# Compound of twenty octahedra with rotational freedom

Compound of twenty octahedra with rotational freedom
Rank3
TypeUniform
Notation
Elements
Components20 octahedra
Faces40+120 triangles
Edges120+120
Vertices120
Vertex figureSquare, 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
Related polytopes
ArmySemi-uniform Grid
DualCompound of twenty cubes with rotational freedom
ConjugateCompound of twenty octahedra with rotational freedom
Abstract & topological properties
Schläfli type{3,4}
OrientableYes
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

The altered disnub icosahedron, addasi, or compound of twenty octahedra with rotational freedom is a uniform polyhedron compound. It consists of 40+120 triangles, with 4 triangles joining at each vertex.

This compound has rotational freedom, represented by an angle θ. The 20 octahedra can each be thought of as triangular antiprisms, and we rotate them in pairs in opposite directions. This compound goes through the following phases as θ increases:

• θ = 0°: Double-cover of the snub icosahedron
• 0° < θ < ${\displaystyle \arccos \left({\sqrt {\frac {-1+3{\sqrt {5}}+3{\sqrt {10{\sqrt {5}}-22}}}{8}}}\right)}$: General phase sometimes known as the outer disnub icosahedron or oddasi
• θ = ${\displaystyle \arccos \left({\sqrt {\frac {-1+3{\sqrt {5}}+3{\sqrt {10{\sqrt {5}}-22}}}{8}}}\right)\approx 14.33033^{\circ }}$: Vertices coincide by pairs to form the disnub icosahedron
• ${\displaystyle \arccos \left({\sqrt {\frac {-1+3{\sqrt {5}}+3{\sqrt {10{\sqrt {5}}-22}}}{8}}}\right)}$ < θ < ${\displaystyle \arccos \left({\frac {\sqrt {10}}{4}}\right)}$: General phase sometimes known as the inner disnub icosahedron or iddasi
• θ = ${\displaystyle \arccos \left({\frac {\sqrt {10}}{4}}\right)\approx 37.76124^{\circ }}$: Octahedra coincide by four, forming a quadruple-cover of the small icosicosahedron
• ${\displaystyle \arccos \left({\frac {\sqrt {10}}{4}}\right)}$ < θ < 60°: General phase known as the great disnub icosahedron or giddasi
• θ = 60°: Double-cover of the great snub icosahedron

## Vertex coordinates

The vertices of an altered disnub icosahedron of edge length 1 and rotation angle θ are given by all even permutations of:

• ${\displaystyle \left(\pm {\frac {{\sqrt {3}}\sin(\theta )}{3}},\,\pm {\frac {{\sqrt {10}}-{\sqrt {2}}+2(1+{\sqrt {5}})\cos(\theta )}{12}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {10}}-2({\sqrt {5}}-1)\cos(\theta )}{12}}\right),}$
• ${\displaystyle \left(\pm {\frac {2{\sqrt {2}}-(3-{\sqrt {5}})\cos(\theta )+({\sqrt {15}}-{\sqrt {3}})\sin(\theta )}{12}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {5}}\cos(\theta )+{\sqrt {3}}\sin(\theta )}{6}},\,\pm {\frac {2{\sqrt {2}}+(3-{\sqrt {5}})\cos(\theta )-({\sqrt {3}}+{\sqrt {15}})\sin(\theta )}{12}}\right),}$
• ${\displaystyle \left(\pm {\frac {{\sqrt {10}}-{\sqrt {2}}-(1+{\sqrt {5}})\cos(\theta )-({\sqrt {3}}+{\sqrt {15}})\sin(\theta )}{12}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {10}}+({\sqrt {5}}-1)\cos(\theta )+({\sqrt {15}}-{\sqrt {3}})\sin(\theta )}{12}},\,\pm {\frac {3\cos(\theta )-{\sqrt {3}}\sin(\theta )}{6}}\right),}$
• ${\displaystyle \left(\pm {\frac {{\sqrt {2}}-{\sqrt {10}}+(1+{\sqrt {5}})\cos(\theta )-({\sqrt {3}}+{\sqrt {15}})\sin(\theta )}{12}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {10}}+({\sqrt {5}}-1)\cos(\theta )-({\sqrt {15}}-{\sqrt {3}})\sin(\theta )}{12}},\,\pm {\frac {3\cos(\theta )+{\sqrt {3}}\sin(\theta )}{6}}\right),}$
• ${\displaystyle \left(\pm {\frac {-2{\sqrt {2}}+(3+{\sqrt {5}})\cos(\theta )+({\sqrt {15}}-{\sqrt {3}})\sin(\theta )}{12}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {5}}\cos(\theta )-{\sqrt {3}}\sin(\theta )}{6}},\,\pm {\frac {2{\sqrt {2}}+(3-{\sqrt {5}})\cos(\theta )+({\sqrt {3}}+{\sqrt {15}})\sin(\theta )}{12}}\right).}$