Truncated cuboctahedron

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The truncated cuboctahedron is an Archimedean solid. It has 12 regular square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices and 72 edges. Since each of its faces has point symmetry (equivalently, 180° rotational symmetry), the truncated cuboctahedron is a zonohedron.

Other names

Alternate interchangeable names are:

Rhombitruncated cuboctahedron
Great rhombicuboctahedron
Omnitruncated cuboctahedron

The name truncated cuboctahedron, given originally by Johannes Kepler, is a little misleading. If you truncate a cuboctahedron by cutting the corners off, you do not get this uniform figure: some of the faces will be rectangles. However, the resulting figure is topologically equivalent to a truncated cuboctahedron and can always be deformed until the faces are regular.

The alternative name great rhombicuboctahedron refers to the fact that the 12 square faces lie in the same planes as the 12 faces of the rhombic dodecahedron which is dual to the cuboctahedron. Compare to small rhombicuboctahedron.

One unfortunate point of confusion: There is a nonconvex uniform polyhedron by the same name. See uniform great rhombicuboctahedron.

Area and volume

The area A and the volume V of the truncated cuboctahedron of edge length a are:

A = 12(2+sqrt{2}+sqrt{3}) a^2 approx 61.7551724a^2

V = (22+14sqrt{2}) a^3 approx 41.7989899a^3.

Cartesian coordinates

Cartesian coordinates for the vertices of a truncated cuboctahedron centered at the origin are all permutations of

(±1, ±(1+√2), ±(1+√8)). (edge length = 2)

References

Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 3-9)

External links

The Uniform Polyhedra
Virtual Reality Polyhedra The Encyclopedia of Polyhedra