"The cube has…13 axes of symmetry:

6 *C*_{2} (axes joining midpoints of opposite edges),

4 *C*_{3} (space diagonals), and

3*C*_{4} (axes joining opposite face centroids)."

–Wolfram MathWorld article
on
the
cube

These 13 symmetry axes can be used to illustrate the
interplay between Euclidean and *Galois*
geometry in a cubic model of the 13-point Galois plane.

The geometer's 3×3×3
cube–

27 separate subcubes unconnected

by any Rubik-like mechanism–

The
13 symmetry axes
of the (Euclidean)
cube--

exactly one axis for each pair of opposite

subcubes in the (Galois) 3×3×3 cube–

A closely
related structure–

the finite projective plane

with 13 points and 13 lines–

A later version
of the
13-point plane

by Ed Pegg Jr.–

A group action
on the
3×3×3
cube

as illustrated by a
Wolfram program

by Ed Pegg Jr. (undated, but closely

related to a
March 26, 1985 note

by Steven H. Cullinane)–

The above images
tell a story of sorts.

The moral of the story–

*Galois
projective geometries can be
viewed
in the context of the larger affine geometries
from which they are derived. *

The standard
definition of *points* in a Galois projective plane is
that
they are *lines* through the (arbitrarily chosen) origin in a
corresponding affine 3-space converted to a vector
3-space.

If we choose the
origin as the center cube in coordinatizing the 3×3×3 cube
(See Weyl's* relativity
problem*), then the cube's 13 axes of symmetry can, if the other
26 cubes have properly (Weyl's "objectively") chosen coordinates,
illustrate nicely the 13 projective points derived from the
27 affine points in the cube model.

The 13 *lines*
of the resulting Galois
projective plane may be derived from Euclidean *planes* through
the cube's center point that are perpendicular to the
cube's 13
Euclidean symmetry axes.

The above standard definition of points in a Galois projective plane may of course also be used in a simpler structure– the eightfold cube.

(The eightfold cube also allows a less standard way to picture projective points that is related to the symmetries of "diamond" patterns formed by group actions on graphic designs.)

See also Ed Pegg
Jr. on finite geometry

at the Mathematical Association of America--

The Fano Plane
“One thing in the Fano plane that bothered me
for years (for
years,
I say) is that it had a circle – and it was described as a line. For
me, a line was a straight line, and I didn’t trust curved or wriggly
lines. This distrust kept me away from understanding projective planes,
designs, and finite geometries for a awhile (for years).” |