# Tetragonal disphenoid

Tetragonal disphenoid
Rank3
TypeNoble
SpaceSpherical
Bowers style acronymTedow
Info
Coxeter diagramxo ox&#y
SymmetryBC2×A1+, order 8
ArmyTedow
RegimentTedow
Elements
Vertex figureIsosceles triangle
Faces4 isosceles triangles
Edges2+4
Vertices4
Measures (edge lengths b [base], ℓ [lacing])
Circumradius${\displaystyle \frac{\sqrt{l^2+\frac{b^2}{2}}}{2}}$
Height${\displaystyle \sqrt{l^2-\frac{b^2}{2}}}$
Central density1
Euler characteristic2
Related polytopes
DualTetragonal disphenoid
ConjugateTetragonal disphenoid
Properties
ConvexYes
OrientableYes
NatureTame

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 ${\displaystyle \frac{b\sqrt2}{2}}$ and side edge length ${\displaystyle \sqrt{l^2-\frac{b^2}{2}}}$.

## 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:

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

## 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.