, April 25, 2014

Quilt Geometry

Filed under: General,Geometry — Tags: , — m759 @ 7:55 PM

or: The Dead Hand Shot

Library Thing book list: 'An Awkward Lie' and 'A Piece of Justice'

See also Tumbling Blocks Quilt and Springtime for Vishnu.

Saturday, November 10, 2012

Battlefield Geometry

Filed under: General,Geometry — Tags: — m759 @ 5:24 AM


Click to enlarge.

Related material from Wikipedia— Baseball metaphors for sex.

"Build it…"

Friday, November 9, 2012

Battlefield Geometry

Filed under: General,Geometry — Tags: — m759 @ 3:28 PM

(Continued from Sept. 11, 2007)

CIA Director David Petraeus resigns, cites extramarital affair

Trouble with the curve?

Wednesday, April 28, 2010

Eightfold Geometry

Filed under: General,Geometry — Tags: , , — m759 @ 11:07 AM

Image-- The 35 partitions of an 8-set into two 4-sets

Image-- Analysis of structure of the 35 partitions of an 8-set into two 4-sets

Image-- Miracle Octad Generator of R.T. Curtis

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 7-set [i.e., the 35 partitions of an 8-set into two 4-sets] 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.

Thursday, April 22, 2010

Mere Geometry

Filed under: General,Geometry — Tags: — m759 @ 1:00 PM

Image-- semeion estin ou meros outhen

Image-- Euclid's definition of 'point'

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 non-Euclidean* approach to parts–

Image-- examples from Galois affine geometry

Corresponding non-Euclidean*
projective points —

Image-- The smallest Galois geometries

Richard J. Trudeau in The Non-Euclidean Revolution, chapter on “Geometry and the Diamond Theory of Truth”–

“… Plato and Kant, and most of the philosophers and scientists in the 2200-year 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 non-Euclidean* figures above illustrate concepts from a 1976 monograph, also called “Diamond Theory.”

Although non-Euclidean,* the theorems of the 1976 “Diamond Theory” are also, in Trudeau’s terminology, diamonds.

* “Non-Euclidean” here means merely “other than  Euclidean.” No violation of Euclid’s parallel postulate is implied.

Friday, February 7, 2020


Filed under: General — Tags: , , — m759 @ 1:05 PM

The 15  2-subsets of a 6-set correspond to the 15 points of PG(3,2).
(Cullinane, 1986*)

The 35  3-subsets of a 7-set correspond to the 35 lines of PG(3,2).
(Conwell, 1910)

The 56  3-subsets of an 8-set 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 6-set 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).

Sunday, January 26, 2020

Harmonic-Analysis Building Blocks

Filed under: General — Tags: — m759 @ 1:14 PM

See also The Eightfold Cube.

Duke Blocks

Filed under: General — Tags: — m759 @ 12:38 AM

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 20-21, and …

Tuesday, January 21, 2020

Eye-in-the-Pyramid Points

Filed under: General — Tags: — m759 @ 2:06 PM

"it remains only to choose a pleasing arrangement of {1, 2, … 7}
to label the eye-in-the-pyramid 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.


Monday, January 20, 2020

Dyadic Harmonic Analysis:

Filed under: General — Tags: — m759 @ 8:26 PM

The Fourfold Square and Eightfold Cube

Related material:  A Google image search for "field dream" + log24.

Saturday, December 28, 2019

Caballo Blanco

Filed under: General — Tags: , , , — m759 @ 9:02 AM

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] 3-space is shown to be of order [in modern notation] 8! ….
To every transformation of the 3-space there corresponds
a transformation of the [projective] 5-space. In the 5-space, there are
determined 8 sets of 7 points each, ‘heptads’ ….”

— George M. Conwell, “The 3-space PG (3, 2) and Its Group,”
The Annals of Mathematics , Second Series, Vol. 11, No. 2 (Jan., 1910),
pp. 60-76.

“It must be remarked that these 8 heptads are the key  to an elegant proof….”

— Philippe Cara, “RWPRI Geometries for the Alternating Group A8,” in
Finite Geometries: Proceedings of the Fourth Isle of Thorns Conference
(July 16-21, 2000), Kluwer Academic Publishers, 2001, ed. Aart Blokhuis,
James W. P. Hirschfeld, Dieter Jungnickel, and Joseph A. Thas, pp. 61-97.

Sunday, December 22, 2019

M24 from the Eightfold Cube

Filed under: General — Tags: , — m759 @ 12:01 PM

Exercise:  Use the Guitart 7-cycles below to relate the 56 triples
in an 8-set (such as the eightfold cube) to the 56 triangles in
a well-known Klein-quartic hyperbolic-plane tiling. Then use
the correspondence of the triples with the 56 spreads of PG(3,2)
to construct M24.

Click image below to download a Guitart PowerPoint presentation.

See as well earlier posts also tagged Triangles, Spreads, Mathieu.

Friday, December 20, 2019

Triangles, Spreads, Mathieu…

Filed under: General — Tags: , — m759 @ 1:38 AM


An addendum for the post “Triangles, Spreads, Mathieu” of Oct. 29:

Wednesday, December 11, 2019

Miracle Octad Generator Structure

Filed under: General — Tags: , , — m759 @ 11:44 PM

Miracle Octad Generator — Analysis of Structure

(Adapted from Eightfold Geometry, a note of April 28, 2010.
See also the recent post Geometry of 6 and 8.)

