Log24

Friday, February 17, 2017

Kostant Is Dead

Filed under: General,Geometry — Tags: , — m759 @ 1:10 pm

"Bertram Kostant, professor emeritus of mathematics at MIT,
died at the Hebrew Senior Rehabilitation Center in Roslindale,
Massachusetts, on Thursday, Feb. 2, at the age of 88."

— MIT News, story dated Feb. 16, 2017

See also a search for Kostant in this journal.

Regarding the discussions of symmetries and "facets" found in
that search —

Kostant:

“A word about E(8). In my opinion, and shared by others,
E(8) is the most magnificent ‘object’ in all of mathematics.
It is like a diamond with thousands of facets. Each facet
offering a different view of its unbelievable intricate internal
structure.”

Cullinane:

In the Steiner system S(5, 8, 24) each octad might be
regarded as a "facet," with the order of the system's
automorphism group, the Mathieu group M24 , obtained
by multiplying the number of such facets, 759, by the
order of the octad stabilizer group, 322,560. 

Analogously …

Platonic solids' symmetry groups   

Thursday, February 2, 2023

The Menand Lede

Filed under: General — Tags: — m759 @ 4:29 pm

"When the Washington Post  unveiled the slogan
'Democracy Dies in Darkness,' on February 17, 2017,
people in the news business made fun of it.
'Sounds like the next Batman movie,' the New York Times’ 
executive editor, Dean Baquet, said."

— Louis Menand in The New Yorker ,
"When Americans Lost Faith in the News," Jan. 30, 2023.

See also Darkness  in this  journal.

Not so dark:

A Log24 post from February 17, 2017
regarding that year's Groundhog Day — The dies natalis
(in the Catholic sense) of St. Bertram Kostant.

Friday, November 26, 2021

Facets for Snorri

Filed under: General — Tags: , — m759 @ 12:31 pm

(Who is Snorri? See The Reykjavik Grapevine , Oct. 3, 2018.)

From this  journal yesterday —

"Faced with a larger surface than he had ever provided with facets,
in his desperation he had divided the diamond with imaginary lines,
treating each section as if it were a single small stone and arranging
the clusters of facets so they would interact with one another, as if they
were single facets in a smaller stone.  What if the final result lacked fire?"

— A novel* by Noah Gordon, who reportedly died on Monday, Nov. 22.

Related material from Log24 on Feb. 17, 2017 —

Also on Feb. 17, 2017 —

* The Jerusalem Diamond , Random House, April 1, 1979,
   republished later in German as Der Diamant des Salomon .

Monday, April 23, 2018

Facets

Filed under: General — Tags: , , — m759 @ 12:00 am

Counting symmetries with the orbit-stabilizer theorem

See also the Feb. 17, 2017, post on Bertram Kostant
as well as "Mathieu Moonshine and Symmetry Surfing."

Saturday, February 25, 2017

Impact Statement

Filed under: General — m759 @ 5:29 pm

See also Kostant in this  journal and a link in a 
Log24 post Friday on another mathematical death —

Hollywood Easter Egg (Groundhog Day, 2017).

Groundhog Day was the day Kostant reportedly died.

Monday, April 25, 2011

Poetry and Physics

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

One approach to the storied philosophers' stone, that of Jim Dodge in Stone Junction , was sketched in yesterday's Easter post. Dodge described a mystical "spherical diamond." The symmetries of the sphere form what is called in mathematics a Lie group . The "spherical" of Dodge therefore suggests a review of the Lie group E8  in Garrett Lisi's poetic theory of everything.

A check of the Wikipedia article on Lisi's theory yields…

http://www.log24.com/log/pix11A/110425-WikipediaE8.jpg

       Diamond and E8 at Wikipedia

Related material — E8  as "a diamond with thousands of facets"—

http://www.log24.com/log/pix11A/110425-Kostant.jpg

Also from the New Yorker  article—

“There’s a dream that underlying the physical universe is some beautiful mathematical structure, and that the job of physics is to discover that,” Smolin told me later. “The dream is in bad shape,” he added. “And it’s a dream that most of us are like recovering alcoholics from.” Lisi’s talk, he said, “was like being offered a drink.”

A simpler theory of everything was offered by Plato. See, in the Timaeus , the Platonic solids—

Platonic solids' symmetry groups

Figure from this journal on August 19th, 2008.
See also July 19th, 2008.

