Quasitruncated dodecadodecahedron

Quasitruncated dodecadodecahedron
(3D model) (OFF file)
Rank	3
Type	Uniform
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	${\frac {\sqrt {11}}{2}}\approx 1.65831$
Volume	15
Dihedral angles	10/3–4: $\arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }$
	10–10/3: $\arccos \left({\frac {\sqrt {5}}{5}}\right)\approx 63.43495^{\circ }$
	10–4: $\arccos \left({\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right)\approx 58.28253^{\circ }$
Central density	-3
Number of external pieces	402
Level of complexity	28
Related polytopes
Army	Semi-uniform Grid, edge lengths ${\frac {3-{\sqrt {5}}}{2}}$ (decagons), ${\frac {{\sqrt {5}}-1}{2}}$ (ditrigon-rectangle)
Regiment	Quitdid
Dual	Medial disdyakis triacontahedron
Conjugate	Quasitruncated dodecadodecahedron
Convex core	Dodecahedron
Abstract & topological properties
Flag count	720
Euler characteristic	-6
Orientable	Yes
Genus	4
Properties
Symmetry	H3, order 120
Flag orbits	6
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

• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {3}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5}}{2}},\,\pm {\frac {\sqrt {5}}{2}}\right),$

along with all even permutations of:

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

External links

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

• Klitzing, Richard. "quitdid".
• Wikipedia contributors. "Truncated dodecadodecahedron".
• McCooey, David. "Truncated Dodecadodecahedron"