Tetragonal disphenoid

Tetragonal disphenoid
Rank	3
Type	Noble
Notation
Bowers style acronym	Tedow
Coxeter diagram	xo ox&#y
Elements
Faces	4 isosceles triangles
Edges	2+4
Vertices	4
Vertex figure	Isosceles triangle
Measures (edge lengths b [base], ℓ [lacing])
Circumradius	${\frac {\sqrt {l^{2}+{\frac {b^{2}}{2}}}}{2}}$
Height	${\sqrt {l^{2}-{\frac {b^{2}}{2}}}}$
Central density	1
Related polytopes
Army	Tedow
Dual	Tetragonal disphenoid
Abstract & topological properties
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	(B2×A1)/2, order 8
Convex	Yes
Nature	Tame

Not to be confused with Square 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 ${\frac {b{\sqrt {2}}}{2}}$ 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 sign changes of:

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

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.