Friday, November 22, 2019

Triangles, Spreads, Mathieu …

Filed under: General — Tags: , — m759 @ 4:39 PM

Continued from October 29, 2019.

More illustrations (click to enlarge) —

Thursday, October 31, 2019

56 Triangles

Filed under: General — Tags: , — m759 @ 8:09 AM

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.”

Tuesday, October 29, 2019

Triangles, Spreads, Mathieu

Filed under: General — Tags: , — m759 @ 8:04 PM

There are many approaches to constructing the Mathieu
group M24. The exercise below sketches an approach that
may or may not be new.


It is well-known that

 There are 56 triangles in an 8-set.
There are 56 spreads in PG(3,2).
The alternating group An is generated by 3-cycles.
The alternating group Ais isomorphic to GL(4,2).

Use the above facts, along with the correspondence
described below, to construct M24.

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 —

Monday, October 21, 2019

Algebra and Space… Illustrated.

Filed under: General — Tags: , — m759 @ 4:26 PM

Related entertainment —


   George Steiner

"Perhaps an insane conceit."




See Quantum Tesseract Theorem .


Perhaps Not.


 See Dirac and Geometry .

Wednesday, October 9, 2019

The Joy of Six

Note that in the pictures below of the 15 two-subsets of a six-set,
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).

Anticommuting Dirac matrices as spreads of projective lines

Related narrative The "Quantum Tesseract Theorem."

Sunday, July 14, 2019

Old Pathways in Science:

Filed under: General — Tags: , — m759 @ 12:37 PM

The Quantum Tesseract Theorem Revisited

From page 274 — 

"The secret  is that the super-mathematician 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 E-operators.
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 —

Anticommuting Dirac matrices as spreads of projective lines

Compare and contrast Eddington's use of the word "perpendicular"
with a later use of the word by Saniga and Planat.

Sunday, May 26, 2019

Nine-Dot Patterns

Filed under: General — m759 @ 9:00 PM

Some nine-dot patterns of greater interest:

IMAGE- Actions of the unit quaternions in finite geometry, on a ninefold square and on an eightfold cube

Friday, May 10, 2019

I Ching g6

Filed under: General — Tags: , , , — m759 @ 12:25 PM

For fans of Resonance Science

When the men on the chessboard
get up and tell you where to go ….”

Desperately Seeking Resonance

Filed under: General — Tags: , , — m759 @ 10:46 AM


Also from Fall Equinox 2018 — Looney Tune for Physicists

Thursday, May 9, 2019

Blade and Chalice at the Museum

Filed under: General — Tags: — m759 @ 9:24 PM

(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.

Defense Against the Dark Arts

Filed under: General — Tags: , , — m759 @ 7:00 PM

F. Lanier Graham chess set (king-queen arrangement by the Wachowskis)

Wednesday, May 8, 2019

When the Men

Filed under: General — Tags: — m759 @ 6:33 PM

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.

Saturday, December 22, 2018


Filed under: General,Geometry — Tags: , — m759 @ 12:34 PM

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 —

Wednesday, December 12, 2018

An Inscape for Douthat

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" —

Detail of Feb. 20, 1986, note by Steven H. Cullinane on Weyl's 'relativity problem'

Kibler's 2008 'Variations on a theme' illustrated.

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."

Thursday, December 6, 2018

The Mathieu Cube of Iain Aitchison

This journal ten years ago today —

Surprise Package

Santa and a cube
From a talk by a Melbourne mathematician on March 9, 2018 —

The Mathieu group cube of Iain Aitchison (2018, Hiroshima)

The source — Talk II below —

Search Results

pdf of talk I  (March 8, 2018)


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)


Iain Aitchison. Hiroshima  University March 2018. (IRA: Hiroshima  03-2018). Highly symmetric objects II.



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 . . .

The Eightfold Cube: The Beauty of Klein's Simple Group

   Image from Christmas Day 2005.

Sunday, November 18, 2018

Space Music

Filed under: General,Geometry — Tags: , , — m759 @ 9:27 AM

'The Eddington Song,' based on 'The Philosophy of Physical Science,' p. 141 (1939)

Update of Nov. 19 —

"Design is how it works." — Steve Jobs

See also www.cullinane.design.

Sunday, October 21, 2018

For Connoisseurs of Bad Art

Filed under: General — Tags: , — m759 @ 7:00 PM

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.

Saturday, October 20, 2018

Study in Blue and Pink

Filed under: General,Geometry — Tags: , — m759 @ 3:00 PM

Related Log24 posts — See Blade + Chalice.

Tuesday, September 25, 2018


Filed under: General — Tags: , , — m759 @ 10:10 AM

See some posts related to three names
associated with Trinity College, Cambridge —

Atiyah + Shaw + Eddington .

Saturday, September 1, 2018

Ron Shaw — D. 21 June 2016

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 —


Sunday, July 1, 2018

Deutsche Ordnung

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

Steiner quadruple system in eightfold cube

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
Jan Zwicky, University of Victoria
09:00-09:40 Friday, April 5, 2013

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

Friday, June 29, 2018

Triangles in the Eightfold Cube

Filed under: General,Geometry — Tags: , — m759 @ 9:10 PM

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 eight-triangle 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 eight-triangle 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

Monday, March 12, 2018

“Quantum Tesseract Theorem?”

