# Tetragonal disphenoid

Tetragonal disphenoid
Rank3
TypeNoble
Notation
Bowers style acronymTedow
Coxeter diagramxo ox&#y
Elements
Faces4 isosceles triangles
Edges2+4
Vertices4
Vertex figureIsosceles triangle
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
Related polytopes
ArmyTedow
DualTetragonal disphenoid
Abstract & topological properties
Flag count24
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
Symmetry(B2×A1)/2, order 8
Flag orbits3
ConvexYes
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 ${\displaystyle {\frac {b{\sqrt {2}}}{2}}}$ and side edge length ${\displaystyle {\sqrt {l^{2}-{\frac {b^{2}}{2}}}}}$.

## Vertex coordinates

The vertices of a tetragonal disphenoid with base edges of lengthb  and side edges of length l  are given by all even sign changes of:

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

## 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.