# Small dodecacronic hexecontahedron

Small dodecacronic hexecontahedron
Rank3
TypeUniform dual
Notation
Coxeter diagramm3/2o5m5*a
Elements
Faces60 darts
Edges60+60
Vertices12+12+20
Vertex figure20 triangles, 12 pentagons, 12 decagons
Measures (edge length 1)
Inradius${\displaystyle {\frac {\sqrt {205\left(19+8{\sqrt {5}}\right)}}{41}}\approx 2.12099}$
Dihedral angle${\displaystyle \arccos \left(-{\frac {19+8{\sqrt {5}}}{41}}\right)\approx 154.12136^{\circ }}$
Central density2
Number of external pieces120
Related polytopes
DualSmall dodecicosidodecahedron
ConjugateGreat dodecacronic hexecontahedron
Convex coreDeltoidal hexecontahedron
Abstract & topological properties
Flag count480
Euler characteristic–16
OrientableYes
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

The small dodecacronic hexecontahedron is a uniform dual polyhedron. It consists of 60 darts.

It appears the same as the small rhombidodecacron.

If its dual, the small dodecicosidodecahedron, has an edge length of 1, then the short edges of the darts will measure ${\displaystyle 2{\frac {\sqrt {65+19{\sqrt {5}}}}{11}}\approx 1.88500}$, and the long edges will be ${\displaystyle 2{\frac {\sqrt {2\left(5+2{\sqrt {5}}\right)}}{3}}\approx 2.90167}$. ​The dart faces will have length ${\displaystyle {\frac {\sqrt {10\left(157+31{\sqrt {5}}\right)}}{33}}\approx 1.44160}$, and width ${\displaystyle {\sqrt {5}}+1\approx 3.23607}$. ​The darts have two interior angles of ${\displaystyle \arccos \left({\frac {5+{\sqrt {5}}}{8}}\right)\approx 25.24283^{\circ }}$, one of ${\displaystyle \arccos \left({\frac {-5+9{\sqrt {5}}}{40}}\right)\approx 67.78301^{\circ }}$, and one of ${\displaystyle 360^{\circ }-\arccos \left(-{\frac {5+2{\sqrt {5}}}{20}}\right)\approx 241.73132^{\circ }}$.

## Vertex coordinates

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

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