Filed under: General,Geometry — Tags: , , — m759 @ 11:00 AM

Remarks related to a recent film and a not-so-recent film.

For some historical background, see Dirac and Geometry in this journal.

Also (as Thas mentions) after Saniga and Planat —

The Saniga-Planat paper was submitted on December 21, 2006.

Excerpts from this  journal on that date —

A Halmos tombstone and the tale of HAL and the pod bay doors

     "Open the pod bay doors, HAL."

Saturday, February 17, 2018

The Binary Revolution

Michael Atiyah on the late Ron Shaw

Phrases by Atiyah related to the importance in mathematics
of the two-element 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 year-end 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:

God’s Dice

Filed under: General,Geometry — Tags: , — m759 @ 10:45 AM

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 —


Tuesday, October 10, 2017

Another 35-Year Wait

Filed under: G-Notes,General,Geometry — Tags: , — m759 @ 9:00 PM

The title refers to today's earlier post "The 35-Year 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 Lévi-Strauss.

The Lévi-Strauss 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 —

Koen Thas, 'Unextendible Mututally Unbiased Bases' (2016)

See also Inscape in this  journal and posts tagged Dirac and Geometry.

Tuesday, September 12, 2017


Filed under: General,Geometry — Tags: — m759 @ 12:18 PM

"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.

Monday, September 11, 2017

New Depth

Filed under: General,Geometry — Tags: — m759 @ 9:48 PM

A sentence from the New York Times Wire  discussed in the previous post

NYT Wire on Len Wein: 'Through characters like Wolverine and Swamp Thing, he helped bring a new depth to his art form.'

"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 …

Monday, June 26, 2017

Upgrading to Six

This post was suggested by the previous post — Four Dots —
and by the phrase "smallest perfect" in this journal.

Related material (click to enlarge) —

Detail —

From the work of Eddington cited in 1974 by von Franz —

See also Dirac and Geometry and Kummer in this journal.

Updates from the morning of June 27 —

Ron Shaw on Eddington's triads "associated in conjugate pairs" —

For more about hyperbolic  and isotropic  lines in PG(3,2),
see posts tagged Diamond Theorem Correlation.

For Shaw, in memoriam — See Contrapuntal Interweaving and The Fugue.

Monday, June 12, 2017


Filed under: General,Geometry — m759 @ 1:19 PM

The "bubble" passage in the previous post suggests a review of
a post from December 21, 2006, with the following images —


Update of 11:01 PM ET the same day, June 12, 2017 —

Related material for the Church of Synchronology

From a tech-article series that began on Halloween 2006 and
ended on the date of the above Geometry's Tombstones post —

Compare and contrast (from a post of Feb. 27, 2017) —

“Lord Arglay had a suspicion that the Stone would be
purely logical.  Yes, he thought, but what, in that sense,
were the rules of its pure logic?”

Many Dimensions  (1931), by Charles Williams

See also "The Geometry of Logic:
Finite Geometry and the 16 Boolean Connectives
by Steven H. Cullinane in 2007.

Sunday, March 26, 2017

Seagram Studies

Filed under: General,Geometry — Tags: — m759 @ 9:00 PM

From a search in this journal for Seagram + Tradition

Related art:  Saturday afternoon's Twin Pillars of Symmetry.

Saturday, March 25, 2017

Twin Pillars of Symmetry

Filed under: General,Geometry — Tags: — m759 @ 1:00 PM

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 3-dimensional simplex and the {4, 3, 3},
see my note from May 21, 2014, on the tetrahedron and the tesseract.

Like Decorations in a Cartoon Graveyard

Filed under: General,Geometry — Tags: — m759 @ 4:00 AM

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.

Friday, March 17, 2017

Narrative for Westworld

Filed under: General,Geometry — Tags: — m759 @ 12:12 PM

“That corpse you planted
          last year in your garden,
  Has it begun to sprout?
          Will it bloom this year?  
  Or has the sudden frost
          disturbed its bed?”

— T. S. Eliot, “The Waste Land

Coxeter exhuming Geometry

Ball and Coxeter, 'Mathematical Recreations,' Twelfth Edition

Click the book for a video.

Friday, February 3, 2017

Raiders of the Lost Chalice

Filed under: General — Tags: — m759 @ 9:30 AM

Personally, I prefer
the religious symbolism
of Hudson Hawk .

Sunday, December 25, 2016

Credit Where Due

Filed under: General,Geometry — Tags: — m759 @ 12:00 PM

See also Robert M. Pirsig in this journal on Dec. 26, 2012.

Saturday, December 24, 2016

Early X Piece

Filed under: General,Geometry — Tags: — m759 @ 11:00 AM

In memory of an American artist whose work resembles that of
the Soviet constructivist Karl Ioganson (c. 1890-1929).

The American artist reportedly died on Thursday, Dec. 22, 2016.

"In fact, the (re-)discovery of this novel structural principle was made in 1948-49 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 GoughOCTOBER  Magazine, MIT,
      Issue 84, Spring 1998, pp. 91-117

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.

Thursday, December 22, 2016

The Laugh-Hospital

Filed under: General,Geometry — Tags: — m759 @ 12:00 PM

Constructivism in mathematics and the laughing academy

See also, from the above publication date, Hudson's Inscape.
The inscape is illustrated in posts now tagged Laughing Academy.

Constructivist Witness

Filed under: General,Geometry — Tags: — m759 @ 1:00 AM

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.

