Medial icosacronic hexecontahedron

Medial icosacronic hexecontahedron
(3D model) (OFF file)
Rank	3
Type	Uniform dual
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	${\frac {3{\sqrt {7}}}{7}}\approx 1.13389$
Dihedral angle	$\arccos \left(-{\frac {5}{7}}\right)\approx 135.58469^{\circ }$
Central density	4
Number of external pieces	180
Related polytopes
Dual	Icosidodecadodecahedron
Conjugate	Medial icosacronic hexecontahedron
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 ${\frac {7{\sqrt {6}}-{\sqrt {30}}}{11}}\approx 1.06084$, and the long edges will be ${\frac {7{\sqrt {6}}+{\sqrt {30}}}{11}}\approx 2.05670$. ​The dart faces will have length ${\frac {6{\sqrt {7}}}{11}}\approx 1.44314$, and width 2. ​The darts have two interior angles of $\arccos \left({\frac {3}{4}}\right)\approx 41.40962^{\circ }$, one of $\arccos \left({\frac {-3+7{\sqrt {5}}}{24}}\right)\approx 58.18445^{\circ }$, and one of $360^{\circ }-\arccos \left(-{\frac {3+7{\sqrt {5}}}{24}}\right)\approx 218.99631^{\circ }$.

Vertex coordinates

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

• $\left(\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,0\right),$
• $\left(\pm 3{\frac {7+{\sqrt {5}}}{22}},\,\pm 3{\frac {3{\sqrt {5}}-1}{22}},\,0\right),$
• $\left(\pm 3{\frac {1+3{\sqrt {5}}}{22}},\,\pm 3{\frac {7-{\sqrt {5}}}{22}},\,0\right),$
• $\left(\pm 1,\,\pm 1,\,\pm 1\right).$

External links

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