See as well the post "Advanced Study" of October 2, 2017, and …
Update . . .
https://x.com/LGBTMath/status/1641817634349133824 —
Sunday, March 9, 2025
Coxeter’s “Pleasantly Discursive”
“What Is Truth?” . . . Continues.
Comments Off on Coxeter’s “Pleasantly Discursive”
“What Is Truth?” . . . Continues.
“What Is Truth?” . . . Continues.
Sunday, August 7, 2022
Friday, August 30, 2019
The Coxeter Aleph
The previous post displayed part of a page from
a newspaper published the day Olivia Newton-John
turned 21 — Friday, September 26, 1969.
A meditation, with apologies to Coleridge:
In Xanadu did Newton-John
A stately pleasure-square decree
Where Aleph the sacred symbol ran
Through subsquares measureless to man.
A related video —
Beware, beware, her flashing eyes, her floating hair:
Set design —
As opposed to block design —
Comments Off on The Coxeter Aleph
Monday, March 31, 2014
Art Wars for Coxeter
Geometer H. S. M. Coxeter died on this date in 2003.
This evening’s daily number from the Keystone state: 822.
Comments Off on Art Wars for Coxeter
Monday, February 20, 2012
Coxeter and the Relativity Problem
In the Beginning…
"As is well known, the Aleph is the first letter of the Hebrew alphabet."
– Borges, "The Aleph" (1945)
From some 1949 remarks of Weyl—
"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, "Relativity Theory as a Stimulus in Mathematical Research," Proceedings of the American Philosophical Society , Vol. 93, No. 7, Theory of Relativity in Contemporary Science: Papers Read at the Celebration of the Seventieth Birthday of Professor Albert Einstein in Princeton, March 19, 1949 (Dec. 30, 1949), pp. 535-541
Weyl in 1946—:
"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
Coxeter in 1950 described the elements of the Galois field GF(9) as powers of a primitive root and as ordered pairs of the field of residue-classes modulo 3—
"… the successive powers of the primitive root λ or 10 are
λ = 10, λ2 = 21, λ3 = 22, λ4 = 02,
λ5 = 20, λ6 = 12, λ7 = 11, λ8 = 01.
These are the proper coordinate symbols….
(See Fig. 10, where the points are represented in the Euclidean plane as if the coordinate residue 2 were the ordinary number -1. This representation naturally obscures the collinearity of such points as λ4, λ5, λ7.)"
Coxeter's Figure 10 yields...
The Aleph
The details:
Coxeter's phrase "in the Euclidean plane" obscures the noncontinuous nature of the transformations that are automorphisms of the above linear 2-space over GF(3).
Comments Off on Coxeter and the Relativity Problem
Friday, January 7, 2011
Coxeter and the Aleph
In a nutshell —
Epigraph to "The Aleph," a 1945 story by Borges:
O God! I could be bounded in a nutshell,
and count myself a King of infinite space…
— Hamlet, II, 2
The story in book form, 1949
A 2006 biography of geometer H.S.M. Coxeter:
The Aleph (implicit in a 1950 article by Coxeter):
The details:
Related material: Group Actions, 1984-2009.
Comments Off on Coxeter and the Aleph
Thursday, July 8, 2010
Coxeter vs. Fano
The following excerpts from Coxeter's Projective Geometry
sketch his attitude toward geometry in characteristic two.
"… we develop a self-contained account… made
more 'modern' by allowing the field to be general
(though not of characteristic 2) instead of real or complex."
The "modern" in quotation marks may have been an oblique
reference to Segre's Lectures on Modern Geometry (1948, 1961).
(See Coxeter's reference 15 below.)
"It is interesting to see what happens…."
Another thing that happens if 1 + 1 = 0 —
It is no longer true that every finite reflection group
is a Coxeter group (provided we use Chevalley's
fixed-hyperplane definition of "reflection").
Comments Off on Coxeter vs. Fano
Monday, July 27, 2026
Note for a Jain Temple:
Magic Objects in Every Dimension
From ICM Invited Lectures & Panels for July 27, 2026:
|
Bhargava — "Inscribed on a wall near the entrance of the majestic Parshvanath Jain Temple in Khajuraho, India (built around 960 CE) is a 4 × 4 magic square with remarkable arithmetic properties. We will explain why this magic square is the unique one of its kind, up to certain transformations. This answers a question posed by the legendary geometer HSM Coxeter in 1938. We will also explain why there is a unique such magic object in every dimension." |
Related reading
-
Bhargava in this journal
-
How are algebraic, combinatorial, and geometric
structures unified within the Brick Space model?
