Wednesday, January 16, 2019

Permutahedron Dream

Filed under: General — Tags: , , , — m759 @ 3:21 PM

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 —


Bourgain reportedly died on December 22.

An image from this journal on that date

Related poetic meditations —

IMAGE- Herbert John Ryser, 'Combinatorial Mathematics' (1963), page 1

Tuesday, March 10, 2020

Labeling a Cuboctahedron

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

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 —

IMAGE- 'Permutahedron of Opposites'-- 24 graphic patterns arranged in space as 12 pairs of opposites

Click the above permutahedron for an analysis of its structure.

Monday, March 9, 2020

“Archimedes at Hiroshima” Continues.

Filed under: General — Tags: , , — m759 @ 7:34 PM

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.

Saturday, November 23, 2019

Diamond Globe

Filed under: General — Tags: , , — m759 @ 7:23 PM

An image from All Souls' Day 2010 —

IMAGE- 'Permutahedron of Opposites'-- 24 graphic patterns arranged in space as 12 pairs of opposites

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

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

Thursday, August 8, 2019

The Embedding

Filed under: General — m759 @ 1:33 PM

(Continued from Nov. 28, 2010)

Saturday, June 1, 2019

Cuboctahedron Labeling Update

Filed under: General — Tags: , , — m759 @ 9:01 PM

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 —"

Friday, May 31, 2019

Working Sketch of Aitchison’s Mathieu Cuboctahedron

Filed under: General — Tags: , , — m759 @ 5:33 AM

Cuboctahedron with its 24 edges labeled by the 24 permutations of a 4-set. By Cullinane on 5/31/19.

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


Friday, May 10, 2019

Walpurgisnacht Riddle

Filed under: General — Tags: — m759 @ 10:04 PM

From a search for Absolut Riddle in this  journal —

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 ….”

Thursday, May 9, 2019

Defense Against the Dark Arts

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

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

Sunday, April 28, 2019


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

"This time-defying preservation of selves,
this dream of plenitude without loss,
is like a snow globe from heaven,
a vision of Eden before the expulsion.
Mathematically demonstrable
but emotionally impossible,
it’s dangled just in front of us
like a bauble we can’t have
but can’t stop reaching for." 

— Judith Shulevitz on Siri Hustvedt in
The New York Times  Sunday Book Review
of March 31, 2019, under the headline
"The Time of Her Life."

A different self-symbolizing bauble appeared in this  journal on that Sunday.

A line for Letterman — "Bauble, Babel.  Babel, Bauble."

'The Tower of Babel,' a 1963 play by Arthur J. Morey with music by Robert A. Paul

Sunday, January 13, 2019

Into the Upside Down

Filed under: General — Tags: — m759 @ 3:09 PM

Images from today's 3:17 AM search for Clifford

See as well some snow news from Europe.

Thursday, January 10, 2019

Archimedes at Hiroshima

Filed under: General,Geometry — Tags: , , , — m759 @ 7:35 PM

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?


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.

Monday, May 7, 2018

Fish Babel

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

Stanley Fish in the online New York Times  today —

". . . Because it is an article of their faith that politics are bad
and the unmediated encounter with data is good,
internet prophets will fail to see the political implications
of what they are trying to do, for in their eyes political implications
are what they are doing away with.

Indeed, their deepest claim — so deep that they are largely
unaware of it — is that politics can be eliminated. They don’t
regard politics as an unavoidable feature of mortal life but as
an unhappy consequence of the secular equivalent of the
Tower of Babel: too many languages, too many points of view.
Politics (faction and difference) will just wither away when
the defect that generates it (distorted communication) has
been eliminated by unmodified data circulated freely among
free and equal consumers; everyone will be on the same page,
reading from the same script and apprehending the same
universal meanings. Back to Eden!"

The final page, 759, of the Harry Potter saga —

"Talk about magical thinking!" — Fish, ibidem .

See also the above Harry Potter page 
in this  journal Sunday morning.

Wednesday, December 20, 2017

A Snow Ball for Clifford Irving (1930-2017)

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

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.

Sunday, October 6, 2013

Church with Josefine*

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

(Continued from last Sunday)

IMAGE- 'Permutahedron of Opposites'-- 24 graphic patterns arranged in space as 12 pairs of opposites

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.)

Saturday, January 21, 2012

The Oxford Murders

Filed under: General — Tags: , — m759 @ 10:09 AM


Blame It on Trajan

Wikipedia on the 2008 film The Oxford Murders

IMAGE- Tall column of images from Log24, headed by permutahedron pictures

Christmas Eve image search
suggested by Stevens's phrase
"diamond globe."

