Great stellapentakis dodecahedron

Great stellapentakis dodecahedron
Rank3
TypeUniform dual
Notation
Coxeter diagramo5/2m3m ()
Elements
Faces60 isosceles triangles
Edges30+60
Vertices12+20
Vertex figure12 pentagrams, 20 hexagons
Measures (edge length 1)
Inradius${\displaystyle 9{\frac {\sqrt {109\left(17-6{\sqrt {5}}\right)}}{218}}\approx 0.81594}$
Dihedral angle${\displaystyle \arccos \left(-{\frac {80-9{\sqrt {5}}}{109}}\right)\approx 123.32007^{\circ }}$
Central density7
Number of external pieces120
Related polytopes
DualTruncated great icosahedron
ConjugatePentakis dodecahedron
Convex coreNon-Catalan deltoidal hexecontahedron
Abstract & topological properties
Flag count360
Euler characteristic2
OrientableYes
Genus0
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

The great stellapentakis dodecahedron is a uniform dual polyhedron. It consists of 60 isosceles triangles.

If its dual, the truncated great icosahedron, has an edge length of 1, then the lateral edges of the triangles will measure ${\displaystyle 9{\frac {1+2{\sqrt {5}}}{19}}\approx 2.59206}$, and the base edges will be ${\displaystyle 3{\frac {1+{\sqrt {5}}}{2}}\approx 4.85410}$. The triangles have two interior angles of ${\displaystyle \arccos \left({\frac {9+{\sqrt {5}}}{12}}\right)\approx 20.55444^{\circ }}$, and one of ${\displaystyle \arccos \left(-{\frac {7+9{\sqrt {5}}}{36}}\right)\approx 138.89111^{\circ }}$.

Vertex coordinates

A great stellapentakis dodecahedron with dual edge length 1 has vertex coordinates given by all even permutations of:

• ${\displaystyle \left(\pm 3{\frac {{\sqrt {5}}-1}{4}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,0\right)}$,
• ${\displaystyle \left(\pm 9{\frac {9-{\sqrt {5}}}{76}},\,\pm 9{\frac {5{\sqrt {5}}-7}{76}},\,0\right)}$,
• ${\displaystyle \left(\pm {\frac {3}{2}},\,\pm {\frac {3}{2}},\,\pm {\frac {3}{2}}\right)}$.