Monday, December 19, 2016


Filed under: General,Geometry — Tags: , — m759 @ 10:25 PM

See also all posts now tagged Memory, History, Geometry.

Tetrahedral Cayley-Salmon Model

Filed under: General,Geometry — Tags: , — m759 @ 9:38 AM

The figure below is one approach to the exercise
posted here on December 10, 2016.

Tetrahedral model (minus six lines) of the large Desargues configuration

Some background from earlier posts —

IMAGE- Geometry of the Six-Set, Steven H. Cullinane, April 23, 2013

Click the image below to enlarge it.

Polster's tetrahedral model of the small Desargues configuration

Sunday, December 18, 2016

Two Models of the Small Desargues Configuration

Filed under: General,Geometry — Tags: — m759 @ 12:00 PM

Click image to enlarge.

Polster's tetrahedral model of the small Desargues configuration

See also the large  Desargues configuration in this journal.

Saturday, December 17, 2016

Tetrahedral Death Star

Filed under: General,Geometry — Tags: , — m759 @ 10:00 PM

Continuing the "Memory, History, Geometry" theme
from yesterday

See Tetrahedral,  Oblivion,  and Tetrahedral Oblivion.

IMAGE- From 'Oblivion' (2013), the Mother Ship

"Welcome home, Jack."

Friday, December 16, 2016

Read Something That Means Something

Filed under: General,Geometry — Tags: — m759 @ 5:29 PM

Rothko’s Swamps

Filed under: General,Geometry — Tags: — m759 @ 2:45 AM

"… 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

Tuesday, December 13, 2016

The Thirteenth Novel

Filed under: General,Geometry — Tags: — m759 @ 4:00 PM

John Updike on Don DeLillo's thirteenth novel, Cosmopolis

" DeLillo’s post-Christian search for 'an order at some deep level'
has brought him to global computerization:
'the zero-oneness 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 Non-Euclidean Revolution

Saturday, December 10, 2016

Folk Etymology

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 small-configuration model's striking symmetry?  
(Note: The related figure below from May 21, 2014, is not
necessarily very helpful. Try the Wolfram Demonstrations
, 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 38-39 in the
poem by Marsha de la O titled "A Natural History of Light."

Cézanne's Greetings.

Wednesday, December 7, 2016

Spreads and Conwell’s Heptads

Filed under: General,Geometry — Tags: — m759 @ 7:11 PM

For a concise historical summary of the interplay between
the geometry of an 8-set and that of a 16-set that is
involved in the the Miracle Octad Generator approach
to the large Mathieu group M24, see Section 2 of 

Alan R. Prince
A near projective plane of order 6 (pp. 97-105)
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.

Tuesday, September 13, 2016

Parametrizing the 4×4 Array

Filed under: General,Geometry — Tags: , , , — m759 @ 10:00 PM

The previous post discussed the parametrization of 
the 4×4 array as a vector 4-space over the 2-element 
Galois field GF(2).

The 4×4 array may also be parametrized by the symbol
0  along with the fifteen 2-subsets of a 6-set, 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 2-element 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:

IMAGE- Geometry of the Six-Set, Steven H. Cullinane, April 23, 2013

Click images for further details.

Monday, September 12, 2016

The Kummer Lattice

Filed under: General,Geometry — Tags: , , , — m759 @ 2:00 PM

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 4-space 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 M24 

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 two-subsets of a six-set. See the post of Sept. 13.

Tuesday, July 12, 2016

Group Elements and Skew Lines

Filed under: General,Geometry — Tags: — m759 @ 12:00 AM

The following passage by Igor Dolgachev (Good Friday, 2003
seems somewhat relevant (via its connection to Kummer's 166 )
to previous remarks here on Dirac matrices and geometry

Note related remarks from E. M. Bruins in 1959 —

First page of 'Configurations in Quantum Mechanics,' by E.M. Bruins, 1959

Friday, June 3, 2016

Bruins and van Dam

Filed under: General,Geometry — Tags: , — m759 @ 8:00 AM

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:

Wednesday, May 25, 2016


Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

"Studies of spin-½ 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 19771978, and 1986 versions by 
Steven H. Cullinane,   the 1987 version by R. T. Curtis, and
the 1988 Conway-Sloane version illustrated below —

Cullinane, 1977

IMAGE- Hypercube and 4x4 matrix from the 1976 'Diamond Theory' preprint, as excerpted in 'Computer Graphics and Art'

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.

Kummer and Dirac

From "Projective Geometry and PT-Symmetric 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 spin-½ 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 Plucker-Klein correspondence between lines of
a three-dimensional projective space and points of a quadric
in a five-dimensional 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 166 configuration . . . ."


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 166 configuration . . . .

See that structure in this  journal, for instance —

See as well yesterday morning's post.

Tuesday, May 24, 2016

Rosenhain and Göpel Revisited

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 many-faceted 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 4-space over
the two-element 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)

IMAGE- Desargues's theorem in light of Galois geometry

Göpel tetrads as 15 of the 35 projective lines in PG(3,2)

Anticommuting Dirac matrices as spreads of projective lines

Related terminology describing the Göpel tetrads above

Ron Shaw on symplectic geometry and a linear complex in PG(3,2)

Monday, February 8, 2016

A Game with Four Letters

Filed under: General,Geometry — Tags: , — m759 @ 2:56 PM

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 ).

