Log24

Monday, February 1, 2016

The Hiroshima Preprint

Filed under: General,Geometry — m759 @ 2:00 PM

This morning at 11:44 I happened upon

This was published as

Toshiyuki Katsura, Shigeyuki Kondo, Ichiro Shimada,
"On the supersingular K3 surface in characteristic 5 with Artin invariant 1,"
Michigan Mathematical Journal , vol. 63, issue 4 (Dec. 2014), 803–844.

Related material from later today —

See also earlier Log24 remarks on the Hoffman-Singleton graph
and a remark on geometry for Princeton.

Wednesday, November 5, 2014

Dark Fields…

Filed under: General — m759 @ 1:06 AM

Continues.

From the first of previous Log24 posts tagged “Dark Fields”—

“A link in memory of Donald G. Higman,
dead on Feb. 13, 2006,
the day after Lincoln’s birthday:

On the Graphs of Hoffman-Singleton and Higman-Sims.

His truth is marching on.”

See also Foundation Square (October 25, 2014).

Saturday, October 25, 2014

Foundation Square

Filed under: General,Geometry — Tags: — m759 @ 2:56 PM

In the above illustration of the 3-4-5 Pythagorean triangle,
the grids on each side may be regarded as figures of
Euclidean  geometry or of Galois  geometry.

In Euclidean geometry, these grids illustrate a property of
the inner triangle.

In elementary Galois geometry, ignoring the connection with
the inner triangle, the grids may be regarded instead as
illustrating vector spaces over finite (i.e., Galois) fields.
Previous posts in this journal have dealt with properties of
the 3×3 and 4×4 grids.  This suggests a look at properties of
the next larger grid, the 5×5 array, viewed as a picture of the
two-dimensional vector space (or affine plane) over the finite
Galois field GF(5) (also known as ℤ5).

The 5×5 array may be coordinatized in a natural way, as illustrated
in (for instance) Matters Mathematical , by I.N. Herstein and
Irving Kaplansky, 2nd ed., Chelsea Publishing, 1978, p. 171:

See Herstein and Kaplansky for the elementary Galois geometry of
the 5×5 array.

For 5×5 geometry that is not so elementary, see…

Hafner's abstract:

We describe the Hoffman-Singleton graph geometrically, showing that
it is closely related to the incidence graph of the affine plane over ℤ5.
This allows us to construct all automorphisms of the graph.

The remarks of Brouwer on graphs connect the 5×5-related geometry discussed
by Hafner with the 4×4 geometry related to the Steiner system S(5,8,24).
(See the Miracle Octad Generator of R. T. Curtis and the related coordinatization
by Cullinane of the 4×4 array as a four-dimensional vector space over GF(2).)

Sunday, May 10, 2009

Sunday May 10, 2009

Filed under: General,Geometry — m759 @ 6:29 PM
Mother’s Day
at MAA

Rick’s Tricky Six Puzzle:
S5 Sits Specially in S6
by Alex Fink and Richard Guy

Abstract. Rick Wilson identified a sliding block puzzle, the Tricky Six puzzle, in which a uniquely small fraction of the possible scrambled arrangements of the six moving pieces can be restored to the solved state. The permutations one can perform form the abstract group S5, the symmetric group on five letters, but surprisingly they aren’t any of the “obvious” copies of S5 in S6 that fix a single point and allow the other five to be permuted arbitrarily. This special S5 comes from the outer automorphism of S6, a remarkable group-theoretic map whose presence is felt in several combinatorial objects. We track down this outer automorphism in the Tricky Six puzzle as well as the projective plane of order 4, the Hoffman-Singleton graph, the Steiner system S(5,6,12), and a couple of error-correcting codes.

Meanwhile:

'Wizard of Id,' Mother's Day 2009-- Royal carriage with 'FINK ON BOARD' sign

Click to enlarge.

Background:

A pair of matronly women
gave readings of
bad mathematical poetry
on April 28 at

Carriage House Conference Center of the Mathematical Association of American in Washington, D.C

the MAA’s Carriage House
Conference Center in
Washington, D.C.

Saturday, July 29, 2006

Saturday July 29, 2006

Filed under: General — Tags: — m759 @ 5:01 PM

Dark Fields
of the Republic

Today’s birthday: Ken Burns

Charley Reese on the republic:

“The republic died at Appomattox, and it’s been empire ever since.”

Charley Reese on Lincoln:

“Washington and Jefferson created the republic; Lincoln destroyed it.”

In closing…

A link in memory of Donald G. Higman, dead on Feb. 13, 2006, the day after Lincoln’s birthday:

On the Graphs of Hoffman-Singleton and Higman-Sims (pdf)

His truth is marching on.

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