Tetragonal disphenoid

The tetragonal disphenoid or tedow is a type of tetrahedron with four identical isosceles triangles for faces. It can also be considered a digonal antiprism. Tetragonal disphenoids, being digonal antiprisms, are related to rhombic disphenoids, which are digonal gyroprisms.

The general tetragonal disphenoid can be obtained as the alternation of a square prism. If the tetragonal disphenoid's base edges are of length b and its side edges are of length l, the corresponding square prism has base edge length $$\frac{b\sqrt2}{2}$$ and side edge length $$\sqrt{l^2-\frac{b^2}{2}}$$.

Vertex coordinates
The vertices of a tetragonal disphenoid with base edges of length b and side edges of length l are given by all even permutations of:


 * $$\left(\frac{b\sqrt2}{4},\,\frac{b\sqrt2}{4},\,\frac{\sqrt{l^2-\frac{b^2}{2}}}{2}\right).$$

In vertex figures
Tetragonal disphenoids occur as vertex figures in 3 noble uniform polychora: the decachoron, the tetracontoctachoron, and the great tetracontoctachoron. They also appear as the vertex figure of any duoprism of two identical polygons.