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

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 snub
dodecahedron Its mirror image

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:

The snub
dodecahedron, showing 20 triangles of the first kind

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 snub
dodecahedron, showing 60 triangles of the second kind

Projections

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

Projection Envelope Description
Irregular icosagon

Pentagon-centered parallel projection. Note the chiral pentagonal symmetry of the projection.

Octadecagon

Parallel projection centered on a triangular face of the first kind.

Dodecagon

Parallel projection centered on an edge shared between two triangles of the second kind.

Coordinates

The Cartesian coordinates for the snub dodecahedron, centered on the origin, are all even permutations of coordinates and even changes of sign of:

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.

Occurrences

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

Credits

The coordinates of the snub dodecahedron were obtained from Wikipedia's snub dodecahedron page.


Last updated 08 Aug 2012.

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