The Snub Cube


The snub cube is a 3D uniform polyhedron bounded by 6 squares and 8+24=32 triangles. It is constructed by alternating the vertices of a suitably-proportioned, non-uniform great rhombicuboctahedron.

The snub cube

The snub cube is chiral: its mirror image is distinct from itself. Its two forms are its enantiomorphs, shown below:

One enantiomorph of
the snub cube The other enantiomorph
of the snub cube

The 32 triangular faces of the snub cube are of two kinds. The first kind, consisting of 8 triangles, corresponds with the vertices of the cube. These are shown below in yellow:

The snub cube,
highlighting 8 triangles corresponding to cube vertices

The remaining 24 triangular faces come in 12 pairs, corresponding with the 12 edges of the cube. The triangles in each pair share an edge with each other. These 24 faces are shown below:

The snub cube,
highlighting 24 triangles corresponding to 12 cube edges

To help identify the two kinds of triangular faces, we will henceforth color the triangles of the first kind slightly differently:

The snub cube, with
triangles of the first kind colored differently

Projections

The following are projections of the snub cube from some representative viewpoints:

Projection Envelope Description
Octagon

Parallel projection centered on a square face.

Irregular dodecagon

Parallel projection centered on a triangular face of the first kind. Note the chiral trigonal symmetry of the projection.

Octagon

Parallel projection centered on an edge between a pair of triangles of the second kind.

Irregular nonagon

Vertex-first parallel projection.

Coordinates

The Cartesian coordinates of the snub cube are all even permutations of coordinates with an even number of negative signs of:

along with the odd permutations of coordinates with an odd number of negative signs of the same, where ξ is the solution to the equation:

ξ3 + ξ2 + ξ = 1

The exact value of ξ is:

(∛(17+3√33) − ∛(−17+3√33) − 1)/3

Numerically, ξ is approximately 0.54368901.

A very nice explanation of how these coordinates are derived may be found on Paul Scott's snub cube page.

Occurrences

The snub cube does not occur as cells in any uniform 4D polytopes except its own prism, the snub cube prism.

Credits

The Cartesian coordinates of the snub cube are described on Wikipedia snub cube page.


Last updated 02 Aug 2012.

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