Compound of twenty octahedra with rotational freedom

Compound of twenty octahedra with rotational freedom
(3D model) (OFF file)
Rank	3
Type	Uniform
Notation
Bowers style acronym	Addasi
Elements
Components	20 octahedra
Faces	40+120 triangles
Edges	120+120
Vertices	120
Vertex figure	Square, edge length 1
Measures (edge length 1)
Circumradius	${\frac {\sqrt {2}}{2}}\approx 0.70711$
Inradius	${\frac {\sqrt {6}}{6}}\approx 0.40825$
Volume	${\frac {20{\sqrt {2}}}{3}}\approx 9.42809$
Dihedral angle	$\arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }$
Central density	20
Related polytopes
Army	Semi-uniform Grid
Regiment	Addasi
Dual	Compound of twenty cubes with rotational freedom
Conjugate	Compound of twenty octahedra with rotational freedom
Abstract & topological properties
Schläfli type	{3,4}
Orientable	Yes
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

Not to be confused with Compound of twenty octahedra.

The altered disnub icosahedron (OBSA: 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.

Vertex coordinates

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

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

Rotational variants

This compound has rotational freedom, represented by an angle θ . The 20 octahedra can each be thought of as triangular antiprisms, and can be rotated in pairs going opposite directions while still maintaining vertex-transitivity. This compound goes through the following named phases as θ  increases:

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

External links

• Bowers, Jonathan. "Polyhedron Category C9: Octahedral Continuums" (#63, 63b, 64, 65).

• Klitzing, Richard. "addasi".
• Wikipedia contributors. "Compound of twenty octahedra with rotational freedom".