The geometric object of the title appears in a post mentioning Bourgain
in this journal. Bourgain appears also in today's online New York Times —
https://www.nytimes.com/2019/01/16/
obituaries/jean-bourgain-dead.html .
Bourgain reportedly died on December 22.
An image from this journal on that date —
Related poetic meditations —
Two examples from the Wikipedia article "Archimedean solid" —
Iain Aitchison said in a 2018 talk at Hiroshima that
the Mathieu group M24 can be represented as permuting
naturally the 24 edges of the cuboctahedron.
The 24 vertices of the truncated octahedron are labeled
naturally by the 24 elements of S4 in a permutahedron —
Can M24 be represented as permuting naturally
the 24 vertices of the truncated octahedron?
Related material from the day Orson Welles and Yul Brynner died —
In memory of Hale Trotter, a mathematician who reportedly
died at Princeton, N.J., on Jan. 17, 2022.
Other perspectives —
“The carnival is an incredibly close-knit, hermetic society.”
— Guillermo del Toro, director and co-writer of
the new remake of "Nightmare Alley"
Dialogue from that remake —
STAN — How do you ever get a guy to geek?
CLEM — Oh- I ain’t going to crap you up. It ain’t easy.
"There is such a thing as a four-set."
— Saying adapted from a 1962 young-adult novel
The above arrangement of graphic images on cube faces is purely
decorative and static, and of little mathematical interest.
(A less static, but structurally chaotic, artifact might be made by
pasting the above 24 graphic images in the "Cosets in S4" picture
above onto the 24 faces of a 2x2x2 Rubik cube. This suggests the
reflection below on the poet Wallace Stevens, whose "Connoisseur
of Chaos" first appeared on page 90 of Twentieth Century Verse ,
Numbers 12-13, October 1938.)
If mathematically interesting permutations of the graphic images
are to be done, the images should be imagined as situated on
parallel planes, as in the permutahedron below —
Click the above permutahedron for an analysis of its structure.
The title is from a post of January 10, 2019.
A figure from this journal on June 1, 2019 —
The following figure may help relate labelings of the
truncated octahedron ("permutahedron") to labelings
of its fellow Archimedean solid, the cuboctahedron.
See as well other posts tagged Aitchison.
An image from All Souls' Day 2010 —
This is from earlier posts tagged Permutahedron.
See also
Wallace Stevens:
A World of Transforming Shapes.
From that book (click to enlarge) —
"Before time began, there was the Cube."
— Optimus Prime.
Also from earlier posts tagged Permutahedron —
See this evening's update to the May 31 post
"Working Sketch of Aitchison’s Mathieu Cuboctahedron" —
". . . And then of course there is the obvious labeling derived from
the … permutahedron —"
The above sketch indicates one way to apply the elements of S4
to the Aitchison cuboctahedron . It is a rough sketch illustrating a
correspondence between four edge-hexagons and four label-sets.
The labeling is not as neat as that of a permutahedron by S4
shown below, but can perhaps be improved.
Permutahedron labeled by S4 .
Update of 9 PM EDT June 1, 2019 —
. . . And then of course there is the obvious labeling derived from
the above permutahedron —
Two examples from the Wikipedia article "Archimedean solid" —
Iain Aitchison said in a talk last year at Hiroshima that
the Mathieu group M24 can be represented as permuting
naturally the 24 edges of the cuboctahedron.
The 24 vertices of the truncated octahedron are labeled
naturally by the 24 elements of S4 in a permutahedron —
Can M24 be represented as permuting naturally
the 24 vertices of the truncated octahedron?
William Grimes in The New York Times this evening —
"Clifford Irving, who perpetrated one of the biggest literary hoaxes
of the 20th century in the early 1970s when he concocted a
supposedly authorized autobiography of the billionaire Howard Hughes
based on meetings and interviews that never took place, died on Tuesday
at a hospice facility near his home in Sarasota, Fla. He was 87."
A figure reproduced here on Tuesday —
A related figure —
See too the 1973 Orson Welles film "F for Fake."
Some background on the second figure above —
posts tagged April 8-11, 2016.
Some background on the first figure above —
today's previous post, January 2018 AMS Notices.
(Continued from last Sunday)
For some background, see Permutahedron in this journal.
See also…
* Jews may prefer to retitle this post "Sunday Shul with Josefine"
and stage it as a SNL sketch, "Norwegian Disco," with
The Sunshine Girls. (For the Norwegian part, see Kristen Wiig,
of Norwegian ancestry. For the disco part, see Amy Adams,
who stars in a new disco-era movie.)
Blame It on Trajan
Wikipedia on the 2008 film The Oxford Murders—
Christmas Eve image search
suggested by Stevens's phrase
"diamond globe."
(Larger version: 2 MB)
From the Crafoord Prize website—
Related meta -mathematical image from Diamond Theory—
Mathematical image related to combinatorics—
See also permutahedron in this journal.
Yesterday's "diamond globe" post linked to a picture by Prof. Mike Zabrocki
of York University in Toronto. Here is the picture itself—
Some related material from 2004—
Prof Talk: Thursday, April 1st, from 2:30pm to 3:30pm in North Ross 638.
Speaker: Prof. Mike Zabrocki
Title: "Gems of Algebra: The Secret Life of the Symmetric Group"
Prof. Zabrocki's talk was enjoyable and accessible. One of the notable aspects of the talk was that Prof. Zabrocki presented some open problems related to the topics he was speaking about. Unfortunately, there were some technical problems that resulted in some images not appearing in Prof. Zabrocki's PowerPoint presentation, but Prof. Zabrocki easily made up for the problem by some work at the chalk board. Please feel free to take a look at Prof. Zabrocki's PowerPoint presentation, as well as the pictures of the permutahedron for n=4 and the permutahedron for n=5.
