# Disdyakis dodecahedron

Disdyakis dodecahedron
Rank3
TypeUniform dual
Notation
Bowers style acronymSiddykid
Coxeter diagramm4m3m ()
Conway notationmC
Elements
Faces48 scalene triangles
Edges24+24+24
Vertices6+8+12
Vertex figure6 octagons, 8 hexagons, 12 squares
Measures
Dihedral angle${\displaystyle \arccos \left(-{\frac {71+12{\sqrt {2}}}{97}}\right)\approx 155.08218^{\circ }}$
Central density1
Number of external pieces48
Level of complexity6
Related polytopes
ArmySiddykid
RegimentSiddykid
DualGreat rhombicuboctahedron
ConjugateGreat disdyakis dodecahedron
Abstract & topological properties
Flag count288
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryB3, order 48
ConvexYes
NatureTame

The disdyakis dodecahedron, also called the small disdyakis dodecahedron, is one of the 13 Catalan solids. It has 48 scalene triangles as faces, with 6 order-8, 8 order-6, and 12 order-4 vertices. It is the dual of the uniform great rhombicuboctahedron.

It can also be obtained as the convex hull of a cube, an octahedron, and a cuboctahedron. If the cube has unit edge length, the octahedron's edge length is ${\displaystyle 3{\frac {2+3{\sqrt {2}}}{14}}\approx 1.33771}$ and the cuboctahedron's edge length is ${\displaystyle 3{\frac {1+2{\sqrt {2}}}{14}}\approx 0.82038}$.

Each face of this polyhedron is a scalene triangle. If the shortest edges have unit edge length, the medium edges have length ${\displaystyle 3{\frac {2+3{\sqrt {2}}}{14}}\approx 1.33771}$ and the longest edges have length ${\displaystyle {\frac {10+{\sqrt {2}}}{7}}\approx 1.63060}$. These triangles have angles measuring ${\displaystyle \arccos \left({\frac {2-{\sqrt {2}}}{12}}\right)\approx 87.20196^{\circ }}$, ${\displaystyle \arccos \left({\frac {6-{\sqrt {2}}}{8}}\right)\approx 55.02470^{\circ }}$, and ${\displaystyle \arccos \left({\frac {1+6{\sqrt {2}}}{12}}\right)\approx 37.77334^{\circ }}$.

## Vertex coordinates

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

• ${\displaystyle \left(\pm 3{\frac {2+3{\sqrt {2}}}{7}},\,0,\,0\right),}$
• ${\displaystyle \left(\pm 3{\frac {1+2{\sqrt {2}}}{7}},\,\pm 3{\frac {1+2{\sqrt {2}}}{7}},\,0\right),}$
• ${\displaystyle \left(\pm {\sqrt {2}},\,\pm {\sqrt {2}},\,\pm {\sqrt {2}}\right).}$