Icosidodecatruncated icosidodecahedron

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.

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

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"