# Great rhombitetratetrahedron

Great rhombitetratetrahedron
Rank3
TypeSemi-uniform
SpaceSpherical
Notation
Bowers style acronymGratet
Coxeter diagramx3y3z
Elements
Faces6 rectangles, 4+4 ditrigons
Edges12+12+12
Vertices24
Vertex figureScalene triangle
Measures (edge lengths a, c (of rectangle), b (between 2 ditrigons))
Circumradius${\displaystyle \sqrt{\frac{3a^2+4b^2+3c^2+4ab+2ac+4bc}{8}}}$
Volume${\displaystyle (a^3+6a^2b+9a^2c+12ab^2+36abc+9ac^2+4b^3+12b^2c+6bc^2+c^3)\frac{\sqrt2}{12}}$
Dihedral angles6–4: ${\displaystyle \arccos\left(-\frac{\sqrt3}{3}\right) ≈ 125.26439°}$
6–6: ${\displaystyle \arccos\left(-\frac13\right) ≈ 109.47122°}$
Central density1
Related polytopes
ArmyGratet
RegimentGratet
DualDisdyakis hexahedron
ConjugateGreat rhombitettratetrahedron
Abstract properties
Euler characteristic2
Topological properties
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryA3, order 24
ConvexYes
NatureTame

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:

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