Rhombitetratetrahedron

Rhombitetratetrahedron
Rank	3
Type	Semi-uniform
Space	Spherical
Notation
Bowers style acronym	Ratet
Coxeter diagram	x3o3y
Elements
Faces	4+4 triangles, 6 rectangles
Edges	12+12
Vertices	12
Vertex figure	Isosceles 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 density	1
Related polytopes
Army	Ratet
Regiment	Ratet
Dual	Deltoidal dodecahedron
Conjugate	Rhombitetratetrahedron
Abstract & topological properties
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	A3, order 24
Convex	Yes
Nature	Tame

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:

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