# Pentakis dodecahedron

Pentakis dodecahedron Rank3
TypeUniform dual
SpaceSpherical
Notation
Bowers style acronymPakid
Coxeter diagramo5m3m (     )
Elements
Faces60 isosceles triangles
Edges30+60
Vertices12+20
Vertex figure12 pentagons, 20 hexagons
Measures (edge length 1)
Dihedral angle$\arccos\left(-\frac{80+9\sqrt5}{109}\right) ≈ 156.71855°$ Central density1
Related polytopes
ArmyPakid
RegimentPakid
DualTruncated icosahedron
ConjugateGreat stellapentakis dodecahedron
Abstract & topological properties
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryH3, order 120
ConvexYes
NatureTame

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 $\arccos\left(\frac{9-\sqrt5}{12}\right) ≈ 55.69064°$ .