# Rhombitetratetrahedron

Rhombitetratetrahedron Rank3
TypeSemi-uniform
SpaceSpherical
Notation
Bowers style acronymRatet
Coxeter diagramx3o3y
Elements
Faces4+4 triangles, 6 rectangles
Edges12+12
Vertices12
Vertex figureIsosceles trapezoid
Measures (edge lengths a, b)
Circumradius$\sqrt{\frac{3a^2+3b^2+2ab}{8}}$ Volume$(a^3+9a^2b+9ab^2+b^3)\frac{\sqrt2}{12}$ Dihedral angle$\arccos\left(-\frac{\sqrt3}{3}\right) ≈ 125.26439^\circ$ Central density1
Related polytopes
ArmyRatet
RegimentRatet
DualDeltoidal dodecahedron
ConjugateRhombitetratetrahedron
Abstract & topological properties
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryA3, order 24
ConvexYes
NatureTame

The rhombitetratetrahedron, or ratet, is a convex semi-uniform polyhedron that is a tetrahedral-symmetric variant of the cuboctahedron. It has 2 sets of 4 triangles of generally different sizes, with 6 rectangles connecting them. Half of the edges are of each size.

A rhombitetratetrahedron is the vertex figure of one uniform polychoron, the grand ditetrahedronary hexacosidishecatonicosachoron.

## Vertex coordinates

A rhombitetratetrahedron with edge lengths a and b has vertices given by all permutations and even sign changes of:

• $\left((a+b)\frac{\sqrt2}{4},\,(a+b)\frac{\sqrt2}{4},\,(a-b)\frac{\sqrt2}{4}\right).$ 