The Pentakis Dodecahedron

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The pentakis dodecahedron is a 3D Catalan solid bounded by 60 isosceles triangles, 90 edges, and 32 vertices. It is the dual of the truncated icosahedron.

The faces are transitive isosceles triangles with 2 short edges and 1 long edge. The edge length ratio is (12+3φ)/19 : 1 (approximately 0.887 : 1), where φ=(1+√5)/2 is the Golden Ratio.

The 32 vertices are of two kinds: 20 vertices where 5 short edges meet, corresponding to an inscribed icosahedron, and 12 vertices where 6 edges meet (3 long, 3 short), corresponding to an inscribed dodecahedron.

Projections

The following are images of the pentakis dodecahedron from various viewpoints:

Projection	Description
	Projection centered on an order-5 vertex.
	Centered on an order-6 vertex.
	Centered on a long edge.
	Centered on a short edge.

Animation

Here's an animation of a pentakis dodecahedron rotating around the vertical axis:

Coordinates

The Cartesian coordinates for the pentakis dodecahedron are all permutations of coordinate and all changes of sign of:

• (1, 1, 1)

along with even permutations of coordinate and all changes of sign of:

• (0, A, Aφ)
• (0, φ, 1/φ)

where φ=(1+√5)/2 is the Golden Ratio, and A=(12+3φ)/19 is the length ratio of short edges to long edges.

The edge length of the corresponding dual truncated icosahedron is 2/(3φ²).