Log24

Friday, September 7, 2018

A Square for Sims

Filed under: General — m759 @ 3:00 AM

The American Mathematical Society on Wednesday, September 5,
reported a death from October 23 last year

AMS obituary for mathematician Charles C. Sims

See also Higman-Sims and 5×5 in this  journal.

Tuesday, August 25, 2020

Geometric Pedigree

Filed under: General — m759 @ 1:50 PM

Curtis on Higman-Sims

Elsewhere . . .

See also Higman-Sims and 5×5 in this  journal.

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).)

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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