Triaugmented truncated dodecahedron

Triaugmented truncated dodecahedron
	(3D model); (OFF file)
Rank	3
Type	CRF
Space	Spherical
Notation
Bowers style acronym	Tautid
Elements
Faces	1+1+3+3+3+6+6+6+6 triangles, 3+6+6 squares, 3 pentagons, 3+3+3 decagons
Edges	9×3+18×6
Vertices	3+3+3+3+3+6+6+6+6+6+6+6+6+6+6
Vertex figures	15 isosceles trapezoids, edge length 1, √2, (1+√5)/2, √2
	30 irregular tetragons, edge length 1, √2, 1, √(5+√5)/2
	30 isosceles triangles, edge lengths 1, √2+√2, √2+√2
Measures (edge length 1)
Volume
Dihedral angles	3–4 join:
	3–4 cupolaic:
	3–10 join:
	4–5:
	3–10 tid:
	10–10:
Central density	1
Related polytopes
Army	Tautid
Regiment	Tautid
Dual	Trirhombirhombistellated triakis icosahedron
Conjugate	Triaugmented quasitruncated great stellated dodecahedron
Abstract & topological properties
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	A2×I, order 6
Convex	Yes
Nature	Tame

The triaugmented truncated dodecahedron is one of the 92 Johnson solids (J₇₁). It consists of 1+1+3+3+3+6+6+6+6 triangles, 3+6+6 squares, 3 pentagons, and 3+3+3 decagons. It can be constructed by attaching pentagonal cupolas to three mutually non-adjacent decagonal faces of the truncated dodecahedron.

Vertex coordinates[edit | edit source]

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

plus the following additional vertices:

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