Paragyrate diminished rhombicosidodecahedron

Paragyrate diminished rhombicosidodecahedron
(3D model) (OFF file)
Rank	3
Type	CRF
Space	Spherical
Notation
Bowers style acronym	Pagydrid
Elements
Faces	5+5+5 triangles, 5+5+5+10 squares, 1+5+5 pentagons, 1 decagon
Edges	5+5+5+5+5+5+5+10+10+10+10+10+10+10
Vertices	5+5+5+10+10+10+10
Vertex figures	5+30 isosceles trapezoids, edge length 1, √2, (1+√5)/2, √2
	10 scalene triangles, edge lengths √2, (1+√5)/2, √(5+√5)/2
	10 irregular tetragons, edge lengths 1, √2, √2, (1+√5)/2
Measures (edge length 1)
Circumradius	$\frac{\sqrt{11+4\sqrt5}}{2} ≈ 2.23295$
Volume	$\frac{115+54\sqrt5}{6} ≈ 39.29128$
Dihedral angles	3–4: $\arccos\left(-\frac{\sqrt3+\sqrt{15}}{6}\right) ≈ 159.09484°$
	3–5: $\arccos\left(-\sqrt{\frac{65-2\sqrt5}{75}}\right) ≈ 153.94242°$
	4–4: $\arccos\left(-\frac{2\sqrt5}{5}\right) ≈ 153.43495°$
	4–5: $\arccos\left(-\sqrt{\frac{5+\sqrt5}{10}}\right) ≈ 148.28253°$
	4–10: $\arccos\left(-\sqrt{\frac{5-\sqrt5}{10}}\right) ≈ 121.71747°$
	5–10: $\arccos\left(-\frac{\sqrt5}{5}\right) ≈ 116.56505°$
Central density	1
Related polytopes
Army	Pagydrid
Regiment	Pagydrid
Dual	Paradeltogyrate stellated deltoidal hexecontahedron
Conjugate	Paragyrate replenished quasirhombicosidodecahedron
Abstract & topological properties
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	H2×I, order 10
Convex	Yes
Nature	Tame

The paragyrate diminished rhombicosidodecahedron is one of the 92 Johnson solids (J₇₇). It consists of 5+5+5 triangles, 5+5+5+10 squares, 1+5+5 pentagons, and 1 decagon. It can be constructed by removing one of the pentagonal cupolaic caps of the small rhombicosidodecahedron, and rotating the opposite cap by 36°.

Vertex coordinates

A paragyrate diminished rhombicosidodecahedron of edge length 1 has vertices given by:

• $\left(±\frac{5+\sqrt5}{4},\,0,\,±\frac{3+\sqrt5}{4}\right),$
• $±\left(0,\,-\frac{3+\sqrt5}{4},\,\frac{5+\sqrt5}{4}\right),$
• $\left(±\frac{3+\sqrt5}{4},\,±\frac{5+\sqrt5}{4},\,0\right),$
• $\left(±\frac12,\,±\frac12,\,±\frac{2+\sqrt5}{2}\right),$
• $\left(±\frac{2+\sqrt5}{2},\,±\frac12,\,±\frac12\right),$
• $±\left(±\frac12,\,-\frac{2+\sqrt5}{2},\,\frac12\right),$
• $\left(±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,±\frac{1+\sqrt5}{2}\right),$
• $\left(±\frac{1+\sqrt5}{2},\,±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{4}\right),$
• $±\left(±\frac{1+\sqrt5}{4},\,-\frac{1+\sqrt5}{2},\,\frac{3+\sqrt5}{4}\right),$
• $\left(±\frac12,\,-\frac{5+4\sqrt5}{10},\,-\frac{10+3\sqrt5}{10}\right),$
• $\left(±\frac{1+\sqrt5}{4},\,-\frac{5+2\sqrt5}{5},\,-\frac{15+\sqrt5}{20}\right),$
• $\left(0,\,-\frac{15+13\sqrt5}{20},\,-\frac{5+\sqrt5}{20}\right).$

External links

• Klitzing, Richard. "pagydrid".

• Quickfur. "The Paragyrate Diminished Rhombicosidodecahedron".

• Wikipedia Contributors. "Paragyrate diminished rhombicosidodecahedron".
• McCooey, David. "Paragyrate Diminished Rhombicosidodecahedron"