Triaugmented dodecahedron

Triaugmented dodecahedron
(3D model) (OFF file)
Rank	3
Type	CRF
Notation
Bowers style acronym	Taud
Elements
Faces	3+6+6 triangles, 3+3+3 pentagons
Edges	3+3+3+3+3+6+6+6+6+6
Vertices	1+1+3+3+3+6+6
Vertex figures	3 pentagons, edge length 1
	15 kites, edge lengths 1 and (1+√5)/2
	5 triangles, edge lengths (1+√5)/2
Measures (edge length 1)
Volume	$5{\frac {7+3{\sqrt {5}}}{8}}\approx 8.56763$
Dihedral angles	3–5: $\arccos \left(-{\sqrt {\frac {65-2{\sqrt {5}}}{75}}}\right)\approx 153.94242^{\circ }$
	3–3: $\arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }$
	5–5: $\arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }$
Central density	1
Number of external pieces	24
Level of complexity	30
Related polytopes
Army	Taud
Regiment	Taud
Dual	Tritruncated icosahedron
Conjugate	Triaugmented great stellated dodecahedron
Abstract & topological properties
Flag count	180
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	A2×I, order 6
Convex	Yes
Nature	Tame

The triaugmented dodecahedron is one of the 92 Johnson solids (J₆₁). It consists of 3+6+6 triangles and 3+3+3 pentagons. It can be constructed by attaching pentagonal pyramids to three mutually non-adjacent faces of the regular dodecahedron.

Vertex coordinates

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

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

As well as:

• $\left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}}\right),$
• $\left(0,\,{\frac {15+{\sqrt {5}}}{20}},\,\pm {\frac {5+4{\sqrt {5}}}{10}}\right),$
• $\left({\frac {15+{\sqrt {5}}}{20}},\,-{\frac {5+4{\sqrt {5}}}{10}},\,0\right).$

External links

• Klitzing, Richard. "taud".
• Quickfur. "The Triaugmented Dodecahedron".

• Wikipedia contributors. "Triaugmented dodecahedron".
• McCooey, David. "Triaugmented Dodecahedron"