(Larger version: 2 MB)

Thursday, January 19, 2012

Mathematical Imagery

Filed under: General,Geometry — Tags: — m759 @ 10:28 PM

Bourgain and Tao

From the Crafoord Prize website

Related meta -mathematical image from Diamond Theory

Mathematical  image related to combinatorics—

See also permutahedron in this journal.

Sunday, December 25, 2011

Xmas Ornaments

Filed under: General — Tags: — m759 @ 5:01 AM

Yesterday's "diamond globe" post linked to a picture by Prof. Mike Zabrocki
of York University in Toronto. Here is the picture itself—
IMAGE- Permutahedron 'diamond globe' by Mike Zabrocki

Some related material from 2004—


2003-2004 Events:

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—

IMAGE- Diamond ornament from slides by Mike Zabrocki

A less academic ornament, from this journal on the date
of the Zabrocki talk—

IMAGE- Rainbow on record label of hymns by Loretta Lynn

Click image for context.

Saturday, December 24, 2011

Stevens for Christmas Eve

Filed under: General — Tags: — m759 @ 11:30 AM

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—

(Click to enlarge)


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.

Tuesday, June 21, 2011


Filed under: General — m759 @ 3:19 AM

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:

Click for background

Permutahedron for the symmetric group on four elements

From artist Steve Richards on January 14, 2010:

Click to enlarge

IMAGE- Interview with Steve Richards, who later contributed to London's 'Piracy Project'

Thursday, November 4, 2010

Annals of Rhetoric

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

Tangled Up In Red


See Siri Hustvedt on the name "Wechsler"
and see the tag "permutahedron" in this journal.

Tuesday, November 2, 2010

Dia de los Muertos

Filed under: General — m759 @ 5:01 PM


Malcolm Lowry, author of
Under the Volcano

Mirror Ball album, Neil Young

Mirror Ball album
by Neil Young

Hey ho away we go
We're on the road to never

Mirror Ball album, Sarah McLachlan

Mirror Ball album
by Sarah McLachlan

Yeah you're working
Building a mystery

Hotel Bella Vista
Gran Baile Noviembre 1938
a Beneficio de la Cruz Roja.
Los Mejores Artistas del radio en accion.
No falte Vd.

A Dozen Pairs of Opposites —

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

24 graphic patterns arranged in space
as 12 pairs of opposites

IMAGE- 'Permutahedron of Opposites'-- 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.

Monday, December 22, 2008

Monday December 22, 2008

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

Part I:

Continued from…

    December 20, 2003

White, Geometric,
   and Eternal

Permutahedron-- a truncated octahedron with vertices labeled by the 24 permutations of four things

Makin' the Changes

(From "Flag Matroids," by
Borovik, Gelfand, and White)

Edward Rothstein,

Edward Rothstein on faith and reason, with snowflakes in an Absolut Vodka ad, NYT 12/20/03

White and Geometric,
 but not Eternal.

Part II:

Cocktail: the logo of the New York Times 'Proof' series

For more information,
click on the cocktail.

Sunday, June 1, 2008

Sunday June 1, 2008

Filed under: General,Geometry — Tags: , — m759 @ 2:14 PM
Yet Another
Cartoon Graveyard

The conclusion of yesterday’s commentary on the May 30-31 Pennsylvania Lottery numbers:

Thomas Pynchon, Gravity’s Rainbow:

“The fear balloons again inside his brain. It will not be kept down with a simple Fuck You…. A smell, a forbidden room, at the bottom edge of his memory. He can’t see it, can’t make it out. Doesn’t want to. It is allied with the Worst Thing.

He knows what the smell has to be: though according to these papers it would have been too early for it, though he has never come across any of the stuff among the daytime coordinates of his life, still, down here, back here in the warm dark, among early shapes where the clocks and calendars don’t mean too much, he knows that’s what haunting him now will prove to be the smell of Imipolex G.

Then there’s this recent dream he is afraid of having again. He was in his old room, back home. A summer afternoon of lilacs and bees and


What are we to make of this enigmatic 286? (No fair peeking at page 287.)

One possible meaning, given The Archivists claim that “existence is infinitely cross-referenced”–

Page 286 of Ernest G. Schachtel, Metamorphosis: On the Conflict of Human Development and the Psychology of Creativity (first published in 1959), Hillsdale NJ and London, The Analytic Press, 2001 (chapter– “On Memory and Childhood Amnesia”):

“Both Freud and Proust speak of the autobiographical [my italics] memory, and it is only with regard to this memory that the striking phenomenon of childhood amnesia and the less obvious difficulty of recovering any past experience may be observed.”

