The Snub Dodecahedron
The snub dodecahedron is a 3D uniform polyhedron bounded by 12 pentagons and 20+60=80 triangles. It is constructed by alternating the vertices of a suitably-proportioned, non-uniform great rhombicosidodecahedron.
The snub dodecahedron, like the snub cube, is chiral: it is not the same as its mirror image. The following images show its two enantiomorphs (mirror images):
The 80 triangular faces of the snub dodecahedron are of two different kinds: the first kind shares all three edges only with other triangles, and the second kind shares an edge with a pentagon, another triangle of the second kind, and a triangle of the first kind. There are 20 triangles of the first kind, shown in cyan below:
These 20 triangles correspond with the faces of the regular icosahedron.
The remaining 60 triangles are of the second kind, and come in 30 pairs, corresponding with the 30 edges of the dodecahedron. The triangles in each pair share an edge with each other. These are shown in purple below:
The following are projections of the snub dodecahedron from some representative viewpoints:
Pentagon-centered parallel projection. Note the chiral pentagonal symmetry of the projection.
Parallel projection centered on a triangular face of the first kind.
Parallel projection centered on an edge shared between two triangles of the second kind.
The Cartesian coordinates for the snub dodecahedron, centered on the origin, are all even permutations of coordinates and even changes of sign of:
- (2A, 2, 2B)
- (A + B/φ + φ, −Aφ + B + 1/φ, A/φ + Bφ − 1)
- (A + B/φ − φ, Aφ − B + 1/φ, A/φ + Bφ + 1)
- (−A/φ + Bφ + 1, −A + B/φ − φ, Aφ + B − 1/φ)
- (−A/φ + Bφ − 1, A − B/φ − φ, Aφ + B + 1/φ)
where φ = (1+√5)/2 is the Golden Ratio, A=ξ−1/ξ, B=ξφ+φ2+φ/ξ, and ξ is the solution of the equation:
ξ3 − 2ξ = φ
The exact value of ξ is:
ξ = ∛(φ/2 + (√(φ−5/27))/2) + ∛(φ/2 − (√(φ−5/27))/2)
Numerically, ξ is approximately 1.715561499697368.
The snub dodecahedron does not occur as cells in any uniform 4D polytopes except its own prism, the snub dodecahedron prism.
The coordinates of the snub dodecahedron were obtained from Wikipedia's snub dodecahedron page.