Great pentakis dodecahedron

Great pentakis dodecahedron
(3D model) (OFF file)
Rank	3
Type	Uniform dual
Notation
Coxeter diagram	m5/3m5o ()
Elements
Faces	60 isosceles triangles
Edges	30+60
Vertices	12+12
Vertex figure	12 pentagons, 12 decagrams
Measures (edge length 1)
Inradius	$5{\frac {\sqrt {41\left(11-4{\sqrt {5}}\right)}}{82}}\approx 0.55980$
Dihedral angle	$\arccos \left({\frac {5{\sqrt {5}}-24}{41}}\right)\approx 108.22049^{\circ }$
Central density	9
Number of external pieces	60
Related polytopes
Dual	Quasitruncated small stellated dodecahedron
Conjugate	Small stellapentakis dodecahedron
Convex core	Deltoidal hexecontahedron
Abstract & topological properties
Flag count	360
Euler characteristic	–6
Orientable	Yes
Genus	4
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

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

If its dual, the quasitruncated small stellated dodecahedron, has an edge length of 1, then the base edges of the triangles will measure ${\frac {3{\sqrt {5}}-5}{2}}\approx 0.85410$, and the lateral edges will be $5{\frac {1+{\sqrt {5}}}{2}}\approx 8.09017$. The triangles have two interior angles of $\arccos \left({\frac {5-2{\sqrt {5}}}{10}}\right)\approx 86.97416^{\circ }$, and one of $\arccos \left({\frac {1+4{\sqrt {5}}}{10}}\right)\approx 6.05169^{\circ }$.

Vertex coordinates

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

• $\left(\pm {\frac {5-{\sqrt {5}}}{4}},\,\pm {\frac {3{\sqrt {5}}-5}{4}},\,0\right),$
• $\left(\pm 5{\frac {3+{\sqrt {5}}}{4}},\,\pm 5{\frac {1+{\sqrt {5}}}{4}},\,0\right).$

External links

• Wikipedia contributors. "Great pentakis dodecahedron".
• McCooey, David. "Great Pentakis Dodecahedron"