Inverted snub dodecadodecahedron

From HandWiki
Short description: Polyhedron with 84 faces


Inverted snub dodecadodecahedron
Type Uniform star polyhedron
Elements F = 84, E = 150
V = 60 (χ = −6)
Faces by sides 60{3}+12{5}+12{5/2}
Wythoff symbol | 5/3 2 5
Symmetry group I, [5,3]+, 532
Index references U60, C76, W114
Dual polyhedron Medial inverted pentagonal hexecontahedron
Vertex figure
3.3.5.3.5/3
Bowers acronym Isdid

File:Inverted snub dodecadodecahedron.stl In geometry, the inverted snub dodecadodecahedron (or vertisnub dodecadodecahedron) is a nonconvex uniform polyhedron, indexed as U60.[1] It is given a Schläfli symbol sr{5/3,5}.

Cartesian coordinates

Cartesian coordinates for the vertices of an inverted snub dodecadodecahedron are all the even permutations of (±2α ,±2 ,±2β ),(±[α+βφ+φ],±[αφ+β+1φ],±[αφ+βφ1]),(±[αφ+βφ+1],±[α+βφφ],±[αφ+β1φ]),(±[αφ+βφ1],±[αβφφ],±[αφ+β+1φ]),(±[α+βφφ],±[αφβ+1φ],±[αφ+βφ+1]),

with an even number of plus signs, where β=  α2φ+φ   αφ1φ , φ=1+52 is the golden ratio, and α is the negative real root of φα4α3+2α2α1φα0.3352090. Taking the odd permutations of the above coordinates with an odd number of plus signs gives another form, the enantiomorph of the other one. Taking α to be the positive root gives the snub dodecadodecahedron.

Medial inverted pentagonal hexecontahedron

Medial inverted pentagonal hexecontahedron
Type Star polyhedron
Face
Elements F = 60, E = 150
V = 84 (χ = −6)
Symmetry group I, [5,3]+, 532
Index references DU60
dual polyhedron Inverted snub dodecadodecahedron

File:Medial inverted pentagonal hexecontahedron.stl The medial inverted pentagonal hexecontahedron (or midly petaloid ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the uniform inverted snub dodecadodecahedron. Its faces are irregular nonconvex pentagons, with one very acute angle.

Proportions

Denote the golden ratio by ϕ, and let ξ0.23699384345 be the largest (least negative) real zero of the polynomial P=8x412x3+5x+1. Then each face has three equal angles of arccos(ξ)103.70918221953, one of arccos(ϕ2ξ+ϕ)3.99013042341 and one of 360arccos(ϕ2ξϕ1)224.88232291799. Each face has one medium length edge, two short and two long ones. If the medium length is 2, then the short edges have length 11ξϕ3ξ0.47412646054, and the long edges have length 1+1ξϕ3ξ37.55187944854. The dihedral angle equals arccos(ξ/(ξ+1))108.09571935234. The other real zero of the polynomial P plays a similar role for the medial pentagonal hexecontahedron.

See also

References