February 2014
When designing this problem, I used four-dimensional space. However, Misha Lavrov gave us a much nicer answer using five-dimensional space, so we present his solution here:
The union of the two polytopes has
Eight vertices
Sixteen edges
Fourteen faces: eight triangles and six squares
Six cells: two tetrahedra and four triangular prisms
The resulting polytope is a polyhedral prism whose base is a tetrahedron. The prism is not right: if the two tetrahedra have vertices ABCD and A'B'C'D', then A'B'C'D' is a translation of ABCD placed so that A is at a unit distance from each of A', B', C', and D'.
1) The convex hull of {A, B, C, D, B', C', D'} is a four-dimensional pyramid whose base is the triangular prism BCDB'C'D' and whose vertex is A. This is our first polytope.
2) The convex hull of {A, A', B', C', D'} is a four-dimensional pyramid whose base is the tetrahedron A'B'C'D' and whose vertex is A. This is our second polytope.
3) The two polytopes are joined along the tetrahedral cell AB'C'D'.
We can give the points A, B, C, D, A', B', C', D' particularly nice coordinates by embedding the four dimensional polytope in five dimensional space while remaining on the hyperplane x1 + x2 + x3 + x4 + x5 = 1. Then we have:
A = (1, 0, 0, 0, 0)
B = (1, -1, 1, 0, 0)
C = (1, -1, 0, 1, 0)
D = (1, -1, 0, 0, 1)
A' = (0, 1, 0, 0, 0)
B' = (0, 0, 1, 0, 0)
C' = (0, 0, 0, 1, 0)
D' = (0, 0, 0, 0, 1)
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