# Disdyakis triacontahedron

Disdyakis triacontahedron
Rank3
TypeUniform dual
Notation
Bowers style acronymSiddykit
Coxeter diagramm5m3m ()
Conway notationmD
Elements
Faces120 scalene triangles
Edges60+60+60
Vertices12+20+30
Vertex figure12 decagons, 20 hexagons, 30 squares
Measures (edge length 1)
Dihedral angle${\displaystyle \arccos \left(-{\frac {179+24{\sqrt {5}}}{241}}\right)\approx 164.88789^{\circ }}$
Central density1
Number of external pieces120
Level of complexity6
Related polytopes
ArmySiddykit
RegimentSiddykit
DualGreat rhombicosidodecahedron
ConjugateGreat disdyakis triacontahedron
Abstract & topological properties
Flag count720
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryH3, order 120
ConvexYes
NatureTame

The disdyakis triacontahedron, also called the small disdyakis triacontahedron, is one of the 13 Catalan solids. It has 120 scalene triangles as faces, with 12 order-10, 20 order-6, and 30 order-4 vertices. It is the dual of the uniform great rhombicosidodecahedron.

It can also be obtained as the convex hull of a dodecahedron, an icosahedron, and an icosidodecahedron. If the dodecahedron has unit edge length, the icosahedron's edge length is ${\displaystyle 3{\frac {3+{\sqrt {5}}}{10}}\approx 1.57082}$ and the icosidodecahedron's edge length is ${\displaystyle 3{\frac {4+{\sqrt {5}}}{22}}\approx 0.85037}$.

Each face of this polyhedron is a scalene triangle. If the shortest edges have unit edge length, the medium edges have length ${\displaystyle 3{\frac {3+{\sqrt {5}}}{10}}\approx 1.57082}$ and the longest edges have length ${\displaystyle {\frac {7+{\sqrt {5}}}{5}}\approx 1.84721}$. These triangles have angles measuring ${\displaystyle \arccos \left({\frac {5-2{\sqrt {5}}}{30}}\right)\approx 88.99180^{\circ }}$, ${\displaystyle \arccos \left({\frac {15-2{\sqrt {5}}}{20}}\right)\approx 58.23792^{\circ }}$, and ${\displaystyle \arccos \left({\frac {9+5{\sqrt {5}}}{24}}\right)\approx 32.77028^{\circ }}$.