Last revision Wednesday 28 August 1996
Determining the properties of the cube is extremely easy. In the diagram
at right, R = Sqrt[3]/2. Finally, it is obvious that _{c}r,
the inradius, is simply half an edge length, or 1/2.
_{c}A rather interesting observation is that the tetrahedron can be inscribed
in a cube. The three twofold axes of the tetrahedron align with the three
fourfold axes of the cube, and similarly, the four threefold axes of the
tetrahedron align with the four threefold axes of the cube. This results
in the sharing of four corners, and the edges of the tetrahedron lie in
the same planes as the faces of the
cube. The resulting compound has tetrahedral symmetry, since the fourfold
symmetry of the square faces is lost. But if we inscribe another tetrahedron
such that its vertices coincide with the remaining four unoccupied vertices
of the cube, we regain the fourfold axes and the resulting compound of
two tetrahedra, known as Kepler's |