- Coxeter, Symmetry, and Diabolic Magic Squares —
Comments Off on Note for a Jain Temple:
Magic Objects in Every Dimension
Magic Objects in Every Dimension
Wednesday, January 14, 2026
Ex Fano Apollinis —
Grok Benchmark
Grok showed admirable persistence and depth in its research, but failed
to completely understand what is meant by "coordinatization of a 3×3 array."
It did, however, indicate a related concept in a suggested further prompt.
For the example that suggested the original prompt, see The Coxeter Aleph.
Comments Off on Ex Fano Apollinis —
Grok Benchmark
Grok Benchmark
Saturday, November 15, 2025
Monday, July 14, 2025
Rosenhain and Göpel in 1950
See also Rosenhain and Göpel in this journal.
Comments Off on Rosenhain and Göpel in 1950
Wednesday, July 9, 2025
Plan 9 from Toronto
This 1950 use of Galois coordinates on a grid
foreshadows their later use by Cullinane to construct
the Miracle Octad Generator of R. T. Curtis and to
illustrate, as below, point-line duality in the Fano plane.
Comments Off on Plan 9 from Toronto
Monday, March 24, 2025
A Combinatorial Configuration
Related art —
From "Self-Dual Configurations and Regular Graphs" by H. S. M. Coxeter,
Bulletin of the American Mathematical Society, Vol. 56 (1950), pp. 413-455
For a related combinatorial configuration, take Oxbury's "16 lines"
to be the the 16 dots above and take the "8 points of intersection"
to be the four squares
234, 1234, 124, 24
23, 123, 12, 2
3, 13, 1, 0
34, 134, 14, 4
along with the four diamonds
234, 23, 3, 34
1234, 123, 13, 134
124, 12, 1, 14
24, 2, 0, 4.
Then each "line" is on two "points" and each "point" on
four "lines."
Note that these eight "points" — the four squares and the four diamonds
of Coxeter's figure — form the rows and columns of the following matrix:
| 234 | 1234 | 124 | 24 |
| 23 | 123 | 12 | 2 |
| 3 | 13 | 1 | 0 |
| 34 | 134 | 14 | 4 |
Related reading: Points with Parts .
Comments Off on A Combinatorial Configuration
Wednesday, December 11, 2024
Sunday, September 8, 2024
Language Game: Sunday in the Park with Steiner
Now added to a Likewise.com list —
The Portage to San Cristobal of A. H.
Related reading —
On a letter from Hebrew, the fentanyl of languages —
Comments Off on Language Game: Sunday in the Park with Steiner
Wednesday, August 9, 2023
“The Seed Crystal” — Plan 9 from Science Direct
Comments Off on “The Seed Crystal” — Plan 9 from Science Direct
Thursday, June 29, 2023
Tomb Raider of Geometry
Comments Off on Tomb Raider of Geometry
Monday, May 1, 2023
Chain of Title: Complete, Clean, Unencumbered!
From some Canadian legal boilerplate —
E. Be able to provide complete, clean, unencumbered
chain of title for the Project, must have all the rights,
releases and clearances necessary to produce, own and
exploit the Project and for deployment of the Project . . . .
Weak Links in the Chain of Title —
A 2006 biography of geometer H.S.M. Coxeter:
The Aleph (implicit in a 1950 article by Coxeter):
Click on images
for further details.
Comments Off on Chain of Title: Complete, Clean, Unencumbered!
Wednesday, April 5, 2023
Annals of Artificial Stupidity:
“A Sort of False Coherence”
The "large language model" approach to AI has yielded
startlingly good results for programmers, but is not so good
for finding out facts . . .
A Google search for harvard mathematician h.s.m. coxeter yields . . .
Readers able to use Google can easily find out who wrote the above
gestalt passage. It was not Coxeter.
Further investigation via Google yields the O'Toole source:
O'Toole, Michael, The Language of Displayed Art ,
Leicester University Press, 1994, p. 4.
