Pentakis dodecahedron

Pentakis dodecahedron
	(3D model); (OFF file)
Rank	3
Type	Uniform dual
Space	Spherical
Notation
Bowers style acronym	Pakid
Coxeter diagram	o5m3m ()
Elements
Faces	60 isosceles triangles
Edges	30+60
Vertices	12+20
Vertex figure	12 pentagons, 20 hexagons
Measures (edge length 1)
Dihedral angle
Central density	1
Related polytopes
Army	Pakid
Regiment	Pakid
Dual	Truncated icosahedron
Conjugate	Great stellapentakis dodecahedron
Abstract & topological properties
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	H3, order 120
Convex	Yes
Nature	Tame

The pentakis dodecahedron is one of the 13 Catalan solids. It has 60 isosceles triangles as faces, with 12 order-5 and 20 order-6 vertices. It is the dual of the uniform truncated icosahedron.

It can also be obtained as the convex hull of a dodecahedron and an icosahedron, where the edges of the icosahedron are $3\frac{7+5\sqrt5}{38} ≈ 1.43529$ times the length of those of the dodecahedron. Using an icosahedron that is any number less than $\frac{1+\sqrt5}{2} ≈ 1.61803$ times the edge length of the dodecahedron including if the two have the same edge length) gives a fully symmetric variant of this polyhedron.

Each face of this polyhedron is an isosceles triangle with base side length $\frac{9-\sqrt5}{6} ≈ 1.12732$ times those of the side edges. These triangles have apex angle $\arccos\left(\frac{9\sqrt5-7}{36}\right) ≈ 68.61873°$ and base angles $\arccos\left(\frac{9-\sqrt5}{12}\right) ≈ 55.69064°$ .