Parabidiminished rhombicosidodecahedron

Parabidiminished rhombicosidodecahedron
(3D model) (OFF file)
Rank	3
Type	CRF
Notation
Bowers style acronym	Pabidrid
Elements
Faces	10 triangles, 10+10 squares, 10 pentagons, 2 decagons
Edges	10+10+10+20+20+20
Vertices	10+20+20
Vertex figures	30 isosceles trapezoids, edge length 1, √2, (1+√5)/2, √2
	20 scalene triangles, edge lengths √2, (1+√5)/2, √(5+√5)/2
Measures (edge length 1)
Circumradius	${\frac {\sqrt {11+4{\sqrt {5}}}}{2}}\approx 2.23295$
Volume	$5{\frac {11+5{\sqrt {5}}}{3}}\approx 36.96723$
Dihedral angles	3–4: $\arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }$
	4–5: $\arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }$
	4–10: $\arccos \left(-{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right)\approx 121.71747^{\circ }$
	5–10: $\arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }$
Central density	1
Number of external pieces	42
Level of complexity	18
Related polytopes
Army	Pabidrid
Regiment	Pabidrid
Dual	Parabistellated deltoidal hexecontahedron
Conjugate	Parabireplenished quasirhombicosidodecahedron
Abstract & topological properties
Flag count	360
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	(I2(10)×A1)/2, order 20
Convex	Yes
Nature	Tame

The parabidiminished rhombicosidodecahedron is one of the 92 Johnson solids (J₈₀). It consists of 10 triangles, 10+10 squares, 10 pentagons, and 2 decagons. It can be constructed by removing two opposite pentagonal cupolaic caps of the small rhombicosidodecahedron.

Vertex coordinates

A parabidiminished rhombicosidodecahedron of edge length 1 has vertices given by:

• $\left(\pm {\frac {5+{\sqrt {5}}}{4}},\,0,\,\pm {\frac {3+{\sqrt {5}}}{4}}\right)$,
• $\pm \left(0,\,-{\frac {3+{\sqrt {5}}}{4}},\,{\frac {5+{\sqrt {5}}}{4}}\right)$,
• $\left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,0\right)$,
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}}\right)$,
• $\left(\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right)$,
• $\pm \left(\pm {\frac {1}{2}},\,-{\frac {2+{\sqrt {5}}}{2}},\,{\frac {1}{2}}\right)$,
• $\left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}}\right)$,
• $\left(\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}}\right)$,
• $\pm \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,-{\frac {1+{\sqrt {5}}}{2}},\,{\frac {3+{\sqrt {5}}}{4}}\right)$.

External links

• Klitzing, Richard. "pabidrid".
• Quickfur. "The Parabidiminished Rhombicosidodecahedron".

• Wikipedia contributors. "Parabidiminished rhombicosidodecahedron".
• McCooey, David. "Parabidiminished Rhombicosidodecahedron"