# Pentakis dodecahedron

Pentakis dodecahedron
Rank3
TypeUniform dual
Notation
Bowers style acronymPakid
Coxeter diagramo5m3m ()
Conway notationkD
Elements
Faces60 isosceles triangles
Edges30+60
Vertices12+20
Vertex figure12 pentagons, 20 hexagons
Measures (edge length 1)
Dihedral angle${\displaystyle \arccos \left(-{\frac {80+9{\sqrt {5}}}{109}}\right)\approx 156.71855^{\circ }}$
Central density1
Number of external pieces60
Level of complexity3
Related polytopes
ArmyPakid
RegimentPakid
DualTruncated icosahedron
ConjugateGreat stellapentakis dodecahedron
Abstract & topological properties
Flag count360
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryH3, order 120
Flag orbits3
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 ${\displaystyle 3{\frac {7+5{\sqrt {5}}}{38}}\approx 1.43529}$ times the length of those of the dodecahedron. Using an icosahedron that is any number less than ${\displaystyle {\frac {1+{\sqrt {5}}}{2}}\approx 1.61803}$ times the edge length of the dodecahedron gives a fully symmetric variant of this polyhedron. The lower limit is ${\displaystyle {\frac {5+3{\sqrt {5}}}{10}}}$ times that of the cube, where the icosahedron's vertices will coincide with the dodecahedron's face centers.

Each face of this polyhedron is an isosceles triangle with base side length ${\displaystyle {\frac {9-{\sqrt {5}}}{6}}\approx 1.12732}$ times those of the side edges. These triangles have apex angle ${\displaystyle \arccos \left({\frac {9{\sqrt {5}}-7}{36}}\right)\approx 68.61873^{\circ }}$ and base angles ${\displaystyle \arccos \left({\frac {9-{\sqrt {5}}}{12}}\right)\approx 55.69064^{\circ }}$.