# Quasitruncated small stellated dodecahedron

Quasitruncated small stellated dodecahedron
Rank3
TypeUniform
Notation
Bowers style acronymQuit sissid
Coxeter diagramx5/3x5o ()
Elements
Faces12 pentagons, 12 decagrams
Edges30+60
Vertices60
Vertex figureIsosceles triangle, edge lengths (1+5)/2, (5–5)/2, (5–5)/2
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {17-5{\sqrt {5}}}{8}}}\approx 0.85291}$
Volume${\displaystyle 11{\frac {3{\sqrt {5}}-5}{4}}\approx 4.69756}$
Dihedral angles10/3–10/3: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}$
5–10/3: ${\displaystyle \arccos \left({\frac {\sqrt {5}}{5}}\right)\approx 63.43495^{\circ }}$
Central density9
Number of external pieces132
Level of complexity11
Related polytopes
ArmySrid, edge length ${\displaystyle {\frac {3-{\sqrt {5}}}{2}}}$
RegimentQuit sissid
DualGreat pentakis dodecahedron
ConjugateTruncated great dodecahedron
Convex coreDodecahedron
Abstract & topological properties
Flag count360
Euler characteristic–6
OrientableYes
Properties
SymmetryH3, order 120
Flag orbits3
ConvexNo
NatureTame

The quasitruncated small stellated dodecahedron, or quit sissid, also called the small stellated truncated dodecahedron, is a uniform polyhedron. It consists of 12 pentagons and 12 decagrams. Each vertex joins one pentagon and two decagrams. As the name suggests, it can be obtained by the quasitruncation of the small stellated dodecahedron.

## Vertex coordinates

A quasitruncated small stellated dodecahedron of edge length 1 has vertex coordinates given by all permutations of:

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

together with all even permutations of:

• ${\displaystyle \left(0,\,\pm {\frac {1}{2}},\,\pm {\frac {5-{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {{\sqrt {5}}-1}{2}}\right).}$