# Tridiminished icosahedron

Tridiminished icosahedron
Rank3
TypeCRF
Notation
Bowers style acronymTeddi
Coxeter diagramxfo3oox&#xt
Stewart notationY5-3I5
J63
Elements
Faces1+1+3 triangles, 3 pentagons
Edges3+3+3+6
Vertices3+3+3
Vertex figures3 isosceles trapezoids, edge length 1, 1, 1, (1+5)/2
3+3 isosceles triangles, edge lengths 1, (1+5)/2, (1+5)/2
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {5+{\sqrt {5}}}{8}}}\approx 0.95106}$
Volume${\displaystyle {\frac {15+7{\sqrt {5}}}{24}}\approx 1.27719}$
Dihedral angles3-3: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }}$
3-5: ${\displaystyle \arccos \left(-{\sqrt {\frac {5-2{\sqrt {5}}}{15}}}\right)\approx 100.81232^{\circ }}$
5-5: ${\displaystyle \arccos \left({\frac {\sqrt {5}}{5}}\right)\approx 63.43495^{\circ }}$
Central density1
Number of external pieces8
Level of complexity10
Related polytopes
ArmyTeddi
RegimentTeddi
DualTri-tridiminished icosahedron
ConjugateTrireplenished great icosahedron
Abstract & topological properties
Flag count60
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryA2×I, order 6
ConvexYes
NatureTame

The tridiminished icosahedron is one of the 92 Johnson solids (J63). It consists of 1+1+3 triangles and 3 pentagons. It can be constructed by removing 3 mutually non-adjacent vertices from a regular icosahedron.

Its 9 vertices fall in three parallel planes in sets of 3. The outer planes contain the extreme triangles, while the plane between them intersects with the figure in another triangle with an edge length ${\displaystyle {\frac {1+{\sqrt {5}}}{2}}}$ times the edge length of the polyhedron. This observation led to a generalization known as the ursatopes, which have vertices falling in three hyperplanes of any dimension; some ursatopes are CRF as the tridiminished icosahedron is.

## Vertex coordinates

A tridiminished icosahedron of edge length 1 has the following vertices:

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

These are the vertices of an icosahedron, but with three missing.

An alternate set of coordinates can be given in a way that positions the tridiminished icosahedron within the symmetry axis:

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

## Related polyhedra

A tetrahedron can be attached to the tridiminished icosahedron at the triangular face surrounded by pentagons to form the augmented tridiminished icosahedron.

## In vertex figures

The tridiminished icosahedron is the vertex figure of the uniform snub disicositetrachoron.