Compound of three cubes

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Compound of three cubes
Rank3
TypeUniform
Notation
Bowers style acronymRah
Elements
Components3 cubes
Faces6+12 squares
Edges12+24
Vertices24
Vertex figureEquilateral triangle, edge length 2
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {3}}{2}}\approx 0.70711}$
Inradius${\displaystyle {\frac {1}{2}}=0.5}$
Volume3
Dihedral angle90°
Central density3
Number of external pieces120
Level of complexity17
Related polytopes
ArmySemi-uniform Toe, edge lengths ${\displaystyle {\frac {\sqrt {2}}{2}}}$ (squares), ${\displaystyle {\frac {2-{\sqrt {2}}}{2}}}$ (between ditrigons)
RegimentRah
DualCompound of three octahedra
ConjugateCompound of three cubes
Convex coreChamfered cube
Abstract & topological properties
Flag count144
OrientableYes
Properties
SymmetryB3, order 48
Flag orbits3
ConvexNo
NatureTame

The rhombihexahedron, rah, or compound of three cubes is a uniform polyhedron compound. It consists of 6+12 squares, with three faces joining at a vertex.

A complete double cover of this compound is a special case of the general rhombisnub dishexahedron.

The cube components each have square prismatic symmetry. In fact this compound can be constructed by starting with three fully coincident cubes and rotating one 45º along each of the 4-fold symmetry axes.

Its quotient prismatic equivalent is the square axial prismatic trigyroprism, which is five-dimensional.

It is one of three uniform polyhedron compounds that is kaleidoscopic, along with the stella octangula and truncated stella octangula.

Vertex coordinates

The vertices of a rhombihexahedron of edge length 1 are given by all permutations of:

• ${\displaystyle \left(\pm {\frac {\sqrt {2}}{2}},\,\pm {\frac {1}{2}},\,0\right)}$.