Compound of twenty octahedra with rotational freedom

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Compound of twenty octahedra with rotational freedom
Rank3
TypeUniform
Notation
Bowers style acronymAddasi
Elements
Components20 octahedra
Faces40+120 triangles
Edges120+120
Vertices120
Vertex figureSquare, edge length 1
Measures (edge length 1)
Circumradius
Inradius
Volume
Dihedral angle
Central density20
Related polytopes
ArmySemi-uniform Grid
RegimentAddasi
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 < : General phase sometimes known as the outer disnub icosahedron or oddasi
  • θ = : Vertices coincide by pairs to form the disnub icosahedron
  • < θ < : General phase sometimes known as the inner disnub icosahedron or iddasi
  • θ = : Octahedra coincide by four, forming a quadruple-cover of the small icosicosahedron
  • < θ < 60°: General phase known as the great disnub icosahedron or giddasi
  • θ = 60°: Double-cover of the great snub icosahedron

Vertex coordinates[edit | edit source]

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

External links[edit | edit source]