# Small dodecicosacron

Small dodecicosacron Rank3
TypeUniform dual
SpaceSpherical
Elements
Faces60 bowties
Edges60+60
Vertices20+12
Vertex figure20 hexagons, 12 decagons
Measures (edge length 1)
Inradius$3\frac{\sqrt{305\left(9+2\sqrt5\right)}}{122} ≈ 1.57627$ Dihedral angle$\arccos\left(-\frac{44+3\sqrt5}{61}\right) ≈ 146.23066°$ Central densityeven
Number of external pieces120
Related polytopes
DualSmall dodecicosahedron
Abstract & topological properties
Flag count480
Euler characteristic–28
OrientableNo
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

The small dodecicosacron is a uniform dual polyhedron. It consists of 60 bowties.

It appears the same as the small ditrigonal dodecacronic hexecontahedron.

If its dual, the small dodecicosahedron, has an edge length of 1, then the short edges of the bowties will measure $\frac{\sqrt{6\left(5+\sqrt5\right)}}{2} ≈ 3.29456$ , and the long edges will be $\sqrt{3\left(5+2\sqrt5\right)} ≈ 5.33070$ . The bowties have two interior angles of $\arccos\left(\frac{5}{12}+\frac{\sqrt5}{4}\right) ≈ 12.66108°$ , and two of $\arccos\left(-\frac34+\frac{\sqrt5}{20}\right) ≈ 129.65748°$ . The intersection has an angle of $\arccos\left(\frac{1}{12}+\frac{19\sqrt5}{60}\right) ≈ 37.68145°$ .

## Vertex coordinates

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

• $\left(±3\frac{3+\sqrt5}{4},\,±3\frac{1+\sqrt5}{4},\,0\right),$ • $\left(±\frac{5-\sqrt5}{4},\,±\frac{5+\sqrt5}{4},\,0\right),$ • $\left(±\frac{\sqrt5}{2},\,±\frac{\sqrt5}{2},\,±\frac{\sqrt5}{2}\right).$ 