Small dodecicosacron

Small dodecicosacron
(3D model) (OFF file)
Rank	3
Type	Uniform dual
Space	Spherical
Elements
Faces	60 bowties
Edges	60+60
Vertices	20+12
Vertex figure	20 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 density	even
Number of external pieces	120
Related polytopes
Dual	Small dodecicosahedron
Abstract & topological properties
Flag count	480
Euler characteristic	–28
Orientable	No
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

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:

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

External links

• Wikipedia Contributors. "Small dodecicosacron".
• McCooey, David. "Small Dodecicosacron"