## Conjectures:

If there is a tiling in four dimensional space other than the hypercube, it
has yet to be found. This is due to time constraints and our inability to visualize
any four dimensional shapes other than the hypercube and the hypertetrahedron.
It is apparent the hyperoctahedron and the hypercube are always duals- by their
very nature. It seems that in every dimension, tiling cubes can be replaced
by inscribed octahedrons, and then the cubes removed, and perhaps the left-over
space can be filled with more octahedrons. It is clear, however, that the case
of the tetrahedron fitting inside the cube (tetrahedron’s edges as diagonals
of cube faces) was a special case, where the number of faces on the cube and
the number of edges on the tetrahedron coincided. This will never happen again
because, as dimension increase, the tetrahedron’s edges increase slower than
the cube’s faces, cells, etc. However, the hyperoctahedron has 16 tetrahedral
cells, and the hypercube has 8 cubical cells, so it is conceivable that the
hyperoctahedron fits inside the hypercube in much the same way that the tetrahedron
fits inside the cube, with each cubical cell of the hypercube housing two tetrahedral
cells, (each in one of two orientations). Possibly if the researchers could
visualize this, then perhaps they could discover strange and beautiful tilings
in higher dimensional spaces, although it is realized that visualization should
not be necessary. Since the two dimensional analogues of the cube and octahedron,
the square and a diamond, both tile, it seems conceivable that in even numbered
dimensions, cubes tile and octahedrons tile, while in odd numbered dimensions,
cubes tile and octahedrons with tetrahedrons tile.