"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 projective plane.

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

exactly one axis for each pair of opposite

subcubes in the (Galois) 3×3×3 cube below --

The geometer's 3
×3×3
cube --

27 separate subcubes unconnected

by any Rubik-like mechanism --

A closely
related structure --

the finite projective plane

with 13 points and 13 lines --

The 13 *lines*
of the Galois
projective plane

may be derived from 13 Galois affine planes

through the Galois cube's center.

(One such plane is illustrated above.)

These 13 Galois affine
planes may in turn be derived from 13

Euclidean planes
through
the Euclidean
cube's center point

perpendicular to the 13 Euclidean
axes of symmetry.

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)–

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).” |

The moral of the story --

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

A summary of the story --

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 perpendicular to the 13 axes of symmetry.

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.)

Update of January 13, 2013

Summary of how the cube's 13 symmetry planes* are related to

the finite projective plane of order 3, with 13 points and 13 lines--

* This is not the standard terminology. Most sources count only the 9 planes

fixed pointwise under reflections as "symmetry planes." This of course

obscures the connection with finite geometry.