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Snub dodecahedron

 

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

In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more ty...
.

The snub dodecahedron has 92 faces, of which 12 are pentagonPentagon

In geometry, a pentagon is any five-sided polygon....
s and the other 80 are equilateral triangleEquilateral triangle

In geometry, an equilateral triangle is a triangle in which all three sides have equal lengths....
s. It also has 150 edges, and 60 vertices. It has two distinct forms, which are mirror imageMirror Image

Mirror Image is an episode of the television series The Twilight Zone. ...
s (or "enantiomorphsChirality (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 i...
") of each other.

Geometric relations

The snub dodecahedron can be generated by taking the twelve pentagonPentagon

In geometry, a pentagon is any five-sided polygon....
al faces of the dodecahedronDodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid comp...
, pulling them outwardExpansion (geometry)

In geometry, expansion is a polytope operation where facets are separated and moved radially apart, and new facets are forme...
 so they no longer touch. At a proper distance this can create the rhombicosidodecahedronRhombicosidodecahedron

The rhombicosidodecahedron, or small rhombicosidodecahedron, is an Archimedean solid....
 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
Rhombicosidodecahedron
(Expanded dodecahedron)


ArchimedesArchimedes

Archimedes was an ancient Greek mathematician, physicist, engineer, astronomer, and philosopher born in the seaport colony...
, 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 meanFacts About Golden mean

Golden mean may mean:*Doctrine of the Golden Mean, a chapter in Li Ji, one of the "four books" of classical Confuci...
 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 enantiomorphChirality (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 i...
 of the other one.

See also

  • and spinning snub dodecahedron
  • Snub cubeSnub cube

    The snub cube, or snub cuboctahedron, is an Archimedean solid....
  • Snub hexagonal tilingSnub hexagonal tiling

    In geometry, the Snub hexagonal tiling is a semiregular tiling of the Euclidean plane....


External links

  • The Encyclopedia of Polyhedra