The Snub Cube
The snub cube is chiral: its mirror image is distinct from itself. Its two forms are its enantiomorphs, shown below:
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 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:
To help identify the two kinds of triangular faces, we will henceforth color the triangles of the first kind slightly differently:
The following are projections of the snub cube from some representative viewpoints:
Parallel projection centered on a square face.
Parallel projection centered on a triangular face of the first kind. Note the chiral trigonal symmetry of the projection.
Parallel projection centered on an edge between a pair of triangles of the second kind.
Vertex-first parallel projection.
The Cartesian coordinates of the snub cube are all even permutations of coordinates with an even number of negative signs of:
- (1, ξ, 1/ξ)
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.
The snub cube does not occur as cells in any uniform 4D polytopes except its own prism, the snub cube prism.
The Cartesian coordinates of the snub cube are described on Wikipedia snub cube page.