The Pentakis Dodecahedron

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 pentakis

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.


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.


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

triacontahedron rotating


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

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

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).

Last updated 03 Feb 2023.

Powered by Apache Runs on Debian GNU/Linux Viewable on any browser Valid CSS Valid HTML 5! Proud to be Microsoft-free