# Compound of twenty octahedra

(Redirected from Disnub icosahedron)
Compound of twenty octahedra
Rank3
TypeUniform
Notation
Bowers style acronymDasi
Elements
Components20 octahedra
Faces40+120 triangles
Edges120+120
Vertices60
Vertex figureStellated octagon, edge length 1
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$
Inradius${\displaystyle {\frac {\sqrt {6}}{6}}\approx 0.40825}$
Volume${\displaystyle {\frac {20{\sqrt {2}}}{3}}\approx 9.42809}$
Dihedral angle${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
Central density20
Number of external pieces900
Level of complexity46
Related polytopes
ArmySemi-uniform srid, edge lengths ${\displaystyle {\sqrt {\frac {3-{\sqrt {5}}-{\sqrt {10{\sqrt {5}}-22}}}{2}}}}$ (pentagons), ${\displaystyle {\sqrt {\frac {{\sqrt {5}}-1-2{\sqrt {{\sqrt {5}}-2}}}{2}}}}$ (triangles)
RegimentGidrid
DualCompound of twenty cubes
ConjugateCompound of twenty octahedra
Convex coreIcosatruncated disdyakis triacontahedron
Abstract & topological properties
Flag count960
Schläfli type{3,4}
OrientableYes
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

The disnub icosahedron, dasi, or compound of twenty octahedra is a uniform polyhedron compound. It consists of 40+120 triangles. The vertices coincide in pairs, and thus eight triangles join at each vertex.

This compound is a special case of the more general altered disnub icosahedron, with θ = ${\displaystyle \arccos \left({\sqrt {\frac {-1+3{\sqrt {5}}+3{\sqrt {-22+10{\sqrt {5}}}}}{8}}}\right)\approx 14.33033^{\circ }}$. It has the same edges as the uniform great dirhombicosidodecahedron.

Its quotient prismatic equivalent is the triangular antiprismatic icosagyroprism, which is 22-dimensional.

## Vertex coordinates

The vertices of a disnub icosahedron of edge length 1 are given by all even permutations of:

• ${\displaystyle \left(\pm {\sqrt {\frac {{\sqrt {5}}-1-2{\sqrt {{\sqrt {5}}-2}}}{8}}},\,\pm {\sqrt {\frac {3-{\sqrt {5}}-{\sqrt {10{\sqrt {5}}-22}}}{8}}},\,\pm {\sqrt {\frac {2+{\sqrt {2{\sqrt {5}}-2}}}{8}}}\right),}$
• ${\displaystyle \left(0,\,\pm {\frac {\sqrt {3-{\sqrt {5}}}}{2}},\,\pm {\frac {\sqrt {{\sqrt {5}}-1}}{2}}\right),}$
• ${\displaystyle \left(\pm {\sqrt {\frac {3-{\sqrt {5}}+{\sqrt {10{\sqrt {5}}-22}}}{8}}},\,\pm {\sqrt {\frac {2-{\sqrt {2{\sqrt {5}}-2}}}{8}}},\,\pm {\sqrt {\frac {{\sqrt {5}}-1+2{\sqrt {{\sqrt {5}}-2}}}{8}}}\right).}$