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Snub dodecahedron
 
 
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The snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid
Archimedean solid

In geometry an Archimedean solid is a highly symmetric, semi-regular convex set polyhedron composed of two or more types of regular polygons meeting in identical vertex ....
.

The snub dodecahedron has 92 faces, of which 12 are pentagon
Pentagon

In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?....
s and the other 80 are equilateral triangle
Equilateral triangle

In geometry, an equilateral triangle is a triangle in which all three sides are equal. In traditional or Euclidean geometry, equilateral triangles are also Equiangular polygon; that is, all three internal angles are also congruent to each other and are each 60?....
s. It also has 150 edges, and 60 vertices. It has two distinct forms, which are mirror image
Mirror Image

"Mirror Image" is an episode of the television series The Twilight Zone ....
s (or "enantiomorphs
Chirality (mathematics)

In geometry, a figure is chiral if it is not identical to its mirror image, or more particularly if it cannot be mapped to its mirror image by rotations and translations alone....
") of each other.
snub dodecahedron can be generated by taking the twelve pentagon
Pentagon

In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?....
al faces of the dodecahedron
Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
, pulling them outward
Expansion (geometry)

In geometry, expansion is a polytope operation where Facet are separated and moved radially apart, and new facets are formed at separated elements ....
 so they no longer touch.






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The snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid
Archimedean solid

In geometry an Archimedean solid is a highly symmetric, semi-regular convex set polyhedron composed of two or more types of regular polygons meeting in identical vertex ....
.

The snub dodecahedron has 92 faces, of which 12 are pentagon
Pentagon

In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?....
s and the other 80 are equilateral triangle
Equilateral triangle

In geometry, an equilateral triangle is a triangle in which all three sides are equal. In traditional or Euclidean geometry, equilateral triangles are also Equiangular polygon; that is, all three internal angles are also congruent to each other and are each 60?....
s. It also has 150 edges, and 60 vertices. It has two distinct forms, which are mirror image
Mirror Image

"Mirror Image" is an episode of the television series The Twilight Zone ....
s (or "enantiomorphs
Chirality (mathematics)

In geometry, a figure is chiral if it is not identical to its mirror image, or more particularly if it cannot be mapped to its mirror image by rotations and translations alone....
") of each other.

Geometric relations

The snub dodecahedron can be generated by taking the twelve pentagon
Pentagon

In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?....
al faces of the dodecahedron
Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
, pulling them outward
Expansion (geometry)

In geometry, expansion is a polytope operation where Facet are separated and moved radially apart, and new facets are formed at separated elements ....
 so they no longer touch. At a proper distance this can create the rhombicosidodecahedron
Rhombicosidodecahedron

The rhombicosidodecahedron, or small rhombicosidodecahedron, is an Archimedean solid. It has 20 regular triangle faces, 30 square faces, 12 regular pentagonal faces, 60 vertices and 120 edges....
 by filling in square faces between the divided edges and triangle faces between the divided vertices. But for the snub form, only add the triangle faces and leave the square gaps empty. Then apply an equal rotation to the centers of the pentagons and triangles, continuing the rotation until the gaps can be filled by two equilateral triangles.

Dodecahedron

Dodecahedron
Small Rhombicosidodecahedron

Rhombicosidodecahedron
(Expanded dodecahedron)


Archimedes
Archimedes

Archimedes of Syracuse was a Greek mathematics, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity....
, an ancient Greek who showed major interest in polyhedral shapes wrote a treatise on thirteen semi-regular solids. Snub-Dodecahedron belongs to the thirteen semi-regular solids.

Cartesian coordinates

Cartesian coordinates for the vertices of a snub dodecahedron 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
α = ξ-1/ξ
and
β = ξτ+τ2+τ/ξ,
where τ = (1+√5)/2 is the golden ratio
Golden ratio

In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller....
 and ξ is the real solution to ξ3-2ξ=τ, which is the beautiful number or approximately 1.7155615. 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 particularly if it cannot be mapped to its mirror image by rotations and translations alone....
 of the other one.

See also

  • ccw and cw spinning snub dodecahedron
  • Snub cube
    Snub cube

    The snub cube, or snub cuboctahedron, is an Archimedean solid.The snub cube has 38 faces, 6 of which are square s and the other 32 are equilateral triangles....
  • Snub hexagonal tiling
    Snub hexagonal tiling

    In geometry, the Snub hexagonal tiling is a semiregular tiling of the Euclidean plane. There are four triangles and one hexagon on each vertex ....


External links

  • The Encyclopedia of Polyhedra