Parabiaugmented truncated dodecahedron

Parabiaugmented truncated dodecahedron
(3D model) (OFF file)
Rank	3
Type	CRF
Space	Spherical
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	$\frac{515+251\sqrt5}{12} ≈ 89.68776$
Dihedral angles	3–4 join: $\arccos\left(-\sqrt{\frac{23+3\sqrt5}{30}}\right) ≈ 174.34011^\circ$
	3–4 cupolaic: $\arccos\left(-\frac{\sqrt3+\sqrt{15}}{6}\right) ≈ 159.09484^\circ$
	3–10 join: $\arccos\left(-\sqrt{\frac{65-2\sqrt5}{75}}\right) ≈ 153.94242^\circ$
	4–5: $\arccos\left(-\sqrt{\frac{5+\sqrt5}{10}}\right) ≈ 148.28253^\circ$
	3–10 tid: $\arccos\left(-\sqrt{\frac{5+2\sqrt5}{15}}\right) ≈ 142.62263^\circ$
	10–10: $\arccos\left(-\frac{\sqrt5}{5}\right) ≈ 116.56505^\circ$
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:

• $\left(0,\,±\frac12,\,±\frac{5+3\sqrt5}{4}\right),$
• $\left(±\frac12,\,±\frac{3+\sqrt5}{4},\,±\frac{3+\sqrt5}{2}\right),$
• $\left(±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{2},\,±\frac{2+\sqrt5}{2}\right),$

plus the following additional vertices:

• $±\left(±\frac12,\,\frac{15+13\sqrt5}{20},\,3\frac{5+\sqrt5}{10}\right),$
• $±\left(±\frac{1+\sqrt5}{4},\,\frac{25+13\sqrt5}{20},\,\frac{25+\sqrt5}{20}\right),$
• $±\left(0,\,\frac{10+9\sqrt5}{10},\,\frac{15+\sqrt5}{20}\right).$

External links

• Klitzing, Richard. "pabautid".

• Quickfur. "The Parabiaugmented Truncated Dodecahedron".

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