Chamfered cube

Chamfered cube
(OFF file)
Rank	3
Type	Near-miss
Notation
Conway notation	cC
Elements
Faces	6 squares, 12 rectangular-symmetric hexagons
Edges	24+24
Vertices	8+24
Vertex figures	8 equilateral triangles
	24 isosceles triangles
Measures (edge length 1)
Dihedral angles	4-6: 135°
	6-6: 120°
Central density	1
Number of external pieces	18
Level of complexity	4
Related polytopes
Army	Chamfered cube
Regiment	Chamfered cube
Dual	Tetrakis cuboctahedron
Conjugate	None
Abstract & topological properties
Flag count	192
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	B3, order 48
Flag orbits	4
Convex	Yes
Nature	Tame

The chamfered cube, also known as the order-4 truncated rhombic dodecahedron, is one of the near-miss Johnson solids. It has 6 squares and 12 hexagons as faces, and 32 order-3 vertices divided into two sets of 8 and 24 each. It can be made equilateral, but the hexagons have two angle sizes.

Assuming they are equilateral, the hexagonal faces have angles of $\arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }$ on one pair of opposite vertices, and angles of $\arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }$ on the four remaining vertices.

The canonical variant with midradius 1 has two edge lengths: one of length ${\sqrt {2-{\sqrt {3}}}}\approx 0.51764$ and the other of length ${\frac {\sqrt {6}}{4}}\approx 0.61237$ with the same dihedral angles as the equilateral variant.

It can also be viewed as an order-4-truncated rhombic dodecahedron, or as an octahedrally-symmetric Goldberg polyhedron of index (2,0).

It is the convex core of the uniform quasirhombicuboctahedron and of the compound rhombihexahedron.

Vertex coordinates

The vertices of an equilateral chamfered cube can be given by all changes of sign of:

• $\left({\frac {1}{2}}+{\frac {\sqrt {3}}{3}},\,{\frac {1}{2}}+{\frac {\sqrt {3}}{3}},\,{\frac {1}{2}}+{\frac {\sqrt {3}}{3}}\right)$,

along with all permutations and changes of sign of:

• $\left({\frac {1}{2}},\,{\frac {1}{2}},\,{\frac {1}{2}}+{\frac {2{\sqrt {3}}}{3}}\right)$.

External links

• Klitzing, Richard. "Chamfered cube".
• Wikipedia contributors. "Chamfered cube".
• McCooey, David. "Chamfered Cube (all edges equal)"