Medial icosacronic hexecontahedron

Medial icosacronic hexecontahedron
(3D model) (OFF file)
Rank	3
Type	Uniform dual
Space	Spherical
Notation
Coxeter diagram	o5/3m3m5*a
Elements
Faces	60 darts
Edges	60+60
Vertices	12+12+20
Vertex figure	12 pentagons, 12 pentagrams, 20 hexagons
Measures (edge length 1)
Inradius	$\frac{3\sqrt7}{7} ≈ 1.13389$
Dihedral angle	$\arccos\left(-\frac57\right) ≈ 135.58469°$
Central density	4
Number of external pieces	180
Related polytopes
Dual	Icosidodecadodecahedron
Abstract & topological properties
Flag count	480
Euler characteristic	–16
Orientable	Yes
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

The medial icosacronic hexecontahedron is a uniform dual polyhedron. It consists of 60 darts.

If its dual, the icosidodecadodecahedron, has an edge length of 1, then the short edges of the darts will measure $\frac{7\sqrt6-\sqrt{30}}{11} ≈ 1.06084$, and the long edges will be $\frac{7\sqrt6+\sqrt{30}}{11} ≈ 2.05670$. ​The dart faces will have length $\frac{6\sqrt7}{11} ≈ 1.44314$, and width 2. ​The darts have two interior angles of $\arccos\left(\frac34\right) ≈ 41.40962°$, one of $\arccos\left(-\frac18+\frac{7\sqrt5}{24}\right) ≈ 58.18445°$, and one of $360°-\arccos\left(-\frac18-\frac{7\sqrt5}{24}\right) ≈ 218.99631°$.

Vertex coordinates

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

• $\left(±\frac{\sqrt5-1}{2},\,±\frac{1+\sqrt5}{2},\,0\right),$
• $\left(±3\frac{7+\sqrt5}{22},\,±3\frac{3\sqrt5-1}{22},\,0\right),$
• $\left(±3\frac{1+3\sqrt5}{22},\,±3\frac{7-\sqrt5}{22},\,0\right),$
• $\left(±1,\,±1,\,±1\right).$

External links

• Wikipedia Contributors. "Medial icosacronic hexecontahedron".
• McCooey, David. "Medial Icosacronic Hexecontahedron"