# Quasitruncated dodecadodecahedron

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Quasitruncated dodecadodecahedron
Rank3
TypeUniform
Notation
Bowers style acronymQuitdid
Coxeter diagramx5/3x5x ()
Elements
Faces30 squares, 12 decagons, 12 decagrams
Edges60+60+60
Vertices120
Vertex figureScalene triangle, edge lengths 2, (5+5)/2, (5–5)/2
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {11}}{2}}\approx 1.65831}$
Volume15
Dihedral angles10/3–4: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }}$
10–10/3: ${\displaystyle \arccos \left({\frac {\sqrt {5}}{5}}\right)\approx 63.43495^{\circ }}$
10–4: ${\displaystyle \arccos \left({\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right)\approx 58.28253^{\circ }}$
Central density-3
Number of external pieces402
Level of complexity28
Related polytopes
ArmySemi-uniform Grid, edge lengths ${\displaystyle {\frac {3-{\sqrt {5}}}{2}}}$ (decagons), ${\displaystyle {\frac {{\sqrt {5}}-1}{2}}}$ (ditrigon-rectangle)
RegimentQuitdid
DualMedial disdyakis triacontahedron
ConjugateQuasitruncated dodecadodecahedron
Convex coreDodecahedron
Abstract & topological properties
Flag count720
Euler characteristic-6
OrientableYes
Genus4
Properties
SymmetryH3, order 120
Flag orbits6
ConvexNo
NatureTame

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

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

along with all even permutations of:

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