Great rhombitetratetrahedron

Great rhombitetratetrahedron
Rank3
TypeSemi-uniform
Notation
Bowers style acronymGratet
Coxeter diagramx3y3z
Elements
Faces6 rectangles, 4+4 ditrigons
Edges12+12+12
Vertices24
Vertex figureScalene triangle
Measures (edge length 1)
Dihedral angles6–4: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
6–6: ${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
Central density1
Related polytopes
ArmyGratet
RegimentGratet
DualDisdyakis hexahedron
ConjugateGreat rhombitettratetrahedron
Abstract & topological properties
Flag count144
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryA3, order 24
Flag orbits6
ConvexYes
NatureTame

The great rhombitetratetrahedron, or gratet, is a convex semi-uniform polyhedron. It is the result of relaxing the truncated octahedron so that it only needs to possess tetrahedral symmetry rather than cubic symmetry. 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:

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

Measures

Letting a , b , and c  be as before, we have the following measures:

• Circumradius = ${\displaystyle {\sqrt {\frac {3a^{2}+4b^{2}+3c^{2}+4ab+2ac+4bc}{8}}}}$
• Volume = ${\displaystyle (a^{3}+6a^{2}b+9a^{2}c+12ab^{2}+36abc+9ac^{2}+4b^{3}+12b^{2}c+6bc^{2}+c^{3}){\frac {\sqrt {2}}{12}}}$