Log24

Monday, September 28, 2015

Cracker Jack Prize

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

From a post of July 24, 2011

Mira Sorvino in 'The Last Templar'

A review —

“The story, involving the Knights Templar, the Vatican, sunken treasure,
the fate of Christianity and a decoding device that looks as if it came out of 
a really big box of medieval Cracker Jack, is the latest attempt to combine
Indiana Jones derring-do with ‘Da Vinci Code’ mysticism.”

— The New York Times

A feeble attempt at a purely mathematical "decoding device"
from this journal earlier this month

Image that may or may not be related to the extended binary Golay code and the large Witt design

For some background, see a question by John Baez at Math Overflow
on Aug. 20, 2015.

The nonexistence of a 24-cycle in the large Mathieu group
might discourage anyone hoping for deep new insights from
the above figure.

See Marston Conder's "Symmetric Genus of the Mathieu Groups" —

Saturday, September 19, 2015

Geometry of the 24-Point Circle

Filed under: General,Geometry — Tags: — m759 @ 1:13 AM

The latest Visual Insight  post at the American Mathematical
Society website discusses group actions on the McGee graph,
pictured as 24 points arranged in a circle that are connected
by 36 symmetrically arranged edges.

Wikipedia remarks that

"The automorphism group of the McGee graph
is of order 32 and doesn't act transitively upon
its vertices: there are two vertex orbits of lengths
8 and 16."

The partition into 8 and 16 points suggests, for those familiar
with the Miracle Octad Generator and the Mathieu group M24,
the following exercise:

Arrange the 24 points of the projective line
over GF(23) in a circle in the natural cyclic order
, 1, 2, 3,  , 22, 0 ).  Can the McGee graph be
modeled by constructing edges in any natural way?

Image that may or may not be related to the extended binary Golay code and the large Witt design

In other words, if the above set of edges has no
"natural" connection with the 24 points of the
projective line over GF(23), does some other 
set of edges in an isomorphic McGee graph
have such a connection?

Update of 9:20 PM ET Sept. 20, 2015:

Backstory: A related question by John Baez
at Math Overflow on August 20.

Tuesday, June 17, 2014

Finite Relativity

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

Continued.

Anyone tackling the Raumproblem  described here
on Feb. 21, 2014 should know the history of coordinatizations
of the 4×6 Miracle Octad Generator (MOG) array by R. T. Curtis
and J. H. Conway. Some documentation:

The above two images seem to contradict a statement by R. T. Curtis
in a 1989 paper.  Curtis seemed in that paper to be saying, falsely, that
his original 1973 and 1976 MOG coordinates were those in array M below—

This seemingly false statement involved John H. Conway's supposedly
definitive and natural canonical coordinatization of the 4×6 MOG
array by the symbols for the 24 points of the projective line over GF(23)—
{∞, 0, 1, 2, 3… , 21, 22}:

An explanation of the apparent falsity in Curtis's 1989 paper:

By "two versions of the MOG" Curtis seems to have meant merely that the
octads , and not the projective-line coordinates , in his earlier papers were
mirror images of the octads  that resulted later from the Conway coordinates,
as in the images below.

Friday, February 21, 2014

Raumproblem*

Filed under: General,Geometry — Tags: , — m759 @ 7:01 PM

Despite the blocking of Doodles on my Google Search
screen, some messages get through.

Today, for instance —

"Your idea just might change the world.
Enter Google Science Fair 2014"

Clicking the link yields a page with the following image—

IMAGE- The 24-triangle hexagon

Clearly there is a problem here analogous to
the square-triangle coordinatization problem,
but with the 4×6 rectangle of the R. T. Curtis
Miracle Octad Generator playing the role of
the square.

I once studied this 24-triangle-hexagon
coordinatization problem, but was unable to
obtain any results of interest. Perhaps
someone else will have better luck.

* For a rather different use of this word,
see Hermann Weyl in the Stanford
Encyclopedia of Philosophy.

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