Snub dodecadodecahedron
Encyclopedia
In geometry
, the snub dodecadodecahedron is a nonconvex uniform polyhedron
, indexed as U40. It is given a Schläfli symbol s{5/2,5}, as a snub
great dodecahedron.
with an even number of plus signs, where
where τ = (1+√5)/2 is the golden mean
and
α is the positive real root of τα4−α3+2α2−α−1/τ, or approximately 0.7964421.
Taking the odd permutations of the above coordinates with an odd number of plus signs gives another form, the enantiomorph
of the other one.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, the snub dodecadodecahedron is a nonconvex uniform polyhedron
Nonconvex uniform polyhedron
In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting...
, indexed as U40. It is given a Schläfli symbol s{5/2,5}, as a snub
Snub (geometry)
In geometry, an alternation is an operation on a polyhedron or tiling that removes alternate vertices. Only even-sided polyhedra can be alternated, for example the zonohedra. Every 2n-sided face becomes n-sided...
great dodecahedron.
Cartesian coordinates
Cartesian coordinates for the vertices of a snub dodecadodecahedron are all the even permutations of- (±2α, ±2, ±2β),
- (±(α+β/τ+τ), ±(-ατ+β+1/τ), ±(α/τ+βτ-1)),
- (±(-α/τ+βτ+1), ±(-α+β/τ-τ), ±(ατ+β-1/τ)),
- (±(-α/τ+βτ-1), ±(α-β/τ-τ), ±(ατ+β+1/τ)) and
- (±(α+β/τ-τ), ±(ατ-β+1/τ), ±(α/τ+βτ+1)),
with an even number of plus signs, where
- β = (α2/τ+τ)/(ατ−1/τ),
where τ = (1+√5)/2 is the golden mean
Golden mean
Golden mean may refer to:*Doctrine of the Golden Mean , a chapter in Li Ji, one of the Four Books of Confucianism*Golden mean , the felicitous middle between the extremes of excess and deficiency...
and
α is the positive real root of τα4−α3+2α2−α−1/τ, or approximately 0.7964421.
Taking the odd permutations of the above coordinates with an odd number of plus signs gives another form, the enantiomorph
Chirality (mathematics)
In geometry, a figure is chiral if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. For example, a right shoe is different from a left shoe, and clockwise is different from counterclockwise.A chiral object...
of the other one.