Pentakis dodecahedron

Pentakis dodecahedron
(3D model) (OFF file)
Rank	3
Type	Uniform dual
Notation
Bowers style acronym	Pakid
Coxeter diagram	o5m3m ()
Conway notation	kD
Elements
Faces	60 isosceles triangles
Edges	30+60
Vertices	12+20
Vertex figure	12 pentagons, 20 hexagons
Measures (edge length 1)
Dihedral angle	$\arccos \left(-{\frac {80+9{\sqrt {5}}}{109}}\right)\approx 156.71855^{\circ }$
Central density	1
Number of external pieces	60
Level of complexity	3
Related polytopes
Army	Pakid
Regiment	Pakid
Dual	Truncated icosahedron
Conjugate	Great stellapentakis dodecahedron
Abstract & topological properties
Flag count	360
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	H3, order 120
Flag orbits	3
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 $3{\frac {7+5{\sqrt {5}}}{38}}\approx 1.43529$ times the length of those of the dodecahedron. Using an icosahedron that is any number less than ${\frac {1+{\sqrt {5}}}{2}}\approx 1.61803$ times the edge length of the dodecahedron gives a fully symmetric variant of this polyhedron. The lower limit is ${\frac {5+3{\sqrt {5}}}{10}}$ 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 ${\frac {9-{\sqrt {5}}}{6}}\approx 1.12732$ times those of the side edges. These triangles have apex angle $\arccos \left({\frac {9{\sqrt {5}}-7}{36}}\right)\approx 68.61873^{\circ }$ and base angles $\arccos \left({\frac {9-{\sqrt {5}}}{12}}\right)\approx 55.69064^{\circ }$.

External links

• Klitzing, Richard. "pakid".
• Wikipedia contributors. "Pentakis dodecahedron".
• McCooey, David. "Pentakis Dodecahedron"