“It’s all in Plato, all in Plato:
bless me, what do  they
teach them at these schools!”
— C. S. Lewis

Tuesday, August 19, 2008

Tuesday August 19, 2008

Filed under: General,Geometry — Tags: , , — m759 @ 8:30 am
Three Times

"Credences of Summer," VII,

by Wallace Stevens, from
Transport to Summer (1947)

"Three times the concentred
     self takes hold, three times
The thrice concentred self,
     having possessed
The object, grips it
     in savage scrutiny,
Once to make captive,
     once to subjugate
Or yield to subjugation,
     once to proclaim
The meaning of the capture,
     this hard prize,
Fully made, fully apparent,
     fully found."
Stevens does not say what object he is discussing.

One possibility —

Bertram Kostant, Professor Emeritus of Mathematics at MIT, on an object discussed in a recent New Yorker:

"A word about E(8). In my opinion, and shared by others, E(8) is the most magnificent 'object' in all of mathematics. It is like a diamond with thousands of facets. Each facet offering a different view of its unbelievable intricate internal structure."

Another possibility —
 

The 4x4 square

  A more modest object —
the 4×4 square.

Update of Aug. 20-21 —

Symmetries and Facets

Kostant's poetic comparison might be applied also to this object.

The natural rearrangements (symmetries) of the 4×4 array might also be described poetically as "thousands of facets, each facet offering a different view of… internal structure."

More precisely, there are 322,560 natural rearrangements– which a poet might call facets*— of the array, each offering a different view of the array's internal structure– encoded as a unique ordered pair of symmetric graphic designs. The symmetry of the array's internal structure is reflected in the symmetry of the graphic designs. For examples, see the Diamond 16 Puzzle.

For an instance of Stevens's "three times" process, see the three parts of the 2004 web page Ideas and Art.

* For the metaphor of rearrangements as facets, note that each symmetry (rearrangement) of a Platonic solid corresponds to a rotated facet: the number of symmetries equals the number of facets times the number of rotations (edges) of each facet–

Platonic solids' symmetry groups

The metaphor of rearrangements as facets breaks down, however, when we try to use it to compute, as above with the Platonic solids, the number of natural rearrangements, or symmetries, of the 4×4 array. Actually, the true analogy is between the 16 unit squares of the 4×4 array, regarded as the 16 points of a finite 4-space (which has finitely many symmetries), and the infinitely many points of Euclidean 4-space (which has infinitely many symmetries).

If Greek geometers had started with a finite space (as in The Eightfold Cube), the history of mathematics might have dramatically illustrated Halmos's saying (Aug. 16) that

"The problem is– the genius is– given an infinite question, to think of the right finite question to ask. Once you thought of the finite answer, then you would know the right answer to the infinite question."

The Greeks, of course, answered the infinite questions first– at least for Euclidean space. Halmos was concerned with more general modern infinite spaces (such as Hilbert space) where the intuition to be gained from finite questions is still of value.
 

Saturday, July 19, 2008

Saturday July 19, 2008

Filed under: General,Geometry — Tags: , , — m759 @ 2:00 pm
Hard Core

(continued from yesterday)

Bertram Kostant, Professor Emeritus of Mathematics at MIT, on an object discussed in this week's New Yorker:

"A word about E(8). In my opinion, and shared by others, E(8) is the most magnificent 'object' in all of mathematics. It is like a diamond with thousands of facets. Each facet offering a different view of its unbelievable intricate internal structure."

Hermann Weyl on the hard core of objectivity:

"Perhaps the philosophically most relevant feature of modern science is the emergence of abstract symbolic structures as the hard core of objectivity behind– as Eddington puts it– the colorful tale of the subjective storyteller mind." (Philosophy of Mathematics and Natural Science, Princeton, 1949, p. 237)


Steven H. Cullinane on the symmetries of a 4×4 array of points:

A Structure-Endowed Entity

"A guiding principle in modern mathematics is this lesson: Whenever you have to do with a structure-endowed entity S, try to determine its group of automorphisms, the group of those element-wise transformations which leave all structural relations undisturbed.  You can expect to gain a deep insight into the constitution of S in this way."

— Hermann Weyl in Symmetry

Let us apply Weyl's lesson to the following "structure-endowed entity."

4x4 array of dots

What is the order of the resulting group of automorphisms?

The above group of
automorphisms plays
a role in what Weyl,
following Eddington,
  called a "colorful tale"–

The Diamond 16 Puzzle

The Diamond 16 Puzzle

This puzzle shows
that the 4×4 array can
also be viewed in
thousands of ways.

"You can make 322,560
pairs of patterns. Each
 pair pictures a different
symmetry of the underlying
16-point space."

— Steven H. Cullinane,
July 17, 2008

For other parts of the tale,
see Ashay Dharwadker,
the Four-Color Theorem,
and Usenet Postings.

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