Icosidodecatruncated icosidodecahedron

Icosidodecatruncated icosidodecahedron
(3D model) (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Idtid
Coxeter diagram	x5/3x3x5*a ()
Elements
Faces	20 hexagons, 12 decagons, 12 decagrams
Edges	60+60+60
Vertices	120
Vertex figure	Scalene triangle, edge lengths √3, √(5+√5)/2, √(5–√5)/2
Measures (edge length 1)
Circumradius	2
Volume	80
Dihedral angles	10/3–6: $\arccos\left(-\sqrt{\frac{5+2\sqrt5}{15}}\right) ≈ 142.62263^\circ$
	10–10/3: $\arccos\left(-\frac{\sqrt5}{5}\right) ≈ 116.56505^\circ$
	10–6: $\arccos\left(-\sqrt{\frac{5-2\sqrt5}{15}}\right) ≈ 100.81232^\circ$
Central density	4
Number of external pieces	152
Level of complexity	15
Related polytopes
Army	Semi-uniform Grid, edge lengths $\frac{3-\sqrt5}{2}$ (decagons), 1 (ditrigon-rectangle)
Regiment	Idtid
Dual	Tridyakis icosahedron
Conjugate	Icosidodecatruncated icosidodecahedron
Convex core	Dodecahedron
Abstract & topological properties
Flag count	720
Euler characteristic	-16
Orientable	Yes
Genus	9
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

The icosidodecatruncated icosidodecahedron or idtid, also called the icositruncated dodecadodecahedron, is a uniform polyhedron. It consists of 12 decagrams, 12 decagons, and 20 hexagons, with one of each type of face meeting per vertex.

It can be alternated into the snub icosidodecadodecahedron after equalizing edge lengths.

Vertex coordinates

An icosidodecatruncated icosidodecahedron of edge length 1 has vertex coordinates given by all even permutations of:

• $\left(±\frac12,\,±\frac{3-\sqrt5}{4},\,±\frac{1+3\sqrt5}{4}\right),$
• $\left(±\frac12,\,±\frac{5+\sqrt5}{4},\,±\frac{5-\sqrt5}{4}\right),$
• $\left(±1,\,±\frac{\sqrt5-1}{2},\,±\frac{1+\sqrt5}{2}\right),$
• $\left(±\frac32,\,±\frac{3-\sqrt5}{4},\,±\frac{3+\sqrt5}{4}\right),$
• $\left(±\frac12,\,±\frac{3\sqrt5-1}{4},\,±\frac{3+\sqrt5}{4}\right).$

External links

• Bowers, Jonathan. "Polyhedron Category 5: Omnitruncates" (#63).

• Klitzing, Richard. "idtid".

• Wikipedia Contributors. "Icositruncated dodecadodecahedron".
• McCooey, David. "Icositruncated Dodecadodecahedron"