## 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.