Saturday, November 21, 2015

The Zero System

Filed under: General,Geometry — Tags: — m759 @ 12:00 AM

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 zero-system polarity in PG(3,2).  For the zero system 
   itself, see diamond-theorem correlation.

Friday, November 20, 2015

Anticommuting Dirac Matrices as Skew Lines

Filed under: General,Geometry — Tags: , — m759 @ 11:45 PM

(Continued from November 13)

The work of Ron Shaw in this area, ca. 1994-1995, 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 3-space over the 2-element Galois field.

Here is an explicit picture —

Anticommuting Dirac matrices as spreads of projective lines


Arfken, George B., Mathematical Methods for Physicists , Third Edition,
Academic Press, 1985, pages 213-214

Cullinane, Steven H., Notes on Groups and Geometry, 1978-1986

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 more-recent related material from the Slovak school of
finite geometry and quantum theory —

Saniga, 'Finite Projective Spaces, Geometric Spreads of Lines and Multi-Qubits,' excerpt

The matrices underlying the Saniga paper are those of Pauli, not
those of Dirac, but these two sorts of matrices are closely related.

Thursday, November 19, 2015

Highlights of the Dirac-Mathieu Connection

Filed under: General,Geometry — Tags: — m759 @ 12:00 PM

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 —

Friday, November 13, 2015

A Connection between the 16 Dirac Matrices and the Large Mathieu Group

Note that the six anticommuting sets of Dirac matrices listed by Arfken
correspond exactly to the six spreads in the above complex of 15 projective
lines of PG(3,2) fixed under a symplectic polarity (the diamond theorem
). As I noted in 1986, this correlation underlies the Miracle
Octad Generator of R. T. Curtis, hence also the large Mathieu group.


Arfken, George B., Mathematical Methods for Physicists , Third Edition,
Academic Press, 1985, pages 213-214

Cullinane, Steven H., Notes on Groups and Geometry, 1978-1986

Related material:

The 6-set 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 1994-1995)—

Thursday, March 26, 2015

The Möbius Hypercube

Filed under: General,Geometry — Tags: , — m759 @ 12:31 AM

The incidences of points and planes in the
Möbius 8 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 face-planes 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 point-plane
incidences, as dotless, and some as hollow dots.  The figure,
"Gallucci's version of Möbius's 84," 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 (x4, x3, x2, x1) over the two-element
Galois field.  In that context, the 4×4 array may be called, instead
of a Möbius hypercube , a Galois tesseract .

*  "Self-Dual Configurations and Regular Graphs," 
    Bulletin of the American Mathematical Society,
    Vol. 56 (1950), pp. 413-455

The subscripts' usual 1-2-3-4 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 16-element Galois field GF(24).  See the Log24 post
     Vector Addition in a Finite Field (Jan. 5, 2013).

Monday, March 23, 2015

Gallucci’s Möbius Configuration

Filed under: General,Geometry — Tags: — m759 @ 12:05 PM

From H. S. M. Coxeter's 1950 paper
"Self-Dual 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."

Wednesday, August 13, 2014

Symplectic Structure continued

Filed under: General,Geometry — Tags: , , , — m759 @ 12:00 PM

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) —

Sunday, August 3, 2014

The Omega Matrix

Filed under: General,Geometry — Tags: , , — m759 @ 10:31 PM

Shown below is the matrix Omega from notes of Richard Evan Schwartz.
See also earlier versions (1976-1979) by Steven H. Cullinane.

IMAGE- The matrix Omega from notes of Richard Evan Schwartz. See also earlier versions (1977-1979) 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.)

Thursday, July 31, 2014

Zero System

Filed under: General,Geometry — Tags: , — m759 @ 6:11 PM

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. 903-906.

An example in PG(3,2), the projective 3-space over the
two-element Galois field GF(2):

IMAGE- The natural symplectic polarity in PG(3,2), illustrating a symplectic structure

See also the 10 AM ET post of Sunday, June 8, 2014, on this topic.

Friday, March 21, 2014

Three Constructions of the Miracle Octad Generator

Filed under: General,Geometry — Tags: , , — m759 @ 12:24 PM

IMAGE- Two constructions, by Turyn/Curtis, and by Conway, of the Miracle Octad Generator

See also a Log24 post on this subject from Dec. 14, 2013,
especially (scroll down) the update of March 9, 2014.

Related material on the Turyn-Curtis construction
from the University of Cambridge —

— Slide by “Dr. Parker” — Apparently Richard A. Parker —
Lecture 4, “Discovering M24,” in slides for lectures 1-8 from lectures
at Cambridge in 2010-2011 on “Sporadic and Related Groups.”
See also the Parker lectures of 2012-2013 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+4-partitions of an 8-set with the 35 lines of the projective 3-space
over the 2-element field, PG(3, 2), is essentially the same correspondence
as that constituting Curtis’s 1976 MOG.

See The Diamond Theorem,  Finite RelativityGalois Space,
Generating the Octad Generator, and The Klein Correspondence.

Update of March 22-March 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
S3 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 “by-hand” 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 —

IMAGE- Discovery of the S_3 action on bricks in the Conwell-Cullinane 'by-hand' approach to octad-building

Thursday, March 20, 2014

Classical Galois

Filed under: General,Geometry — Tags: , , — m759 @ 12:26 PM

IMAGE- The large Desargues configuration and Desargues's theorem in light of Galois geometry

Click image for more details.

