truncated cuboctahedron

It is the dual of the uniform great truncated cuboctahedron.

The projection envelope is in the shape of a non-uniform truncated cuboctahedron.

This model shares the name with the convex great rhombicuboctahedron, also called the truncated cuboctahedron.

In geometry, the truncated cuboctahedron is an Archimedean solid.

Its convex hull is a nonuniform truncated cuboctahedron.

The name truncated cuboctahedron, given originally by Johannes Kepler, is a little misleading.

However, the resulting figure is topologically equivalent to a truncated cuboctahedron and can always be deformed until the faces are regular.

The bifurcating diagram form can be drawn with two types (colors) of truncated cuboctahedron cells alternating.

Thus, the omnitruncated tesseract may be thought of as another analogue of the truncated cuboctahedron in 4 dimensions.

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

For example, this class uniform polyhedron with octahedral symmetry exist as degenerate forms of the truncated cuboctahedron (4.6.8).

Great rhombicuboctahedron is an alternative name for a truncated cuboctahedron, whose faces are parallel to those of the (small) rhombicuboctahedron.

The truncated cuboctahedron is one of a family of uniform polyhedra related to the cube and regular octahedron.

The other two truncated cuboctahedral cells project to a truncated cuboctahedron inscribed in the projection envelope.

Since each of its faces has point symmetry (equivalently, 180 rotational symmetry), the truncated cuboctahedron is a zonohedron.

In the truncated cuboctahedron first parallel projection into 3 dimensions, the cells of the cantitruncated tesseract are laid out as follows:

Truncated cuboctahedron (Johannes Kepler)

The Cartesian coordinates for the vertices of a truncated cuboctahedron having edge length 2 and centered at the origin are all permutations of:

If the original truncated cuboctahedron has edge length 1, its dual disdyakis dodecahedron has edge lengths , and .

In geometry, a disdyakis dodecahedron, or hexakis octahedron, is a Catalan solid and the dual to the Archimedean truncated cuboctahedron.

These are connected to the central truncated cuboctahedron via 6 octagonal prisms, which are the images of the octagonal prism cells, a pair to each image.

The truncated cuboctahedron and the great truncated cuboctahedron form isomorphic graphs despite their different geometric structure.

The vertex arrangement of this compound is shared by a convex nonuniform truncated cuboctahedron, having rectangular faces, alongside irregular hexagons and octagons, each alternating with two edge lengths.

The Archimedean solids with some hexagonal faces are the truncated tetrahedron, truncated octahedron, truncated icosahedron (of soccer ball and fullerene fame), truncated cuboctahedron and the truncated icosidodecahedron.

Finally, the 8 volumes between the hexagonal faces of the projection envelope and the hexagonal faces of the central truncated cuboctahedron are the images of the 16 truncated octahedra, two cells to each image.