Log24

Thursday, March 6, 2008

Thursday March 6, 2008

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

This note is prompted by the March 4 death of Richard D. Anderson, writer on geometry, President (1981-82) of the Mathematical Association of America (MAA), and member of the MAA's Icosahedron Society.

Royal Road

"The historical road
from the Platonic solids
to the finite simple groups
is well known."

— Steven H. Cullinane,
November 2000,
Symmetry from Plato to
the Four-Color Conjecture

Euclid is said to have remarked that "there is no royal road to geometry." The road to the end of the four-color conjecture may, however, be viewed as a royal road from geometry to the wasteland of mathematical recreations.* (See, for instance, Ch. VIII, "Map-Colouring Problems," in Mathematical Recreations and Essays, by W. W. Rouse Ball and H. S. M. Coxeter.) That road ended in 1976 at the AMS-MAA summer meeting in Toronto– home of H. S. M. Coxeter, a.k.a. "the king of geometry."

See also Log24, May 21, 2007.

A different road– from Plato to the finite simple groups– is, as I noted in November 2000, well known. But new roadside attractions continue to appear. One such attraction is the role played by a Platonic solid– the icosahedron– in design theory, coding theory, and the construction of the sporadic simple group M24.

"By far the most important structure in design theory is the Steiner system S(5, 8, 24)."

— "Block Designs," by Andries E. Brouwer (Ch. 14 (pp. 693-746) of Handbook of Combinatorics, Vol. I, MIT Press, 1995, edited by Ronald L. Graham, Martin Grötschel, and László Lovász, Section 16 (p. 716))

This Steiner system is closely connected to M24 and to the extended binary Golay code. Brouwer gives an elegant construction of that code (and therefore of  M24):

"Let N be the adjacency matrix of the icosahedron (points: 12 vertices, adjacent: joined by an edge). Then the rows of the 12×24 matrix (I  J-N) generate the extended binary Golay code." [Here I is the identity matrix and J is the matrix of all 1's.]

— Op. cit., p. 719

Related material:

Finite Geometry of
the Square and Cube
and
Jewel in the Crown

"There is a pleasantly discursive
treatment of Pontius Pilate's
unanswered question
'What is truth?'"
— H. S. M. Coxeter, 1987,
introduction to Trudeau's
"story theory" of truth

Those who prefer stories to truth
may consult the Log24 entries
 of March 1, 2, 3, 4, and 5.

They may also consult
the poet Rubén Darío:

… Todo lo sé por el lucero puro
que brilla en la diadema de la Muerte.


* For a road out of this wasteland, back to geometry, see The Kaleidoscope Puzzle and Reflection Groups in Finite Geometry.

Wednesday, March 5, 2008

Wednesday March 5, 2008

Filed under: General,Geometry — Tags: — m759 @ 1:09 pm
(Context: March 2-4)

For CENTRAL
Central Intelligence:

"God does not play dice."
— Paraphrase of a remark
by Albert Einstein

Another Nobel Prize winner,
Isaac Bashevis Singer—

"a God who speaks in deeds,
not in words, and whose
vocabulary is the Cosmos"

From "The Escapist:
The Reality of Fantasy Games"–

Platonic solids as Dungeons & Dragons dice
Dungeons & Dragons Dice

 

From today's New York Times:

NY Times obituaries online, March 5, 2008: Gary Gygax, Wm. F. Buckley, Kaddish ad by Hadassah

A Kaddish for Gygax:

 

 

"I was reading Durant's section on Plato, struggling to understand his theory of the ideal Forms that lay in inviolable perfection out beyond the phantasmagoria. (That was the first, and I think the last, time that I encountered that word.)"


Related material:

For more on the word
"phantasmagoria," see
Log24 on Dec. 12, 2004
and on Sept. 23, 2006.

For phantasmagoria in action,
see Dungeons & Dragons
and Singer's (and others')
Jewish fiction.

For non-phantasmagoria,
see (for instance) the Elements
of Euclid, which culminates
in the construction of the
Platonic solids illustrated above.

See also Geometry for Jews.

Tuesday, March 4, 2008

Tuesday March 4, 2008

Filed under: General,Geometry — Tags: — m759 @ 1:00 pm
… And for a
    Swiftly Tilting
       Shadowed Planet …

Wm. F. Buckley as Archimedes, moving the world with a giant pen as lever. The pen's point is applied to southern South America.
John Trever, Albuquerque Journal, 2/29/08

The pen's point:

Log24, Dec. 11, 2006

SINGER, ISAAC:
"Are Children the
Ultimate Literary Critics?"
— Top of the News 29
(Nov. 1972): 32-36.

"Sets forth his own aims in writing for children and laments 'slice of life' and chaos in children's literature. Maintains that children like good plots, logic, and clarity, and that they have a concern for 'so-called eternal questions.'"

— An Annotated Listing
of Criticism
by Linnea Hendrickson

"She returned the smile, then looked across the room to her youngest brother, Charles Wallace, and to their father, who were deep in concentration, bent over the model they were building of a tesseract: the square squared, and squared again: a construction of the dimension of time."

— A Swiftly Tilting Planet,
by Madeleine L'Engle

 

Cover of 'A Swiftly Tilting Planet' and picture of tesseract

For "the dimension of time,"
see A Fold in Time,
Time Fold, and
Diamond Theory in 1937. 
 
A Swiftly Tilting Planet  is a fantasy for children set partly in Vespugia, a fictional country bordered by Chile and Argentina.

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