# Antitruncated cube

Antitruncated cube
Rank3
TypeSemi-uniform
Elements
Faces8 triangles, 6 tetrapods
Edges12+24
Vertices24
Vertex figureIsosceles triangle, edge lengths 1, 2-2, 2-2
Measures (edge length 1)
Central density1
Related polytopes
ArmySirco
DualAntiapiculated octahedron
Convex coreCube
Abstract & topological properties
OrientableYes
Properties
SymmetryB3, order 48
ConvexNo
NatureTame

The antitruncated cube or inflected truncated cube is a semi-uniform polyhedron. It has 8 triangles and 6 tetrapods as faces and an isosceles triangle as a vertex figure, having one triangle and two tetrapods meeting at each vertex. It is both the antitruncation (sometimes called inflected truncation) of the cube and a faceting of the small rhombicuboctahedron.

It is isomorphic to the truncated cube.

This polyhedron cannot be optimized using the ratio method, because the solution (with intended minimal ratio 1:${\displaystyle {\sqrt {2}}}$ ≈ 1:1.41421) would yield an octahedron with three stellated squares instead.

## Vertex coordinates

The coordinates of the antitruncated cube are shared by that of the rhombicuboctahedron, being all permutations of:

• ${\displaystyle \left(\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right).}$

## Variations

The antitruncated cube is part of a very similar teepee where the triangles are resized, but the distances between them stay the same.