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.

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.

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"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 M_{24} , obtained

by multiplying the number of such facets, 759, by the

order of the octad stabilizer group, 322,560.

Analogously …

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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 E_{8 }in Garrett Lisi's poetic theory of everything.

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

** Diamond and E _{8} at Wikipedia**

Related material — E_{8 }as "a diamond with thousands of facets"—

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—

**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

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"Credences of Summer," VII,
by Wallace Stevens, from
"Three times the concentred |

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 —

A more modest object —

the 4×4 square.

Update of Aug. 20-21 —

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–

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.

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Hard Core

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: — Hermann Weyl in Let us apply Weyl’s lesson to the following “structure-endowed entity.”
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”–

automorphisms plays

a role in what Weyl,

following Eddington,

called a “colorful tale”–

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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