Great stellapentakis dodecahedron

Great stellapentakis dodecahedron
(3D model) (OFF file)
Rank	3
Type	Uniform dual
Space	Spherical
Notation
Coxeter diagram	o5/2m3m ()
Elements
Faces	60 isosceles triangles
Edges	30+60
Vertices	12+20
Vertex figure	12 pentagrams, 20 hexagons
Measures (edge length 1)
Inradius	$9\frac{\sqrt{109\left(17−6\sqrt5\right)}}{218} \approx 0.81594$
Dihedral angle	$\arccos\left(-\frac{80-9\sqrt5}{109}\right) \approx 123.32007°$
Central density	7
Number of external pieces	120
Related polytopes
Dual	Truncated great icosahedron
Abstract & topological properties
Flag count	360
Euler characteristic	2
Orientable	Yes
Genus	0
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

The great stellapentakis dodecahedron is a uniform dual polyhedron. It consists of 60 isosceles triangles.

If its dual, the truncated great icosahedron, has an edge length of 1, then the short edges of the triangles will measure $9\frac{1+2\sqrt5}{19} ≈ 2.59206$, and the long edges will be $3\frac{1+\sqrt5}{2} ≈ 4.85410$. The triangles have two interior angles of $\arccos\left(\frac34+\frac{\sqrt5}{12}\right) ≈ 20.55444°$, and one of $\arccos\left(-\frac{7}{36}-\frac{\sqrt5}{4}\right) ≈ 138.89111°$.

Vertex coordinates

A great stellapentakis dodecahedron with dual edge length 1 has vertex coordinates given by all even permutations of:

• $\left(±3\frac{\sqrt5-1}{4},\,±3\frac{1+\sqrt5}{4},\,0\right),$
• $\left(±9\frac{9-\sqrt5}{76},\,±9\frac{5\sqrt5-7}{76},\,0\right),$
• $\left(±\frac32,\,±\frac32,\,±\frac32\right).$

External links

• Wikipedia Contributors. "Great stellapentakis dodecahedron".
• McCooey, David. "Great Stellapentakis Dodecahedron"