The object shown above is the Hemi-Cube. It is formed from the cube
by identifying points which are directly opposite each other in the cube.
It is a very special example of a regular map. A map is an
embedding of a graph [in this case K4] in a surface [here, the projective
plane] so that each region is simply-connected (i.e., a polygon). The map
is regular provided that it is very symmetric.
- The study of symmetry in graphs and maps is the part of Combinatorial geometry that
is my own field of research. I am currently engaged (engrossed) in questions about edge-transitive graphs of degree 4. My colleague
Primož Potočnik and I are engaged in finding all of them up to, say, 512 vertices. Currently, we are offering a Mini-Census which goes up to 150 vertices, and is known to be incomplete (though not as incomplete as it used to be!).
Last updated 18 February, 2008
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