Parabiaugmented truncated dodecahedron

Parabiaugmented truncated dodecahedron
(3D model) (OFF file)
Rank	3
Type	CRF
Notation
Bowers style acronym	Pabautid
Elements
Faces	10+10+10 triangles, 10 squares, 2 pentagons, 10 decagons
Edges	10+10+10+10+10+10+20+20+20
Vertices	10+10+10+20+20
Vertex figures	10 isosceles trapezoids, edge length 1, √2, (1+√5)/2, √2
	20 irregular tetragons, edge length 1, √2, 1, √(5+√5)/2
	40 isosceles triangles, edge lengths 1, √2+√2, √2+√2
Measures (edge length 1)
Volume	${\frac {515+251{\sqrt {5}}}{12}}\approx 89.68776$
Dihedral angles	3–4 join: $\arccos \left(-{\sqrt {\frac {23+3{\sqrt {5}}}{30}}}\right)\approx 174.34011^{\circ }$
	3–4 cupolaic: $\arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }$
	3–10 join: $\arccos \left(-{\sqrt {\frac {65-2{\sqrt {5}}}{75}}}\right)\approx 153.94242^{\circ }$
	4–5: $\arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }$
	3–10 tid: $\arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }$
	10–10: $\arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }$
Central density	1
Number of external pieces	52
Level of complexity	24
Related polytopes
Army	Pabautid
Regiment	Pabautid
Dual	Parabirhombirhombistellated triakis icosahedron
Conjugate	Parabiaugmented quasitruncated great stellated dodecahedron
Abstract & topological properties
Flag count	480
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	(I2(10)×A1)/2, order 20
Convex	Yes
Nature	Tame

The parabiaugmented truncated dodecahedron is one of the 92 Johnson solids (J₆₉). It consists of 10+10+10 triangles, 10 squares, 2 pentagons, and 10 decagons. It can be constructed by attaching two pentagonal cupolas to two opposite decagonal faces of the truncated dodecahedron.

Vertex coordinates

A parabiaugmented truncated dodecahedron of edge length 1 has vertices given by all even permutations of:

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

plus the following additional vertices:

• $\pm \left(\pm {\frac {1}{2}},\,{\frac {15+13{\sqrt {5}}}{20}},\,3{\frac {5+{\sqrt {5}}}{10}}\right),$
• $\pm \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\frac {25+13{\sqrt {5}}}{20}},\,{\frac {25+{\sqrt {5}}}{20}}\right),$
• $\pm \left(0,\,{\frac {10+9{\sqrt {5}}}{10}},\,{\frac {15+{\sqrt {5}}}{20}}\right).$

External links

• Klitzing, Richard. "pabautid".
• Quickfur. "The Parabiaugmented Truncated Dodecahedron".

• Wikipedia contributors. "Parabiaugmented truncated dodecahedron".
• McCooey, David. "Parabiaugmented Truncated Dodecahedron"