Log24

Thursday, September 1, 2011

The Galois Tesseract

Filed under: General,Geometry — Tags: , — m759 @ 7:11 PM

Click to enlarge

IMAGE- The Galois Tesseract, 1979-1999

IMAGE- Review of Conway and Sloane's 'Sphere Packings...' by Rota

For Thor’s Day

Filed under: General — Tags: — m759 @ 12:00 PM

If I Had a Hammer
(Eine Alte Geschichte )

How It Works

Filed under: General,Geometry — Tags: , , — m759 @ 11:00 AM

"Design is how it works." — Steven Jobs (See Symmetry and Design.)

"By far the most important structure in design theory is the Steiner system S(5, 8, 24)."
 — "Block Designs," by Andries E. Brouwer

IMAGE- Harvard senior thesis on Mathieu groups, 2010, and supporting material from book 'Design Theory'

The name Carmichael is not to be found in Booher's thesis. In a reference he does  give for the history of S(5,8,24), Carmichael's construction of this design is dated 1937. It should be dated 1931, as the following quotation shows—

From Log24 on Feb. 20, 2010

"The linear fractional group modulo 23 of order 24•23•11 is often represented as a doubly transitive group of degree 24 on the symbols ∞, 0, 1, 2,…, 22. This transitive group contains a subgroup of order 8 each element of which transforms into itself the set ∞, 0, 1, 3, 12, 15, 21, 22 of eight elements, while the whole group transforms this set into 3•23•11 sets of eight each. This configuration of octuples has the remarkable property that any given set of five of the 24 symbols occurs in one and just one of these octuples. The largest permutation group Γ on the 24 symbols, each element of which leaves this configuration invariant, is a five-fold transitive group of degree 24 and order 24•23•22•21•20•48. This is the Mathieu group of degree 24."

– R. D. Carmichael, "Tactical Configurations of Rank Two," in American Journal of Mathematics, Vol. 53, No. 1 (Jan., 1931), pp. 217-240

Epigraph from Ch. 4 of Design Theory , Vol. I:

"Es is eine alte Geschichte,
 doch bleibt sie immer neu
"
 —Heine (Lyrisches Intermezzo  XXXIX)

See also "Do you like apples?"

Boundary (continued*)

Filed under: General — Tags: — m759 @ 12:00 AM

It is now midnight. Yesterday was Odin's Day. Today is Thor's Day.

From a weblog post on Captain America and Thor

"While all this [Captain America] is happening an SS officer, Johann Schmidt (Hugo Weaving), has found a religious artefact called the Tesseract which Schmidt describes as 'the jewel of Odin’s treasure room,' linking it in with the Thor storyline."

That's Entertainment  weblog, August 14, 2011

From Wallace Stevens, "An Ordinary Evening in New Haven," Canto III—

The point of vision and desire are the same.
It is to the hero of midnight that we pray
On a hill of stones to make beau mont thereof.

Captain America opened in the United States on Friday, July 22, 2011.

Thor opened in the United States on Friday, May 6, 2011.

"There is  such a thing as a tesseract." —A Wrinkle in Time

* Continued from August 30.

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