Comments Off on Annals of Artificial Stupidity:
“A Sort of False Coherence”
“A Sort of False Coherence”
Sunday, April 10, 2022
Friday, July 30, 2021
The Secret Subterranean River
The "secret, subterranean river" of Shulevitz is
a flow of thought favorable to the cause of feminism,
but not necessarily to other "revolutionary" ideas.
Compare and contrast:
"Where Alph, the sacred river, ran"
— Coleridge, Kubla Khan
"Where Aleph the sacred symbol ran"
— Cullinane, "The Coxeter Aleph"
For group discussion:
How (if at all) is the "finitude" of Heidegger related to
mathematical finitude and The King of Infinite Space ?
Comments Off on The Secret Subterranean River
Saturday, July 3, 2021
Here, There, and Chicago
The above phrase "the intersection of storytelling and visual arts"
suggests a review . . .
Storytelling —
Visual arts —
"This pattern is a square divided into nine equal parts.
It has been called the 'Holy Field' division and
was used throughout Chinese history for many
different purposes, most of which were connected
with things religious, political, or philosophical."
– The Magic Square: Cities in Ancient China,
by Alfred Schinz, Edition Axel Menges, 1996, p. 71
A Midrash for Michener —
For a connection of the above "Holy Field"
with pure mathematics, see Coxeter's Aleph.
Comments Off on Here, There, and Chicago
Friday, December 4, 2020
Devil Music … Now in Infernovision!
“DEVIL – MUSIC
20 pages of incidental music written at school
for G. K. Chesterton’s play MAGIC
by D. Coxeter.”
See also other posts now tagged Infernovision.
Comments Off on Devil Music … Now in Infernovision!
Friday, July 5, 2019
The Motive for Metaphor
"János Bolyai was a nineteenth-century mathematician who
set the stage for the field of non-Euclidean geometry."
— Transylvania Now , October 26, 2018
From Coxeter and the Relativity Problem —
Desiring the exhilarations of changes:
The motive for metaphor, shrinking from
The weight of primary noon,
The A B C of being,
The ruddy temper, the hammer
Of red and blue, the hard sound—
Steel against intimation—the sharp flash,
The vital, arrogant, fatal, dominant X.
Comments Off on The Motive for Metaphor
Friday, April 19, 2019
Pleasantly Discursive Day
” 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
Comments Off on Pleasantly Discursive Day
Wednesday, January 2, 2019
“Pleasantly Discursive” Continues.
From this journal on December 13th, 2016 —
" 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
Also on December 13th, 2016 —
Comments Off on “Pleasantly Discursive” Continues.
Saturday, December 22, 2018
Cremona-Richmond
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 Cremona-Richmond
Friday, September 28, 2018
ART WARS Midrash
"When times are mysterious
Serious numbers
Will always be heard."
— Paul Simon,
"When Numbers Get Serious"
"There is a pleasantly discursive treatment of
Pontius Pilate's unanswered question 'What is truth?'"
— H. S. M. Coxeter, introduction to Richard J. Trudeau's remarks
on the "story theory" of truth as opposed to the "diamond theory"
of truth in The Non-Euclidean Revolution (1987)
The deaths of Roth and Grünbaum on September 14th,
The Feast of the Holy Cross, along with Douthat's column
today titled "Only the Truth Can Save Us Now," suggest a
review of …
.
Comments Off on ART WARS Midrash
Monday, May 14, 2018
Space
Comments Off on Space
Tuesday, February 13, 2018
A Titan of the Field
|
On the late Cambridge astronomer Donald Lynden-Bell — "As an academic at a time when students listened and lecturers lectured, he had the disconcerting habit of instead picking on a random undergraduate and testing them on the topic. One former student, now a professor, remembered how he would 'ask on-the-spot questions while announcing that his daughter would solve these problems at the breakfast table'. He got away with it because he was genuinely interested in the work of his colleagues and students, and came to be viewed with great affection by them. He also got away with it because he was well established as a titan of the field." — The London Times on Feb. 8, 2018, at 5 PM (British time) |
Related material —
Two Log24 posts from yesteday, Art Wars and The Void.
See as well the field GF(9) …
… and the 3×3 grid as a symbol of Apollo
(an Olympian rather than a Titan) —
.
Comments Off on A Titan of the Field