# Great disdyakis triacontahedron

Great disdyakis triacontahedron
Rank3
TypeUniform dual
Notation
Coxeter diagramm5/3m3m
Elements
Faces120 scalene triangles
Edges60+60+60
Vertices12+20+30
Vertex figure30 squares, 20 hexagons, 12 decagrams
Measures (edge length 1)
Inradius${\displaystyle 3{\frac {\sqrt {1205\left(39-16{\sqrt {5}}\right)}}{241}}\approx 0.77575}$
Dihedral angle${\displaystyle \arccos \left(-{\frac {179-24{\sqrt {5}}}{241}}\right)\approx 121.33625^{\circ }}$
Central density13
Number of external pieces120
Related polytopes
DualGreat quasitruncated icosidodecahedron
ConjugateDisdyakis triacontahedron
Abstract & topological properties
Flag count720
Euler characteristic2
OrientableYes
Genus0
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

The great disdyakis triacontahedron is a uniform dual polyhedron. It consists of 120 scalene triangles.

If its dual, the great quasitruncated icosidodecahedron, has an edge length of 1, then the triangle faces' short edges will measure ${\displaystyle 3{\frac {\sqrt {15\left(65-19{\sqrt {5}}\right)}}{55}}\approx 1.00239}$, the medium edges will be ${\displaystyle 2{\frac {\sqrt {15\left(5+{\sqrt {5}}\right)}}{5}}\approx 4.16732}$, and the long edges will be ${\displaystyle {\frac {\sqrt {15\left(85+31{\sqrt {5}}\right)}}{11}}\approx 4.37382}$. The triangles have one interior angle of ${\displaystyle \arccos \left({\frac {5+2{\sqrt {5}}}{30}}\right)\approx 71.59464^{\circ }}$, one of ${\displaystyle \arccos \left({\frac {15+2{\sqrt {5}}}{20}}\right)\approx 13.19300^{\circ }}$, and one of ${\displaystyle \arccos \left({\frac {9-5{\sqrt {5}}}{24}}\right)\approx 95.21236^{\circ }}$.

## Vertex coordinates

A great disdyakis triacontahedron with dual edge length 1 has vertex coordinates given by all even permutations of:

• ${\displaystyle \left(\pm 3{\frac {4{\sqrt {5}}-5}{11}},\,0,\,0\right),}$
• ${\displaystyle \left(\pm {\frac {5-{\sqrt {5}}}{2}},\,\pm {\frac {5+{\sqrt {5}}}{2}},\,0\right),}$
• ${\displaystyle \left(\pm 3{\frac {5-{\sqrt {5}}}{10}},\,\pm 3{\frac {3{\sqrt {5}}-5}{10}},\,0\right),}$
• ${\displaystyle \left(\pm 3{\frac {25-9{\sqrt {5}}}{44}},\,\pm 3{\frac {4{\sqrt {5}}-5}{22}},\,\pm 3{\frac {15-{\sqrt {5}}}{44}}\right),}$
• ${\displaystyle \left(\pm {\sqrt {5}},\,\pm {\sqrt {5}},\,\pm {\sqrt {5}}\right).}$