# Great dodecacronic hexecontahedron

Great dodecacronic hexecontahedron
Rank3
TypeUniform dual
Notation
Coxeter diagramm5/3m5/2o3*a
Elements
Faces60 kites
Edges60+60
Vertices12+12+20
Vertex figure20 triangles, 12 pentagrams, 12 decagrams
Measures (edge length 1)
Inradius${\displaystyle {\frac {\sqrt {205\left(19-8{\sqrt {5}}\right)}}{41}}\approx 0.36816}$
Dihedral angle${\displaystyle \arccos \left(-{\frac {19-8{\sqrt {5}}}{41}}\right)\approx 91.55340^{\circ }}$
Central density10
Number of external pieces300
Related polytopes
DualGreat dodecicosidodecahedron
ConjugateSmall dodecacronic hexecontahedron
Convex coreNon-Catalan pentakis dodecahedron
Abstract & topological properties
Flag count480
Euler characteristic–16
OrientableYes
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

The great dodecacronic hexecontahedron is a uniform dual polyhedron. It consists of 60 kites.

If its dual, the great dodecicosidodecahedron, has an edge length of 1, then the short edges of the kites will measure ${\displaystyle 2{\frac {\sqrt {2\left(5-2{\sqrt {5}}\right)}}{3}}\approx 0.68499}$, and the long edges will be ${\displaystyle 2{\frac {\sqrt {65-19{\sqrt {5}}}}{11}}\approx 0.86272}$. ​The kite faces will have length ${\displaystyle {\frac {\sqrt {10\left(157-31{\sqrt {5}}\right)}}{33}}\approx 0.89731}$, and width ${\displaystyle {\sqrt {5}}-1\approx 1.23607}$. ​The kites have two interior angles of ${\displaystyle \arccos \left({\frac {5-{\sqrt {5}}}{8}}\right)\approx 69.78820^{\circ }}$, one of ${\displaystyle \arccos \left({\frac {-5+2{\sqrt {5}}}{20}}\right)\approx 91.51239^{\circ }}$, and one of ${\displaystyle \arccos \left(-{\frac {5+9{\sqrt {5}}}{40}}\right)\approx 128.91121^{\circ }}$.

## Vertex coordinates

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

• ${\displaystyle \left(\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm {\frac {3-{\sqrt {5}}}{2}},\,0\right),}$
• ${\displaystyle \left(\pm {\frac {25-9{\sqrt {5}}}{22}},\,\pm {\frac {15-{\sqrt {5}}}{22}},\,0\right),}$
• ${\displaystyle \left(\pm {\frac {5-{\sqrt {5}}}{6}},\,\pm {\frac {3{\sqrt {5}}-5}{6}},\,0\right),}$
• ${\displaystyle \left(\pm {\frac {4{\sqrt {5}}-5}{11}},\,\pm {\frac {4{\sqrt {5}}-5}{11}},\,\pm {\frac {4{\sqrt {5}}-5}{11}}\right).}$