Quasitruncated dodecadodecahedron

Quasitruncated dodecadodecahedron
(3D model) (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Quitdid
Coxeter diagram	x5/3x5x ()
Elements
Faces	30 squares, 12 decagons, 12 decagrams
Edges	60+60+60
Vertices	120
Vertex figure	Scalene triangle, edge lengths √2, √(5+√5)/2, √(5–√5)/2
Measures (edge length 1)
Circumradius	$\frac{\sqrt{11}}{2} ≈ 1.65831$
Volume	15
Dihedral angles	10/3–4: $\arccos\left(-\sqrt{\frac{5+\sqrt5}{10}}\right) ≈ 148.28253°$
	10–10/3: $\arccos\left(\frac{\sqrt5}{5}\right) ≈ 63.43495°$
	10–4: $\arccos\left(\sqrt{\frac{5-\sqrt5}{10}}\right) ≈ 58.28253°$
Central density	-3
Number of pieces	402
Level of complexity	28
Related polytopes
Army	Semi-uniform Grid
Regiment	Quitdid
Dual	Medial disdyakis triacontahedron
Conjugate	Quasitruncated dodecadodecahedron
Convex core	Dodecahedron
Abstract properties
Euler characteristic	-6
Topological properties
Orientable	Yes
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

The quasitruncated dodecadodecahedron or quitdid, also called the truncated dodecadodecahedron, is a uniform polyhedron. It consists of 12 decagrams, 12 decagons, and 30 squares, with one of each type of face meeting per vertex. It can be obtained by quasicantitruncation of the small stellated dodecahedron or great dodecahedron, or equivalently by quasitruncating the vertices of a dodecadodecahedron and then adjusting the edge lengths to be all equal.

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

Vertex coordinates

A quasitruncated dodecadodecahedron of edge length 1 has vertex coordinates given by all permutations of

• $\left(±\frac12,\,±\frac12,\,±\frac32\right),$
• $\left(±\frac12,\,±\frac{\sqrt5}{2},\,±\frac{\sqrt5}{2}\right),$

along with all even permutations of:

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

Related polyhedra

o5/3o5o truncations

Name	OBSA	Schläfli symbol	CD diagram	Picture
Small stellated dodecahedron	sissid	{5/3,5}	x5/3o5o ()
Quasitruncated small stellated dodecahedron	quit sissid	t{5/3,5}	x5/3x5o ()
Dodecadodecahedron	did	r{5,5/3}	o5/3x5o ()
Truncated great dodecahedron	tigid	t{5,5/3}	o5/3x5x ()
Great dodecahedron	gad	{5,5/3}	o5/3o5x ()
Complex ditrigonal rhombidodecadodecahedron (degenerate, ditdid+rhom)	cadditradid	rr{5,5/3}	x5/3o5x ()
Quasitruncated dodecadodecahedron	quitdid	tr{5,5/3}	x5/3x5x ()
Inverted snub dodecadodecahedron	isdid	sr{5,5/3}	s5/3s5s ()

External links

• Bowers, Jonathan. "Polyhedron Category 5: Omnitruncates" (#60).

• Klitzing, Richard. "quitdid".

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