bgcolor=#e7dcc3 colspan=2 | Compound of twelve pentagonal antiprisms with rotational freedom | |
---|---|---|
align=center colspan=2 | ||
Type | Uniform compound | |
Index | UC26 | |
Polyhedra | 12 pentagonal antiprisms | |
Faces | 120 triangles, 24 pentagons | |
Edges | 240 | |
Vertices | 120 | |
Symmetry group | icosahedral (Ih) | |
Subgroup restricting to one constituent | 10-fold improper rotation (S10) |
When θ is 36 degrees, the antiprisms coincide in pairs to yield (two superimposed copies of) the compound of six pentagonal antiprisms (without rotational freedom).
This compound shares its vertices with the compound of twelve pentagrammic crossed antiprisms with rotational freedom.
Cartesian coordinates for the vertices of this compound are all the cyclic permutations of
(±(2τ−1−(2τ+4)cos θ), ±2(√(5τ+10))sin θ, ±(τ+2+(4τ−2)cos θ))
(±(2τ−1−(2τ−1)cos θ−τ(√(5τ+10))sin θ), ±(−5τcos θ+τ−1(√(5τ+10))sin θ),
±(τ+2+(3−τ)cos θ+(√(5τ+10))sin θ))
(±(2τ−1+(1+3τ)cos θ−(√(5τ+10))sin θ), ±(−5cos θ−τ(√(5τ+10))sin θ),
±(τ+2−(τ+2)cos θ+τ−1(√(5τ+10))sin θ))
(±(2τ−1+(1+3τ)cos θ+(√(5τ+10))sin θ), ±(5cos θ−τ(√(5τ+10))sin θ),
±(τ+2−(τ+2)cos θ−τ−1(√(5τ+10))sin θ))
(±(2τ−1−(2τ−1)cos θ+τ(√(5τ+10))sin θ), ±(5τcos θ+τ−1(√(5τ+10))sin θ),
±(τ+2+(3−τ)cos θ−(√(5τ+10))sin θ))
where τ = (1+√5)/2 is the golden ratio (sometimes written φ).