## Response from Prof. B.

These are the polyhedra that we will call "semi-regular" in keeping with commonly accepted usage. Just as a figure is called "regular" only if it has regular polygon faces and additional symmetry constraints at the vertices, we will require that the semi-regular objects have symmetry too, so that every vertex looks like every other vertex, and moreove, so that there is a symmetry of all of space taking the object to itself taking any specified vertex to any other vertex position. The second condition is required in order to rule out some objects that "locally" have the same shape, but are rather asymmetric in other ways.

A key example here is similar to a polyhedron we began discussing a while ago and never completed. We started with a cube and began shaving off the twelve edges, leaving squares (not diamonds) in the center of each face of the cube. We cut the corners off too, giving in general eight equilateral triangles, six squares, and twelve congruent rectangles. By cutting off just the right amount, the rectangles will also be squares and we will have a semi-regular figure with the same symmetries as the original cube, with eighteen squares and eight triangles.

Now consider the following modification of this surface. We take the top square and its adjacent four squares and four triangles, and lift this "cap" up, then turn it forty-five degrees, and place it back down. The resulting figure will not have as much symmetry as the cube, but it will have at every vertex three squares and a triangle, and every vertex will have the same shape as every other. This figure was found early in this century by a Russian mathematician who claimed that he had found a semi-regular figure to add to those already enumerated by Archimedes. Either you accept that as a new discovery, or you go back and change the definition so that only figures with the appropriate symmetry need apply.

In any case, as you examine these semi-regular polyhedra, you should rediscover Archimedes' classification for convex polyhedra in three-space, and see how many of these figures will have analogues in higher dimensions.

Good work. I look forward to seeing such things in your final project. Also good use of sketches, in the spirit of Stringham. We can also develop some fnord figures to accompany the final document (although I would like to see the sketches preserved as well).