Snub dodecahedron

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Snub dodecahedron
Snubdodecahedronccw.jpg
(Click here for rotating model)
Type Archimedean solid
Uniform polyhedron
Elements F = 92, E = 150, V = 60 (χ = 2)
Faces by sides (20+60){3}+12{5}
Conway notation sD
Schläfli symbols sr{5,3} or s\begin{Bmatrix} 5 \\ 3 \end{Bmatrix}
ht0,1,2{5,3}
Wythoff symbol | 2 3 5
Coxeter diagram CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.png
Symmetry group I, ½H3, [5,3]+, (532), order 60
Rotation group I, [5,3]+, (532), order 60
Dihedral Angle 3-3:164°10'31" (164.18°)
3-5:152°55'53" (152.93°)
References U29, C32, W18
Properties Semiregular convex chiral
Snub dodecahedron ccw.png
Colored faces
Snub dodecahedron vertfig.png
3.3.3.3.5
(Vertex figure)
Pentagonalhexecontahedronccw.jpg
Pentagonal hexecontahedron
(dual polyhedron)
Snub dodecahedron flat.svg
Net

In geometry, the snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.

The snub dodecahedron has 92 faces (the most of the 13 Archimedean solids): 12 are pentagons and the other 80 are equilateral triangles. It also has 150 edges, and 60 vertices.

It has two distinct forms, which are mirror images (or "enantiomorphs") of each other. The union of both forms is a compound of two snub dodecahedra, and the convex hull of both forms is a truncated icosidodecahedron.

Kepler first named it in Latin as dodecahedron simum in 1619 in his Harmonices Mundi. H. S. M. Coxeter, noting it could be derived equally from either the dodecahedron or the icosahedron, called it snub icosidodecahedron, with a vertical extended Schläfli symbol s\begin{Bmatrix} 5 \\ 3 \end{Bmatrix}.

Cartesian coordinates[edit]

Cartesian coordinates for the vertices of a snub dodecahedron are all the even permutations of

(±2α, ±2, ±2β),
(±(α+β/ϕ+ϕ), ±(−αϕ+β+1/ϕ), ±(α/ϕ+βϕ−1)),
(±(−α/ϕ+βϕ+1), ±(−α+β/ϕ−ϕ), ±(αϕ+β−1/ϕ)),
(±(−α/ϕ+βϕ−1), ±(α−β/ϕ−ϕ), ±(αϕ+β+1/ϕ)) and
(±(α+β/ϕ−ϕ), ±(αϕ−β+1/ϕ), ±(α/ϕ+βϕ+1)),

with an even number of plus signs, where

α = ξ − 1 / ξ

and

β = ξϕ + ϕ2 + ϕ /ξ,

where ϕ = (1 + √5)/2 is the golden ratio and ξ is the real solution to ξ3 − 2ξ = ϕ, which is the number:

\xi = \sqrt[3]{\frac{\phi}{2} + \frac{1}{2}\sqrt{\phi - \frac{5}{27}}} + \sqrt[3]{\frac{\phi}{2} - \frac{1}{2}\sqrt{\phi - \frac{5}{27}}}

or approximately 1.7155615.

This snub dodecahedron has an edge length of approximately 6.0437380841.

Taking the odd permutations of the above coordinates with an even number of plus signs gives another form, the enantiomorph of the other one. Though it may not be immediately obvious, the figure obtained by taking the even permutations with an even number of plus signs is the same as that obtained by taking the odd permutations with an odd number of plus signs. Similarly, the mirror image has either an odd permutation with an even number of plus signs or an even permutation with an odd number of plus signs.

Surface area and volume[edit]

Transformation from rhombicosidodecahedron to snub dodecahedron

For a snub dodecahedron whose edge length is 1, the surface area is

A = 20\sqrt{3} + 3\sqrt{25+10\sqrt{5}} \approx 55.28674495844515

and the volume is

V= \frac{12\xi^2(3\phi+1)-\xi(36\phi+7)-(53\phi+6)}{6\sqrt{3-\xi^2}^3} \approx 37.61664996273336

where ϕ is the golden ratio.

The snub dodecahedron has the highest sphericity of all Archimedean solids.

Orthogonal projections[edit]

The snub dodecahedron has two special orthogonal projections, centered, on two types of faces: triangles, and pentagons, correspond to the A2 and H2 Coxeter planes.

