Good discussion of the notions of regularity in all these different
dimensions.  Today in class I am going to discuss semi-regularity as well,
something that will give some additional evidence of the richness of the
subject, in three and in higher dimensions.  The notions of duality become
richer as we go up in dimensions too--we can consider the figures defined
by the midpoints of all segments, or all squares, or all cubes of the
hypercube.  The one for the centers of cubes gives the 16-cell, a cube dual
with 16 tetrahedra.  The one determined by the centers of the 32 edges is a
bit more complicated, involving tetrahedra and possibly other figures, the
way the figure in three-space determined by the midpoints of edges is
composed of triangles and squares.  (That sentence is already leading
somewhere).  The figure in four-space determined by the centers of the 24
squares is especially intriguing, and in fact it is my favorite.  I hope we
will have several chances to discuss it over the course of the semester.
        More to come, (from me and from you),

        Prof. B.