# Great disdyakis dodecahedron

Great disdyakis dodecahedron
Rank3
TypeUniform dual
Notation
Coxeter diagramm4/3m3m
Elements
Faces48 scalene triangles
Edges24+24+24
Vertices12+8+6
Vertex figures12 squares
8 hexagons
6 octagrams
Measures (edge length 1)
Inradius${\displaystyle 3{\frac {\sqrt {194(15-8{\sqrt {2}})}}{97}}\approx 0.82708}$
Volume${\displaystyle 144{\frac {{\sqrt {2}}-1}{7}}\approx 8.52096}$
Dihedral angles48 edges: ${\displaystyle \arccos \left(-{\frac {71-12{\sqrt {2}}}{97}}\right)\approx 123.84889^{\circ }}$
24 edges: ${\displaystyle 180^{\circ }+\arccos \left({\frac {71-12{\sqrt {2}}}{97}}\right)\approx 236.15111^{\circ }}$
Central density–1
Number of external pieces48
Related polytopes
DualQuasitruncated cuboctahedron
Abstract & topological properties
Flag count288
Euler characteristic2
OrientableYes
Genus0
Properties
SymmetryB3, order 48
ConvexNo
NatureTame

The great disdyakis dodecahedron is a uniform dual polyhedron. It consists of 48 scalene triangles.

If its dual, the great cubicuboctahedron, has an edge length of 1, then the short edges of the triangles will measure ${\displaystyle 3{\frac {\sqrt {6\left(2-{\sqrt {2}}\right)}}{7}}\approx 0.80347}$, the medium edges will be ${\displaystyle 2{\frac {\sqrt {3\left(10+{\sqrt {2}}\right)}}{7}}\approx 1.67192}$, and the long edges will be ${\displaystyle 2{\frac {\sqrt {6\left(10-{\sqrt {2}}\right)}}{7}}\approx 2.05068}$. The triangles have one interior angle of ${\displaystyle \arccos \left({\frac {3}{4}}+{\frac {\sqrt {2}}{8}}\right)\approx 22.06219^{\circ }}$, one of ${\displaystyle \arccos \left(-{\frac {1}{6}}-{\frac {\sqrt {2}}{12}}\right)\approx 106.53003^{\circ }}$, and one of ${\displaystyle \arccos \left(-{\frac {1}{12}}+{\frac {\sqrt {2}}{2}}\right)\approx 51.40778^{\circ }}$.

## Vertex coordinates

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

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