Tridiminished icosahedron

Tridiminished icosahedron
(3D model) (OFF file)
Rank	3
Type	CRF
Notation
Bowers style acronym	Teddi
Coxeter diagram	xfo3oox&#xt
Stewart notation	Y5-3I5 J63
Elements
Faces	1+1+3 triangles, 3 pentagons
Edges	3+3+3+6
Vertices	3+3+3
Vertex figures	3 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	${\sqrt {\frac {5+{\sqrt {5}}}{8}}}\approx 0.95106$
Volume	${\frac {15+7{\sqrt {5}}}{24}}\approx 1.27719$
Dihedral angles	3-3: $\arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }$
	3-5: $\arccos \left(-{\sqrt {\frac {5-2{\sqrt {5}}}{15}}}\right)\approx 100.81232^{\circ }$
	5-5: $\arccos \left({\frac {\sqrt {5}}{5}}\right)\approx 63.43495^{\circ }$
Central density	1
Number of external pieces	8
Level of complexity	10
Related polytopes
Army	Teddi
Regiment	Teddi
Dual	Tri-tridiminished icosahedron
Conjugate	Trireplenished great icosahedron
Abstract & topological properties
Flag count	60
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	A2×I, order 6
Convex	Yes
Nature	Tame

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 ${\frac {1+{\sqrt {5}}}{2}}$ 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:

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

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:

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

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.

External links

• Bowers, Jonathan. "Batch 2: Ike and Sissid Facetings" (#11 under ike).

• Klitzing, Richard. "teddi".
• Quickfur. "The Tridiminished Icosahedron".

• Wikipedia contributors. "Tridiminished icosahedron".