The name truncated cuboctahedron, given originally by Johannes Kepler, is misleading: an actual truncation of a cuboctahedron has rectangles instead of squares; however, this nonuniform polyhedron is topologically equivalent to the Archimedean solid unrigorously named truncated cuboctahedron.
The Cartesian coordinates for the vertices of a truncated cuboctahedron having edge length 2 and centered at the origin are all the permutations of:
Area and volume
The area A and the volume V of the truncated cuboctahedron of edge length a are:
Dissection
The truncated cuboctahedron is the convex hull of a rhombicuboctahedron with cubes above its 12 squares on 2-fold symmetry axes. The rest of its space can be dissected into 6 square cupolas below the octagons, and 8 triangular cupolas below the hexagons.
A dissected truncated cuboctahedron can create a genus 5, 7, or 11 Stewart toroid by removing the central rhombicuboctahedron, and either the 6 square cupolas, the 8 triangular cupolas, or the 12 cubes respectively. Many other lower symmetry toroids can also be constructed by removing the central rhombicuboctahedron, and a subset of the other dissection components. For example, removing 4 of the triangular cupolas creates a genus 3 toroid; if these cupolas are appropriately chosen, then this toroid has tetrahedral symmetry.[4][5]
Stewart toroids
Genus 3
Genus 5
Genus 7
Genus 11
Uniform colorings
There is only one uniform coloring of the faces of this polyhedron, one color for each face type.
A 2-uniform coloring, with tetrahedral symmetry, exists with alternately colored hexagons.
Orthogonal projections
The truncated cuboctahedron has two special orthogonal projections in the A2 and B2Coxeter planes with [6] and [8] projective symmetry, and numerous [2] symmetries can be constructed from various projected planes relative to the polyhedron elements.
Orthogonal projections
Centered by
Vertex
Edge 4-6
Edge 4-8
Edge 6-8
Face normal 4-6
Image
Projective symmetry
[2]+
[2]
[2]
[2]
[2]
Centered by
Face normal Square
Face normal Octagon
Face Square
Face Hexagon
Face Octagon
Image
Projective symmetry
[2]
[2]
[2]
[6]
[4]
Spherical tiling
The truncated cuboctahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
Like many other solids the truncated octahedron has full octahedral symmetry - but its relationship with the full octahedral group is closer than that: Its 48 vertices correspond to the elements of the group, and each face of its dual is a fundamental domain of the group.
The image on the right shows the 48 permutations in the group applied to an example object (namely the light JF compound on the left). The 24 light elements are rotations, and the dark ones are their reflections.
The edges of the solid correspond to the 9 reflections in the group:
Those between octagons and squares correspond to the 3 reflections between opposite octagons.
Hexagon edges correspond to the 6 reflections between opposite squares.
(There are no reflections between opposite hexagons.)
^Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN0-486-23729-X. (Section 3-9, p. 82)
^Cromwell, P.; Polyhedra, CUP hbk (1997), pbk. (1999). (p. 82)
![{\displaystyle {\begin{aligned}{\text{4-6:}}&\ \arccos {\tfrac {-{\sqrt {6}}}{3}}=144^{\circ }44'08''\\[4pt]{\text{4-8:}}&\ \arccos {\tfrac {-1}{\sqrt {2}}}=135^{\circ }\\{\text{6-8:}}&\ \arccos {\tfrac {-{\sqrt {3}}}{3}}=125^{\circ }15'51''\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad2edada83ede0261a61223952aa4e01f01d5b09)
