The Rhombicuboctahedron

The rhombicuboctahedron, also known as the small rhombicuboctahedron, is a 3D uniform polyhedron bounded by 8 triangles and 6+12=18 squares. It may be constructed by radially expanding the square faces of the cube outwards, or equivalently, radially expanding the triangular faces of the octahedron outwards.


There are two distinct kinds of square faces on the rhombicuboctahedron: the first kind are the axial faces which are surrounded by 4 other square faces. There are six of these faces, and they correspond with the faces of a cube:

axial faces of a rhombicuboctahedron

The second kind of square face are the non-axial faces, which are surrounded by 2 squares and 2 triangles. There are 12 of them, and they correspond with the edges of a cube:

non-axial faces of a rhombicuboctahedron

It is important to distinguish between these two kinds of square faces, because their relative position to the triangular faces gives them different functions when the rhombicuboctahedron is fitted together with other polyhedra into 4D polytopes.


In order to be able to identify the rhombicuboctahedron in various projections of 4D objects, it is useful to know how it appears from various viewpoints. The following are some of the commonly-encountered views:

Projection Envelope Description
Regular octagon

Parallel projection centered on an axial square face.

Regular hexagon

Triangle-centered parallel projection.


Parallel projection centered on non-axial square face.


Vertex-centered parallel projection.


The Cartesian coordinates of the rhombicuboctahedron, centered on the origin and having edge length 2, are all permutations of coordinates and changes of sign of:


The rhombicuboctahedron appears as cells in the following 4D uniform polytopes:

Last updated 26 Feb 2013.

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