Last Wednesday's 11 PM post mentioned the
adjacency-isomorphism relating the 4-dimensional
hypercube over the 2-element Galois field GF(2) to
the 4×4 array made up of 16 square cells, with
opposite edges of the 4×4 array identified.
A web page illustrates this property with diagrams that
enjoy the Karnaugh property— adjacent vertices, or cells,
differ in exactly one coordinate. A brief paper by two German
authors relates the Karnaugh property to the construction
of a magic square like that of Dürer (see last Wednesday).
In a similar way (search the Web for Karnaugh + cube ),
vertex adjacency in the 6-dimensional hypercube over GF(2)
is isomorphic to cell adjacency in the 4x4x4 cube, with
opposite faces of the 4x4x4 cube identified.
The above cube may be used to illustrate some properties
of the 64-point Galois 6-space that are more advanced
than those studied by enthusiasts of "magic" squares
and cubes.
See
- the 4x4x4 cube and An Invariance of Symmetry
- the 4x4x4 cube and the nineteenth-century
geometers' "Solomon's seal" - the 4x4x4 cube and the Kummer surface
- the 4x4x4 cube and the Klein quadric.
Those who prefer narrative to mathematics may
consult posts in this journal containing the word "Cuber."