# Rhombitetratetrahedron

Rhombitetratetrahedron
Rank3
TypeSemi-uniform
Notation
Bowers style acronymRatet
Coxeter diagramx3o3y
Elements
Faces4+4 triangles, 6 rectangles
Edges12+12
Vertices12
Vertex figureIsosceles trapezoid
Measures (edge lengths a, b)
Circumradius${\displaystyle {\sqrt {\frac {3a^{2}+3b^{2}+2ab}{8}}}}$
Volume${\displaystyle (a^{3}+9a^{2}b+9ab^{2}+b^{3}){\frac {\sqrt {2}}{12}}}$
Dihedral angle${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
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:

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