Orthogonal projections
Centered by Face
Triangle
Face
Pentagon
Edge
Image Snub dodecahedron A2.png Snub dodecahedron H2.png Snub dodecahedron e1.png
Projective
symmetry
[3] [5]+ [2]
Dual
image
Dual snub dodecahedron A2.png Dual snub dodecahedron H2.png Dual snub dodecahedron e1.png

Geometric relations[edit]

The snub dodecahedron can be generated by taking the twelve pentagonal faces of the dodecahedron and pulling them outward so they no longer touch. At a proper distance this can create the rhombicosidodecahedron 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.png
Dodecahedron
Small rhombicosidodecahedron.png
Rhombicosidodecahedron
(Expanded dodecahedron)
Snub dodecahedron cw.png
Snub dodecahedron

The snub dodecahedron can also be derived from the truncated icosidodecahedron by the process of alternation. Sixty of the vertices of the truncated icosidodecahedron form a polyhedron topologically equivalent to one snub dodecahedron; the remaining sixty form its mirror-image. The resulting polyhedron is vertex-transitive but not uniform, because its edges are of unequal lengths; some deformation is required to transform it into a uniform polyhedron.

Related polyhedra and tilings[edit]

Family of uniform icosahedral polyhedra
Symmetry: [5,3], (*532) [5,3]+, (532)
Uniform polyhedron-53-t0.png Uniform polyhedron-53-t01.png Uniform polyhedron-53-t1.png Uniform polyhedron-53-t12.png Uniform polyhedron-53-t2.png Uniform polyhedron-53-t02.png Uniform polyhedron-53-t012.png Uniform polyhedron-53-s012.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.png
{5,3} t{5,3} r{5,3} t{3,5} {3,5} rr{5,3} tr{5,3} sr{5,3}
Duals to uniform polyhedra
Icosahedron.svg Triakisicosahedron.jpg Rhombictriacontahedron.svg Pentakisdodecahedron.jpg Dodecahedron.svg Deltoidalhexecontahedron.jpg Disdyakistriacontahedron.jpg Pentagonalhexecontahedronccw.jpg
V5.5.5 V3.10.10 V3.5.3.5 V5.6.6 V3.3.3.3.3 V3.4.5.4 V4.6.10 V3.3.3.3.5

This semiregular polyhedron is a member of a sequence of snubbed polyhedra and tilings with vertex figure (3.3.3.3.n) and Coxeter–Dynkin diagram CDel node h.pngCDel n.pngCDel node h.pngCDel 3.pngCDel node h.png. These figures and their duals have (n32) rotational symmetry, being in the Euclidean plane for n=6, and hyperbolic plane for any higher n. The series can be considered to begin with n=2, with one set of faces degenerated into digons.

n32 symmetry mutations of snub tilings: 3.3.3.3.n
Symmetry
n32
Spherical Euclidean Compact hyperbolic Paracomp.
232 332 432 532 632 732 832 ∞32
Snub
figures
Spherical trigonal antiprism.png Spherical snub tetrahedron.png Spherical snub cube.png Spherical snub dodecahedron.png Uniform tiling 63-snub.png Uniform tiling 73-snub.png Uniform tiling 83-snub.png Uniform tiling i32-snub.png
Config. 3.3.3.3.2 3.3.3.3.3 3.3.3.3.4 3.3.3.3.5 3.3.3.3.6 3.3.3.3.7 3.3.3.3.8 3.3.3.3.∞
Gryro
figures
Uniform tiling 432-t0.png Uniform tiling 532-t0.png Spherical pentagonal icositetrahedron.png Spherical pentagonal hexecontahedron.png Tiling Dual Semiregular V3-3-3-3-6 Floret Pentagonal.svg Ord7 3 floret penta til.png Order-3-infinite floret pentagonal tiling.png
Config. V3.3.3.3.2 V3.3.3.3.3 V3.3.3.3.4 V3.3.3.3.5 V3.3.3.3.6 V3.3.3.3.7 V3.3.3.3.8 V3.3.3.3.∞

Snub dodecahedral graph[edit]

Snub dodecahedral graph
Snub dodecahedral graph.png
5-fold symmetry Schlegel diagram
Vertices 60
Edges 150
Automorphisms 60
Properties Hamiltonian, regular

In the mathematical field of graph theory, a snub dodecahedral graph is the graph of vertices and edges of the snub dodecahedron, one of the Archimedean solids. It has 60 vertices and 150 edges, and is an Archimedean graph.[1]

Orthogonal projections
Snub dodecahedron A2.png Snub dodecahedron H2.png Snub dodecahedron e1.png

See also[edit]

  • Planar polygon to polyhedron transformation animation
  • ccw and cw spinning snub dodecahedron


References[edit]

  1. ^ Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269 
  • Jayatilake, Udaya (March 2005). "Calculations on face and vertex regular polyhedra". Mathematical Gazette 89 (514): 76–81. 
  • Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X.  (Section 3-9)
  • Cromwell, P. (1997). Polyhedra. United Kingdom: Cambridge. pp. 79–86 Archimedean solids. ISBN 0-521-55432-2. 

External links[edit]