Small hexagrammic hexecontahedron

Small hexagrammic hexecontahedron
(3D model) (OFF file)
Rank	3
Type	Uniform dual
Space	Spherical
Notation
Coxeter diagram	p5/2p3/2p3/2*a
Elements
Faces	60 mirror-symmetric unicursal hexagrams
Edges	60+60+60
Vertices	60+20+12
Vertex figure	60 triangles, 20 hexagrams, 12 pentagrams
Measures (edge length 1)
Inradius	${\frac {\sqrt {58\left(103+41{\sqrt {5}}-2{\sqrt {2\left(2239+1005{\sqrt {5}}\right)}}\right)}}{116}}\approx 0.15018$
Dihedral angle	≈ 61.13345°
Central density	38
Number of external pieces	420
Related polytopes
Dual	Small inverted retrosnub icosicosidodecahedron
Conjugate	Small hexagonal hexecontahedron
Convex core	Non-Catalan triakis icosahedron
Abstract & topological properties
Flag count	720
Euler characteristic	–8
Orientable	Yes
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

The small hexagrammic hexecontahedron is a uniform dual polyhedron. It consists of 60 mirror-symmetric unicursal hexagrams.

If its dual, the small inverted retrosnub icosicosidodecahedron, has an edge length of 1, then the two short edges of each hexagram will measure ${\frac {\sqrt {13+7{\sqrt {5}}+{\sqrt {22\left(9+5{\sqrt {5}}\right)}}}}{6}}\approx 1.17524$ , and the four long edges will be ${\frac {\sqrt {9+3{\sqrt {5}}+{\sqrt {2\left(51+23{\sqrt {5}}\right)}}}}{2}}\approx 2.73958$ . The unicursal hexagrams have five interior angles of $\arccos \left(\xi \right)\approx 21.03199^{\circ }$ , and one of $360^{\circ }-\arccos \left(\phi ^{-2}\xi -\phi ^{-1}\right)\approx 254.84055^{\circ }$ , where $\xi ={\frac {1+{\sqrt {1+4\phi }}}{4}}\approx 0.93338$ , and $\phi$ is the golden ratio.

A dihedral angle is equal to $\arccos \left({\frac {\xi }{1+\xi }}\right)\approx 61.13345^{\circ }$ .

External links

• Wikipedia contributors. "Small hexagrammic hexecontahedron".
• McCooey, David. "Small Hexagrammic Hexecontahedron"