To enlarge image, click here.

Thursday, February 6, 2014

The Representation of Minus One

Filed under: General,Geometry — Tags: , , — m759 @ 6:24 AM

For the late mathematics educator Zoltan Dienes.

“There comes a time when the learner has identified
the abstract content of a number of different games
and is practically crying out for some sort of picture
by means of which to represent that which has been
gleaned as the common core of the various activities.”

— Article by “Melanie” at Zoltan Dienes’s website

Dienes reportedly died at 97 on Jan. 11, 2014.

From this journal on that date —


A star figure and the Galois quaternion.

The square root of the former is the latter.

Update of 5:01 PM ET Feb. 6, 2014 —

An illustration by Dienes related to the diamond theorem —

See also the above 15 images in


and versions of the 4×4 coordinatization in  The 4×4 Relativity Problem
(Jan. 17, 2014).

Friday, January 17, 2014

The 4×4 Relativity Problem

Filed under: General,Geometry — Tags: , , , — m759 @ 11:00 PM

The sixteen-dot square array in yesterday’s noon post suggests
the following remarks.

“This is the relativity problem:  to fix objectively a class of
equivalent coordinatizations and to ascertain the group of
transformations S mediating between them.”

— Hermann Weyl, The Classical Groups ,
Princeton University Press, 1946, p. 16

The Galois tesseract  appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977—

IMAGE- Hypercube and 4x4 matrix from the 1976 'Diamond Theory' preprint, as excerpted in 'Computer Graphics and Art'

The 1977 matrix Q is echoed in the following from 2002—

IMAGE- Dolgachev and Keum, coordinatization of the 4x4 array in 'Birational Automorphisms of Quartic Hessian Surfaces,' AMS Transactions, 2002

A different representation of Cullinane’s 1977 square model of the
16-point affine geometry over the two-element Galois field GF(2)
is supplied by Conway and Sloane in Sphere Packings, Lattices and Groups   
(first published in 1988) :

IMAGE- The Galois tesseract as a four-dimensional vector space, from a diagram by Conway and Sloane in 'Sphere Packings, Lattices, and Groups'

Here a, b, c, d   are basis vectors in the vector 4-space over GF(2).
(For a 1979 version of this vector space, see AMS Abstract 79T-A37.)

See also a 2011 publication of the Mathematical Association of America —

From 'Beautiful Mathematics,' by Martin Erickson, an excerpt on the Cullinane diamond theorem (with source not mentioned)

Friday, December 20, 2013

For Emil Artin

Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

(On His Dies Natalis )

An Exceptional Isomorphism Between Geometric and
Combinatorial Steiner Triple Systems Underlies 
the Octads of the M24 Steiner System S(5, 8, 24).

This is asserted in an excerpt from… 

"The smallest non-rank 3 strongly regular graphs
​which satisfy the 4-vertex condition"
by Mikhail Klin, Mariusz Meszka, Sven Reichard, and Alex Rosa,

(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 geometric-combinatorial isomorphism.

Monday, October 14, 2013

Up and Down

Filed under: General — Tags: , — m759 @ 9:29 AM

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

— http://www.heraclitusfragments.com/B60/index.html

IMAGE- Fetzer on ambiguity in Mann's 'Doctor Faustus'

See also Blade and Chalice and, for a less Faustian
approach, Universe of Discourse.

IMAGE- Logic related to 'the arsenal of algebraic analysis tools for fields'

Further context:  Not Theology.

Saturday, September 21, 2013

Geometric Incarnation

The  Kummer 166  configuration  is the configuration of sixteen
6-sets within a 4×4 square array of points in which each 6-set
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 166 to a number
of other mathematical concepts — "geometric incarnation."

Geometric Incarnation in the Galois Tesseract

Related material from finitegeometry.org —

IMAGE- 4x4 Geometry: Rosenhain and Göpel Tetrads and the Kummer Configuration

* Apparently from David Lehavi on March 18, 2007, at Citizendium .

Thursday, September 5, 2013

Moonshine II

Filed under: General,Geometry — Tags: , , , , , — m759 @ 10:31 AM

(Continued from yesterday)

The foreword by Wolf Barth in the 1990 Cambridge U. Press
reissue of Hudson's 1905 classic Kummer's Quartic Surface
covers some of the material in yesterday's post Moonshine.

The distinction that Barth described in 1990 was also described, and illustrated,
in my 1986 note "Picturing the smallest projective 3-space."  The affine 4-space
over the the finite Galois field GF(2) that Barth describes was earlier described—
within a 4×4 array like that pictured by Hudson in 1905— in a 1979 American
Mathematical Society abstract, "Symmetry invariance in a diamond ring."

"The distinction between Rosenhain and Goepel tetrads
is nothing but the distinction between isotropic and
non-isotropic planes in this affine space over the finite field."

The 1990 paragraph of Barth quoted above may be viewed as a summary
of these facts, and also of my March 17, 2013, note "Rosenhain and Göpel
Tetrads in PG(3,2)

Friday, July 5, 2013

Mathematics and Narrative (continued)

Filed under: General,Geometry — Tags: , , — m759 @ 6:01 PM

Short Story — (Click image for some details.)

IMAGE- Andries Brouwer and the Galois Tesseract

Parts of a longer story —

The Galois Tesseract and Priority.

