Great rhombitetratetrahedron

Great rhombitetratetrahedron
Rank	3
Type	Semi-uniform
Space	Spherical
Notation
Bowers style acronym	Gratet
Coxeter diagram	x3y3z
Elements
Faces	6 rectangles, 4+4 ditrigons
Edges	12+12+12
Vertices	24
Vertex figure	Scalene triangle
Measures (edge lengths a, c (of rectangle), b (between 2 ditrigons))
Circumradius	$\sqrt{\frac{3a^2+4b^2+3c^2+4ab+2ac+4bc}{8}}$
Volume	$(a^3+6a^2b+9a^2c+12ab^2+36abc+9ac^2+4b^3+12b^2c+6bc^2+c^3)\frac{\sqrt2}{12}$
Dihedral angles	6–4: $\arccos\left(-\frac{\sqrt3}{3}\right) ≈ 125.26439°$
	6–6: $\arccos\left(-\frac13\right) ≈ 109.47122°$
Central density	1
Related polytopes
Army	Gratet
Regiment	Gratet
Dual	Disdyakis hexahedron
Conjugate	Great rhombitettratetrahedron
Abstract properties
Euler characteristic	2
Topological properties
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	A3, order 24
Convex	Yes
Nature	Tame

The great rhombitetratetrahedron, or gratet, is a convex semi-uniform polyhedron that is a tetrahedral-symmetric variant of the truncated octahedron. It has 2 sets of 4 ditrigons and 6 rectangles as faces. It generally has 3 types of edge lengths, connecting each pair of face types.

It can be alternated into a snub tetrahedron.

Vertex coordinates

A great rhombitetratetrahedron with edges of length a, b, and c, where a and c are the rectangle edges and b edges are between the two types of ditrigons, has vertex coordinates given by all permutation and even sign changes of:

• $\left((a+2b+c)\frac{\sqrt2}{4},\,(a+c)\frac{\sqrt2}{4},\,a\frac{\sqrt2}{4}\right).$