Some slides from the talk—
Detail from the slides—
A less academic ornament, from this journal on the date
of the Zabrocki talk—
Click image for context.
A search for Wallace Stevens ebooks
today at Alibris yielded 24 results.
I selected one to order—
Wallace Stevens: A World of Transforming Shapes .
From that book—
Stevens's phrase "diamond globe" in this context suggests an image search
on permutahedron + stone + log24 .
For the results of that search (2 MB), click here.
Some background for the phrase used in the search—
See a photo by Mike Zabrocki from June 4, 2011.
See also a Log24 image and a generalization of the underlying structure.
This post was suggested by a book advertised
above A. Whitney Ellsworth's obituary in tonight's
online New York Times .
See also the following illustrations—
From this journal on June 1, 2008:
From artist Steve Richards on January 14, 2010:
Tangled Up In Red
| CHANGE FEW CAN BELIEVE IN |
See Siri Hustvedt on the name "Wechsler"
and see the tag "permutahedron" in this journal.
Malcolm Lowry, author of
Under the Volcano
|
Mirror Ball album
Hey ho away we go |
Mirror Ball album
Yeah you're working |
Hotel Bella Vista
Gran Baile Noviembre 1938
a Beneficio de la Cruz Roja.
Los Mejores Artistas del radio en accion.
No falte Vd.
24 graphic patterns arranged in space
as 12 pairs of opposites
Click image for an illustration of how the above labeling was derived.
For further background, see Cases of the Diamond Theorem
and recent art by Josefine Lyche of Norway.
Part I:
Ratio
Continued from…
|
White and Geometric,
but not Eternal. |
Part II:
Fides
For more information,
click on the cocktail.
Happy Birthday, Hassler Whitney
In honor of the late Hassler Whitney, mathematician and mountaineer, here is a link to the five Log24 entries ending with White, Geometric, and Eternal (Dec. 20, 2003).
Related material: the five Log24 entries ending with The Meadow (Dec. 18, 2005) and the five Log24 entries ending with Strange Attractor (Jan. 7, 2006).
The cross and the epiphany star in this last group of entries may interest the symbol-mongers among us.
Those more interested in substance than in symbols may prefer the following (click to enlarge):
This is apparently the original source for
the figure I cited on Dec. 20, 2003, as
from antiquark.com.
The connection with Whitney is
through the theory of matroids,
which Whitney founded in 1935.
See Hassler Whitney,
"On the abstract properties
of linear dependence,"
American Journal of Mathematics,
vol. 57 (1935), 509-533,
Collected Papers, vol. I, 147-171.
White, Geometric, and Eternal
This afternoon's surfing:
Prompted by Edward Rothstein's own Fides et Ratio encyclical in today's NY Times, I googled him.
At the New York Review of Books, I came across the following by Rothstein:
"… statements about TNT can be represented within TNT: the formal system can, in a precise way, 'talk' about itself."
This naturally prompted me to check what is on TNT on this, the feast day of St. Emil Artin. At 5 PM this afternoon, we have Al Pacino in "The Devil's Advocate" — a perfect choice for the festival of an alleged saint.
Preparing for Al, I meditated on the mystical significance of the number 373, as explained in Zen and Language Games: the page number 373 in Robert Stone's theological classic A Flag for Sunrise conveys the metaphysical significance of the phrase "diamonds are forever" — "the eternal in the temporal," according to Stone's Catholic priest. This suggests a check of another theological classic, Pynchon's Gravity's Rainbow. Page 373 there begins with the following description of prewar Berlin:
"white and geometric."
This suggests the following illustration of a white and geometric object related to yesterday's entry on Helmut Wielandt:
Figure 1
(This object, which illustrates the phrase "makin' the changes," also occurs in this morning's entry on the death of a jazz musician.)
A further search for books containing "white" and "geometric" at Amazon.com yields the following:
Figure 2
From Mosaics, by
Fassett, Bahouth, and Patterson:
"A risco fountain in Mexico city, begun circa 1740 and made up of Mexican pottery and Chinese porcelain, including Ming.
The delicate oriental patterns on so many different-sized plates and saucers [are] underlined by the bold blue and white geometric tiles at the base."
Note that the tiles are those of Diamond Theory; the geometric object in figure 1 above illustrates a group that plays a central role in that theory.
Finally, the word "risco" (from Casa del Risco) associated with figure 2 above leads us to a rather significant theological site associated with the holy city of Santiago de Compostela:
Figure 3
Vicente Risco's
Dedalus in Compostela.
Figure 3 shows James Joyce (alias Dedalus), whose daughter Lucia inspired the recent entry Jazz on St. Lucia's Day — which in turn is related, by last night's 2:45 entry and by Figure 1, to the mathematics of group theory so well expounded by the putative saint Emil Artin.
"His lectures are best described as
polished diamonds."
— Fine Hall in its Golden Age,
by Gian-Carlo Rota
If Pynchon plays the role of devil's advocate suggested by his creation, in Gravity's Rainbow, of the character Emil Bummer, we may hope that Rota, no longer in time but now in eternity, can be persuaded to play the important role of saint's advocate for his Emil.
Update of 6:30 PM 12/20/03:
Riddled:
The Absolutist Faith
of The New York Times
White and Geometric, but not Eternal.
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