The Disdyakis Triacontahedron

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The disdyakis triacontahedron is a 3D Catalan solid bounded by 120 scalene triangles, 180 edges (60 short, 60 medium, 60 long), and 62 vertices. It is the dual of the great rhombicuboctahedron.

The 120 triangular faces are transitive but irregular. The ratio of edge lengths is 5√(5−3φ) : 3 : 2√(29−16φ), or approximately 1.910 : 3 : 3.528, where φ=(1+√5)/2 is the Golden Ratio.

The 62 vertices are of three kinds: 12 vertices where 10 edges (5 long, 5 short) meet, corresponding to an inscribed icosahedron, 20 vertices where 6 edges (3 long, 3 short) meet, corresponding to an inscribed dodecahedron, and 30 vertices where 4 edges (2 long, 2 short) meet, corresponding to an inscribed icosidodecahedron.

The triangular faces of the disdyakis triacontahedron have a 1-to-1 correspondence with the fundamental regions of the icosahedral symmetry group.

Projections

The following are images of the disdyakis triacontahedron from various viewpoints:

Projection	Description
	Parallel projection centered on an order-10 vertex.
	Centered on an order-6 vertex.
	Centered on an order-4 vertex.
	Centered on one of the long edges.

Animation

Here's an animation of the disdyakis triacontahedron rotating around one of the axes with 5-fold symmetry:

Coordinates

The Cartesian coordinates for the disdyakis triacontahedron are all permutations of coordinate and changes of sign of:

• (0, 0, 2φ)
• (A, A, A)

as well as even permutations of coordinate and all changes of sign of:

• (0, Aφ, A/φ)
• (0, B, Bφ)
• (1, φ², φ)

where φ=(1+√5)/2 is the Golden Ratio, A=(6φ−4)/3, and B=(6+2φ)/5.

The vertices that have A as a factor correspond with an inscribed dodecahedron of edge length 2A; the vertices containing B correspond with an inscribed icosahedron of edge length 2B. The remaining vertices correspond with an inscribed icosidodecahedron of edge length 2.

The dual great rhombicosidodecahedron corresponding to these coordinates has edge length (9−5φ)/11.