Monday, June 10, 2013

Galois Coordinates

Filed under: General,Geometry — Tags: , — m759 @ 10:30 PM

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 Galois-field 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 pre-1976
coordinatization of a 4×4 array  by the 16 elements
of the vector 4-space  over the Galois field with two
elements, GF(2).

Such coordinatizations are important because of their
close relationship to the Mathieu group 24 .

See a preprint by Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of Kummer
surfaces in the Mathieu group 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.

Saturday, June 1, 2013


Filed under: General,Geometry — Tags: , , — m759 @ 4:00 PM

"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—

IMAGE- The diamond-theorem affine group of order 322,560, published without acknowledgment of its source by the Mathematical Association of America in 2011

Tuesday, May 28, 2013


The hypercube  model of the 4-space over the 2-element Galois field GF(2):

IMAGE- A hyperspace model of the 4D vector space over GF(2)

The phrase Galois tesseract  may be used to denote a different model
of the above 4-space: 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 Galois-tesseract model of the 4-space over GF(2).

IMAGE- Octads within the Curtis MOG, which uses a 4x4-array model of the 4D vector space over GF(2)

The thirty-five 4×4 structures within the MOG:

IMAGE- The 35 square patterns within the Curtis MOG

Curtis himself first described these 35 square MOG patterns
combinatorially, (as his title indicated) rather than
algebraically or geometrically:

IMAGE- R. T. Curtis's combinatorial construction of 4x4 patterns within the Miracle Octad Generator

A later book co-authored by Sloane, first published in 1988,
did  recognize the 4×4 MOG patterns as based on the 4×4
Galois-tesseract 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 MacWilliams-Sloane book was first published):

IMAGE- Hypercube and 4x4 matrix from the 1976 'Diamond Theory' preprint, as excerpted in 'Computer Graphics and Art'

Sunday, May 19, 2013

Priority Claim

Filed under: General,Geometry — Tags: , , , — m759 @ 9:00 AM

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 F24 built on I \ O9. 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 \ O9."

[Cur89] reference:
 R. T. Curtis, "Further elementary techniques using
the miracle octad generator," Proc. Edinburgh
Math. Soc. 
32 (1989), 345-353 (received on
July 20, 1987).

— Anne Taormina and Katrin Wendland,
    "The overarching finite symmetry group of Kummer
      surfaces in the Mathieu group 24 ,"
     arXiv.org > hep-th > 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).

Sunday, April 28, 2013

The Octad Generator

Filed under: General,Geometry — Tags: , , — m759 @ 11:00 PM

… And the history of geometry  
Desargues, Pascal, Brianchon and Galois
in the light of complete n-points 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 five-point."

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 five-point 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 6-point 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 Six-Set (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 M24 —

IMAGE- Geometry of the Six-Set, Steven H. Cullinane, April 23, 2013

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.

Thursday, April 25, 2013

Rosenhain and Göpel Revisited

Filed under: General,Geometry — Tags: , — m759 @ 5:24 PM

Some historical background for today's note on the geometry
underlying the Curtis Miracle Octad Generator (MOG):

IMAGE- Bateman in 1906 on Rosenhain and Göpel tetrads

The above incidence diagram recalls those in today's previous post
on the MOG, which is used to construct the large Mathieu group M24.

For some related material that is more up-to-date, search the Web
for Mathieu + Kummer .

Note on the MOG Correspondence

Filed under: General,Geometry — Tags: , — m759 @ 4:15 PM

In light of the April 23 post "The Six-Set,"
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
2-subsets and 3-subsets of a 6-set. This underlies M24."

A related note from today:

IMAGE- Three-sets in the Curtis MOG

Saturday, April 6, 2013

Pascal via Curtis

Filed under: General,Geometry — Tags: , — m759 @ 9:17 AM

Click image for some background.

IMAGE- The Miracle Octad Generator (MOG) of R.T. Curtis

Shown above is a rearranged version of the
Miracle Octad Generator (MOG) of R. T. Curtis
("A new combinatorial approach to M24,"
Math. Proc. Camb. Phil. Soc., 79 (1976), 25-42.)

The 8-subcell rectangles in the left part of the figure may be
viewed as illustrating (if the top left subcell is disregarded)
the thirty-five 3-subsets of a 7-set.

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 354 Configuration:

IMAGE- Branko Grünbaum on Danzer's configuration

"Combinatorially, Danzer’s configuration can be interpreted
as defined by all 3-sets and all 4-sets that can be formed
by the elements of a 7-element set; each 'point' is represented
by one of the 3-sets, and it is incident with those lines
(represented by 4-sets) that contain the 3-set."

— 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).

Thursday, March 7, 2013

Proof Symbol

Filed under: General,Geometry — m759 @ 8:28 PM

Today's previous post recalled a post
from ten years before yesterday's  date.

The subject of that post was the
Galois tesseract.

Here is a post from ten years before
today's  date

The subject of that  post is the Halmos

"The symbol    is used throughout the entire book
in place of such phrases as 'Q.E.D.' or 'This
completes the proof of the theorem' to signal
the end of a proof."

Measure Theory  (1950)

For exact proportions, click on the tombstone.

For some classic mathematics related
to the proportions, see September 2003.

