For previous remarks on this topic, as it relates to

symmetry axes of the cube, see previous posts tagged Interplay.

The above posts discuss, among other things, the Galois

projective plane of order 3, with 13 points and 13 lines.

These Galois points and lines may be modeled in Euclidean geometry

by the 13 symmetry axes and the 13 rotation planes

of the Euclidean cube. They may also be modeled in Galois geometry

by subsets of the 3x3x3 Galois cube (vector 3-space over GF(3)).

**The 3×3×3 Galois Cube **

**Exercise:** Is there any such analogy between the 31 points of the

order-5 Galois projective plane and the 31 symmetry axes of the

Euclidean dodecahedron and icosahedron? Also, how may the

31 projective *points *be naturally pictured as *lines* within the

5x5x5 Galois cube (vector 3-space over GF(5))?

Update of Nov. 30, 2014 —

For background to the above exercise, see

pp. 16-17 of *A Geometrical Picture Book* ,

by Burkard Polster (Springer, 1998), esp.

the citation to a 1983 article by Lemay.