Great rhombitetratetrahedron

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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:
 6–6:
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[edit | edit source]

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:

  • .

Measures[edit | edit source]

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

  • Circumradius =
  • Volume =