Thursday, February 28, 2013


Filed under: General,Geometry — Tags: , — m759 @ 1:06 PM

A different dodecahedral space (Log24 on Oct. 3, 2011)—

R. T. Curtis, symmetric generation of M12 in a dodecahedron

Wednesday, February 13, 2013


Filed under: General,Geometry — Tags: , — m759 @ 9:29 PM

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—

IMAGE- The Penrose diamond and the Klein quadric

The Klein quadric occurs in the five-dimensional projective space
over a field. If the field is the two-element 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 M24. These properties are also 
relevant to the 1976 "Diamond Theory" monograph.

For some background on the quadric, see (for instance)

IMAGE- Stroppel on the Klein quadric, 2008

See also The Klein Correspondence,
Penrose Space-Time, 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, 43-78.

Those who prefer story to structure may consult 

  1. today's previous post on the Penrose diamond
  2. the remarks of Scott Aaronson on August 17, 2012
  3. the remarks in this journal on that same date
  4. the geometry of the 4×4 array in the context of M24.

Tuesday, February 5, 2013


Filed under: General — Tags: — m759 @ 3:33 PM

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 (1894-1967), 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.



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.

IMAGE- Logic related to 'the arsenal of algebraic analysis tools for fields'

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.

Saturday, January 5, 2013

Vector Addition in a Finite Field

Filed under: General,Geometry — Tags: , — m759 @ 10:18 AM

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 4-space over GF(2)—

The same field, again disguised as an affine 4-space,
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 vector-space 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
Self-Dual Configurations and Regular Graphs.”

Saturday, December 8, 2012

Defining the Contest…

Filed under: General,Geometry — Tags: , , , , — m759 @ 5:48 AM

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 —

IMAGE- Diamond Theory patterns found in a 2012 Royal Society paper

"First steps"?  The mathematics underlying such patterns
was presented 35 years earlier, in Diamond Theory.

* See Gombrich-Douat in this journal.

Monday, November 19, 2012

Poetry and Truth

Filed under: General,Geometry — Tags: , , , , — m759 @ 7:59 PM

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

Spaces as Hypercubes

— 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 M24."

Monday, October 8, 2012

Air America

Filed under: General,Geometry — Tags: , — m759 @ 4:00 AM

Related entertainment—

The song being performed in the above trailer 
for Air America  is "A Horse with No Name."

See  "Instantia Crucis" and "Winning."

Thursday, August 16, 2012

Raiders of the Lost Tesseract

Filed under: General,Geometry — Tags: — m759 @ 8:00 PM

(Continued from August 13. See also Coxeter Graveyard.)

Coxeter exhuming Geometry

Here the tombstone says
"GEOMETRY… 600 BC — 1900 AD… R.I.P."

In the geometry of Plato illustrated below,
"the figure of eight [square] feet" is not , at this point
in the dialogue, the diamond in Jowett's picture.

An 1892 figure by Jowett illustrating Plato's Meno

Jowett's picture is nonetheless of interest for
its resemblance to a figure drawn some decades later
by the Toronto geometer H. S. M. Coxeter.

A similar 1950 figure by Coxeter illustrating a tesseract

For a less scholarly, but equally confusing, view of the number 8,
see The Eight , a novel by Katherine Neville.

Thursday, July 12, 2012

Galois Space

Filed under: General,Geometry — Tags: , — m759 @ 6:01 PM

An example of lines in a Galois space * —

The 35 lines in the 3-dimensional 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 3-set 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  
(Cambridge U. Press, 1960, 488-499.)

(Update of Jan. 5, 2013— This post has been added to finitegeometry.org.)

Monday, June 18, 2012


Filed under: General,Geometry — Tags: , , , — m759 @ 11:00 PM

"Poetry is an illumination of a surface…."

— Wallace Stevens

IMAGE- NY Times online front page, June 18, 2012- New Microsoft 'Surface' computer

Some poetic remarks related to a different surface, Klein's Quartic

This link between the Klein map κ and the Mathieu group M24
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 M24
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—

Sunday, June 17, 2012

Congruent Group Actions

Filed under: General,Geometry — Tags: , , , — m759 @ 9:00 PM

A Google search today yielded no results
for the phrase "congruent group actions."

Places where this phrase might prove useful include—

Saturday, April 14, 2012

Scottish Algebra

Filed under: General,Geometry — Tags: — m759 @ 11:59 PM

Two papers suggested by Google searches tonight—



File Format: PDF/Adobe Acrobat – View as HTML

by RT Curtis2001Related articles

This paper is based on a talk given at the Scottish Algebra Day 1998 in Edinburgh. ……

Curtis discusses the exceptional outer automorphism of S6
as arising from group actions of PGL(2,5).

See also Cameron and Galois on PGL(2,5)—



File Format: PDF/Adobe Acrobat – Quick View

by PJ CAMERON1975Cited by 14Related 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 2-dimensional projective general linear Received 24 October, 1973

Illustration from Cameron (1973)—


Monday, January 23, 2012

How It Works

Filed under: General,Geometry — Tags: , — m759 @ 7:59 PM


J. H. Conway in 1971 discussed the role of an elementary abelian group
of order 16 in the Mathieu group M24. His approach at that time was
purely algebraic, not geometric—

IMAGE- J. H. Conway in 1971 discussed the role of the elementary abelian group of order 16 in the Mathieu group M24. His approach then was purely algebraic, not geometric.

For earlier (and later) discussions of the geometry  (not the algebra )
of that order-16 group (i.e., the group of translations of the affine space
of 4 dimensions over the 2-element field), see The Galois Tesseract.

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