Log24

Sunday, February 28, 2021

Factional Group

Filed under: General — Tags: — m759 @ 2:30 PM

Related tune suggested yesterday by Peter J. Cameron —

The Beatles, “I Me Mine,” from the “Let It Be” album.

Related imagery —

Saturday, February 27, 2021

The Pencil Case

Filed under: General — Tags: , , — m759 @ 11:55 PM

Clue

Here is  a midrash on “desmic,” a term derived from the Greek desmé
( δέσμη: bundle, sheaf , or, in the mathematical sense, pencil —
French faisceau ), which is related to the term desmos , bond …

(The term “desmic,” as noted earlier, is relevant to the structure of
Heidegger’s Sternwürfel .)

“Gadzooks, I’ve done it again!” — Sherlock Hemlock

Quarters

Filed under: General — Tags: , — m759 @ 1:58 PM

From posts tagged “The Empty Quarter” —

The image “http://www.log24.com/log/pix05B/050805-Rag.jpg” cannot be displayed, because it contains errors.

Related tune suggested today by Peter J. Cameron

The Beatles, “I Me Mine,” from the “Let It Be” album.

That album, and an image from Log24 on Feb. 23 —

The bond with reality is cut.

Monday, February 22, 2021

Design Theory

Filed under: General — Tags: , , , , — m759 @ 10:59 PM

A related image —

Related design theory in mathematics

http://m759.net/wordpress/?p=9221

Friday, January 17, 2020

September Morn

Filed under: General — Tags: — m759 @ 10:17 PM

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

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

This epigraph was quoted here earlier on
the morning of September 1, 2011.

Design Theory

Filed under: General — Tags: , , — m759 @ 12:57 PM

On a recently deceased professor emeritus of architecture
at Princeton —

“… Maxwell  ‘established the school as a principal
center of design research, history and theory.’ ”

“This is not the Maxwell you’re looking for.”

Thursday, September 1, 2011

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.  A book he does  cite for the history of S(5,8,24) gives the date of Carmichael’s construction of this design as 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?

Tuesday, July 6, 2010

Window, continued

“Simplicity, simplicity, simplicity!
I say, let your affairs be as two or three,
and not a hundred or a thousand;
instead of a million count half a dozen,
and keep your accounts on your thumb-nail.”
— Henry David Thoreau, Walden

This quotation is the epigraph to
Section 1.1 of Alexandre V. Borovik’s
Mathematics Under the Microscope:
Notes on Cognitive Aspects of Mathematical Practice
(American Mathematical Society,
Jan. 15, 2010, 317 pages).

From Peter J. Cameron’s review notes for
his new course in group theory

http://www.log24.com/log/pix10A/100705-CameronExample.jpg

From Log24 on June 24

Geometry Simplified

Image-- The Four-Point Plane: A Finite Affine Space
(an affine  space with subsquares as points
and sets  of subsquares as hyperplanes)

Image-- The Three-Point Line: A Finite Projective Space
(a projective  space with, as points, sets
of line segments that separate subsquares)

Exercise

Show that the above geometry is a model
for the algebra discussed by Cameron.

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