Great retrosnub icosidodecahedron
Great retrosnub icosidodecahedron | |
---|---|
Type | Uniform star polyhedron |
Elements | F = 92, E = 150 V = 60 (χ = 2) |
Faces by sides | (20+60){3}+12{5/2} |
Wythoff symbol | | 2 3/2 5/3 |
Symmetry group | I, [5,3]+, 532 |
Index references | U74, C90, W117 |
Dual polyhedron | Great pentagrammic hexecontahedron |
Vertex figure | (34.5/2)/2 |
Girsid |
In geometry, the great retrosnub icosidodecahedron or great inverted retrosnub icosidodecahedron or Azathoth is a nonconvex uniform polyhedron, indexed as U74. It has 92 faces (80 triangles and 12 pentagrams), 150 edges, and 60 vertices.[1] It is given a Schläfli symbol sr{3/2,5/3}.
Cartesian coordinates[]
Cartesian coordinates for the vertices of a great retrosnub icosidodecahedron are all the even permutations of
- (±2α, ±2, ±2β),
- (±(α−βτ−1/τ), ±(α/τ+β−τ), ±(−ατ−β/τ−1)),
- (±(ατ−β/τ+1), ±(−α−βτ+1/τ), ±(−α/τ+β+τ)),
- (±(ατ−β/τ−1), ±(α+βτ+1/τ), ±(−α/τ+β−τ)) and
- (±(α−βτ+1/τ), ±(−α/τ−β−τ), ±(−ατ−β/τ+1)),
with an even number of plus signs, where
- α = ξ−1/ξ
and
- β = −ξ/τ+1/τ2−1/(ξτ),
where τ = (1+√5)/2 is the golden mean and ξ is the smaller positive real root of ξ3−2ξ=−1/τ, namely
or approximately 0.3264046. 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 the odd permutations with an even number of plus signs or vice versa results in the same two figures rotated by 90 degrees.
The circumradius for unit edge length is
where is the appropriate root of . The four positive real roots of the sextic in
are the circumradii of the snub dodecahedron (U29), great snub icosidodecahedron (U57), great inverted snub icosidodecahedron (U69), and great retrosnub icosidodecahedron (U74).
See also[]
References[]
- ^ Maeder, Roman. "74: great retrosnub icosidodecahedron". MathConsult.
{{cite web}}
: CS1 maint: url-status (link)
External links[]
- Weisstein, Eric W. "Great retrosnub icosidodecahedron". MathWorld.
- http://gratrix.net/polyhedra/uniform/summary
- Uniform polyhedra
- Polyhedron stubs