# Augmented tridiminished icosahedron

Augmented tridiminished icosahedron
Rank3
TypeCRF
Notation
Bowers style acronymAuteddi
Coxeter diagramoxfo3ooox&#xt
Elements
Faces1+3+3 triangles, 3 pentagons
Edges3+3+3+3+6
Vertices1+3+3+3
Vertex figures1 triangle, edge length 1
3 isosceles trapezoids, edge length 1, 1, 1, (1+5)/2
3 isosceles triangles, edge lengths 1, (1+5)/2, (1+5)/2
3 kites, edge lengths 1 and (1+5)/2
Measures (edge length 1)
Volume${\displaystyle {\frac {15+2{\sqrt {2}}+7{\sqrt {5}}}{24}}\approx 1.39504}$
Dihedral angles3–5 join: ${\displaystyle \arccos \left(-{\sqrt {\frac {5-2{\sqrt {5}}}{15}}}\right)+\arccos \left({\frac {1}{3}}\right)\approx 171.34110^{\circ }}$
3–3 teddi: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }}$
3–5 teddi: ${\displaystyle \arccos \left(-{\sqrt {\frac {5-2{\sqrt {5}}}{15}}}\right)\approx 100.81232^{\circ }}$
3–3 pyramidal: ${\displaystyle \arccos \left({\frac {1}{3}}\right)\approx 70.52878^{\circ }}$
5–5: ${\displaystyle \arccos \left({\frac {\sqrt {5}}{5}}\right)\approx 63.43495^{\circ }}$
Central density1
Number of external pieces10
Level of complexity12
Related polytopes
ArmyAuteddi
RegimentAuteddi
DualBase-truncated tristellated dodecahedron
ConjugatesAugmented trireplenished great icosahedron, Excavated trireplenished great icosahedron
Abstract & topological properties
Flag count72
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryA2×I, order 6
ConvexYes
NatureTame

The augmented tridiminished icosahedron is one of the 92 Johnson solids (J64). It consists of 1+3+3 triangles and 3 pentagons. It can be constructed by attaching a tetrahedron, seen as a triangular pyramid, to the triangular face of the tridiminished icosahedron that is connected only to pentagons.

It is the only Johnson solid that is constructed using both diminishing and augmenting, assuming that no diminishing and augmenting cancel each other out.

## Vertex coordinates

An augmented tridiminished icosahedron of edge length 1 has the following vertices:

• ${\displaystyle \left(0,\,0,\,{\frac {{\sqrt {3}}+2{\sqrt {6}}+{\sqrt {15}}}{6}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right),}$
• ${\displaystyle \left(0,\,{\frac {\sqrt {3}}{3}},\,{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,-{\frac {{\sqrt {3}}+{\sqrt {15}}}{12}},\,0\right),}$
• ${\displaystyle \left(0,\,{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}},\,0\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,{\frac {\sqrt {3}}{6}},\,-{\frac {\sqrt {3}}{3}}\right),}$
• ${\displaystyle \left(0,\,-{\frac {\sqrt {3}}{3}},\,-{\frac {\sqrt {3}}{3}}\right).}$

Alternatively, orienting it so that it is derived from the vertices of a regular icosahedron, we obtain:

• ${\displaystyle \left(0,\,{\frac {1}{2}},\,{\frac {1+{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(0,\,\pm {\frac {1}{2}},\,-{\frac {1+{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left({\frac {1}{2}},\,{\frac {1+{\sqrt {5}}}{4}},\,0\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\frac {1+{\sqrt {5}}}{4}},\,0\right),}$
• ${\displaystyle \left({\frac {1+{\sqrt {5}}}{4}},\,0,\,{\frac {1}{2}}\right),}$
• ${\displaystyle \left(-{\frac {1+{\sqrt {5}}}{4}},\,0,\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left({\frac {3+4{\sqrt {2}}+{\sqrt {5}}}{12}},\,{\frac {3+4{\sqrt {2}}+{\sqrt {5}}}{12}},\,{\frac {3+4{\sqrt {2}}+{\sqrt {5}}}{12}}\right).}$