The concluding “summer afternoon of lilacs and bees” suggests that 286 may also be a chance allusion to the golden afternoon of Disney’s Alice in Wonderland. (Cf. St. Sarah’s Day, 2008)

Some may find the Disney afternoon charming; others may see it as yet another of Paul Simon’s dreaded cartoon graveyards.

More tastefully, there is poem 286 in the 1919 Oxford Book of English Verse– “Love.”

For a midrash on this poem, see Simone Weil, who became acquainted with the poem by chance:

“I always prefer saying chance rather than Providence.”

— Simone Weil, letter of about May 15, 1942

Weil’s brother André might prefer Providence (source of the Bulletin of the American Mathematical Society.)

Andre Weil and his sister Simone, summer of 1922(Photo from Providence)


Related material:

Log24, December 20, 2003–
White, Geometric, and Eternal

A description in Gravity’s Rainbow of prewar Berlin as “white and geometric”  suggested, in combination with a reference elsewhere to “the eternal,” a citation of the following illustration of the concept “white, geometric, and eternal”–

For more on the mathematical significance of this figure, see (for instance) Happy Birthday, Hassler Whitney, and Combinatorics of Coxeter Groups, by Anders Björner and Francesco Brenti, Graduate Texts in Mathematics, vol. 231, Springer, New York, 2005.

This book is reviewed in the current issue (July 2008) of the above-mentioned Providence Bulletin.

The review in the Bulletin discusses reflection groups in continuous spaces.

For a more elementary approach, see Reflection Groups in Finite Geometry and Knight Moves: The Relativity Theory of Kindergarten Blocks.

See also a commentary on
the phrase “as a little child.”

Thursday, March 23, 2006

Thursday March 23, 2006

Filed under: General,Geometry — Tags: — m759 @ 5:55 AM

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):

The image “http://www.log24.com/theory/images/permutahedron-matroid497.gif” cannot be displayed, because it contains errors.

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.

Monday, December 19, 2005

Monday December 19, 2005

Filed under: General,Geometry — Tags: , — m759 @ 2:45 AM
 "There is an
underlying timelessness
in the basic conversation
that is mathematics
Barry Mazur (pdf)

It's Quarter to Three

"I could tell you a lot
but you gotta be
 true to your code."
— Sinatra

Today is the birthday of Helmut Wielandt (Dec. 19, 1910 – Feb. 14, 2001).

From MacTutor:

"In his speech accepting membership of the Heidelberg Academy in 1960 he said:-

It is to one of Schur's seminars that I owe the stimulus to work with permutation groups, my first research area. At that time the theory had nearly died out. It had developed last century, but at about the turn of the century had been so completely superseded by the more generally applicable theory of abstract groups that by 1930 even important results were practically forgotten – to my mind unjustly."

Permutation groups are still not without interest.  See today's updates (Notes [01] and [02]) to Pattern Groups.


Saturday, December 20, 2003

Saturday December 20, 2003

Filed under: General,Geometry — Tags: — m759 @ 5:00 PM

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 RainbowPage 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:

From antiquark.com

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:


The Absolutist Faith
of The New York Times

White and Geometric, but not Eternal.

Saturday December 20, 2003

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

Quarter to Three

"You've got to be true to your code."
Frank Sinatra

In memory of Webster Young,
who died on Saint Lucia's day,
December 13, 2003 —

From my entry of 12/16/03,
Jazz on St. Lucia's Day:


"Now you has jazz."
High Society, 1956

Webster Young was a jazz trumpeter.

In 1957, Young was featured on
saxophonist Jackie McLean's albums
"A Long Drink of the Blues" and
"Makin' the Changes."

Adam Bernstein,
Washington Post, Dec. 18

"One for my baby,
and one more for the road."
— Frank Sinatra


Friday, December 19, 2003

Friday December 19, 2003

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

Happy Birthday, Helmut Wielandt
(wherever you may be)

Cover illustration,
AMS Notices, January 2004

In light of my entry on change-ringing of this date last year, the above AMS Notices cover may serve to illustrate what Heidegger so memorably dubbed the

 "Geheimnis des Glockenturms."

For details on the illustration,
click here and scroll down.

(Wielandt was an expert
on permutation groups.)

Powered by WordPress