Response from Prof. B.

Often the dimensional analogy doesn't show up in beginning analytic geometry, but it should. After all, once you have mastered pairs of numbers, then triples and quadruples come easily and n-tuples follow right along. Somehow it's the fourth dimension where the two ways of desigating "tuples" tend to blur together. We hardly would think of calling a pair a "two-tuple" and "one-tuple" for a single number on the real line is even less thinkable. Yet there would be a logic to it.

Did I forget to define a parallelotope? It is a generalization of a parallelogram in the plane or a parallelepiped in three-space, combinatorially the same as a hypercube, or n-cube, but without the restriction that the segments coming out of each vertex are mutually perpendicular. You still will have n sets of parallel segments, one from each set at each vertex.

Is the stella octangula still difficult to see? Note that when we divide the eight vertices of a cube into two disjoint sets, where neither set contains a pair of points joined by an edge of the cube, then the tetrahedra formed from these four-tuples have their edges along the diagonals of the square faces of the cube, and each diagonal belongs either to one tetrahedron or the other. The intersections of the diagonals are the centers of the square faces, and these six vertices are the vertices of the octahedron. Does that help?

Complex numbers lead to some of the most beautiful patterns around, as in my show at the Providence Art Club.