Snub dodecahedron

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Snub dodecahedron
(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
Wythoff symbol | 2 3 5
Coxeter diagram CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.png
Symmetry group I, 1/2H3, [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
Polyhedron snub 12-20 left max.png
Colored faces
Polyhedron snub 12-20 left vertfig.svg
(Vertex figure)
Polyhedron snub 12-20 left dual max.png
Pentagonal hexecontahedron
(dual polyhedron)
Polyhedron snub 12-20 left net.svg
3D model of a snub dodecahedron

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 and flat Schläfli symbol sr{5,3}.

Cartesian coordinates[edit]

Let ξ0.94315125924 be the real zero of the cubic polynomial x3 + 2x2φ2, where φ is the golden ratio. Let the point p be given by


Let the rotation matrices M1 and M2 be given by


M1 represents the rotation around the axis (0,1,φ) through an angle of 2π/5 counterclockwise, while M2 being a cyclic shift of (x,y,z) represents the rotation around the axis (1,1,1) through an angle of 2π/3. Then the 60 vertices of the snub dodecahedron are the 60 images of point p under repeated multiplication by M1 and/or M2, iterated to convergence. (The matrices M1 and M2 generate the 60 rotation matrices corresponding to the 60 rotational symmetries of a regular icosahedron.) The coordinates of the vertices are integral linear combinations of 1, φ, ξ, φξ, ξ2 and φξ2. The edge length equals

Negating all coordinates gives the mirror image of this snub dodecahedron.

As a volume, the snub dodecahedron consists of 80 triangular and 12 pentagonal pyramids. The volume V3 of one triangular pyramid is given by:

and the volume V5 of one pentagonal pyramid by:

The total volume is

The circumradius equals

The midradius equals ξ. This gives an interesting geometrical interpretation of the number ξ. The 20 "icosahedral" triangles of the snub dodecahedron described above are coplanar with the faces of a regular icosahedron. The midradius of this "circumscribed" icosahedron equals 1. This means that ξ is the ratio between the midradii of a snub dodecahedron and the icosahedron in which it is inscribed.

The triangle–triangle dihedral angle is given by

The triangle–pentagon dihedral angle is given by

Metric properties[edit]

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

Its volume is

Its circumradius is

Its midradius is

There are two inscribed spheres, one touching the triangular faces, and one, slightly smaller, touching the pentagonal faces. Their radii are, respectively:


The four positive real roots of the sextic equation in R2

are the circumradii of the snub dodecahedron (U29), great snub icosidodecahedron (U57), great inverted snub icosidodecahedron (U69), and great retrosnub icosidodecahedron (U74).

The snub dodecahedron has the highest sphericity of all Archimedean solids. If sphericity is defined as the ratio of volume squared over surface area cubed, multiplied by a constant of 36π (where this constant makes the sphericity of a sphere equal to 1), the sphericity of the snub dodecahedron is about 0.947.[1]

Orthogonal projections[edit]

The snub dodecahedron has no point symmetry, so the vertex in the front does not correspond to an opposite vertex in the back.

The snub dodecahedron has two especially symmetric orthogonal projections as shown below, centered on two types of faces: triangles and pentagons, corresponding to the A2 and H2 Coxeter planes.

Orthogonal projections
Centered by Face
Solid Polyhedron snub 12-20 left from yellow max.png Polyhedron snub 12-20 left from red max.png Polyhedron snub 12-20 left from blue max.png
Wireframe Snub dodecahedron A2.png Snub dodecahedron H2.png Snub dodecahedron e1.png
[3] [5]+ [2]
Dual Dual snub dodecahedron A2.png Dual snub dodecahedron H2.png Dual snub dodecahedron e1.png

Geometric relations[edit]

Dodecahedron, rhombicosidodecahedron and snub dodecahedron (animated expansion and twisting)

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, pull the pentagonal faces out slightly less, only add the triangle faces and leave the other gaps empty (the other gaps are rectangles at this point). 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. (The fact that the proper amount to pull the faces out is less in the case of the snub dodecahedron can be seen in either of two ways: the circumradius of the snub dodecahedron is smaller than that of the icosidodecahedron; or, the edge length of the equilateral triangles formed by the divided vertices increases when the pentagonal faces are rotated.)

Uniform alternation of a truncated icosidodecahedron

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.

Related polyhedra and tilings[edit]

Family of uniform icosahedral polyhedra
Symmetry: [5,3], (*532) [5,3]+, (532)
Uniform polyhedron-53-t0.svg Uniform polyhedron-53-t01.svg Uniform polyhedron-53-t1.svg Uniform polyhedron-53-t12.svg Uniform polyhedron-53-t2.svg 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.jpg Triakisicosahedron.jpg Rhombictriacontahedron.jpg Pentakisdodecahedron.jpg Dodecahedron.jpg Deltoidalhexecontahedron.jpg Disdyakistriacontahedron.jpg Pentagonalhexecontahedronccw.jpg
V5.5.5 V3.10.10 V3.5.3.5 V5.6.6 V3. V3.4.5.4 V4.6.10 V3.

This semiregular polyhedron is a member of a sequence of snubbed polyhedra and tilings with vertex figure ( 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:
Spherical Euclidean Compact hyperbolic Paracomp.
232 332 432 532 632 732 832 ∞32
Spherical trigonal antiprism.png Spherical snub tetrahedron.png Spherical snub cube.png Spherical snub dodecahedron.png Uniform tiling 63-snub.svg Snub triheptagonal tiling.svg H2-8-3-snub.svg Uniform tiling i32-snub.png
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 7-3 floret pentagonal tiling.svg H2-8-3-floret.svg Order-3-infinite floret pentagonal tiling.png
Config. V3. V3. V3. V3. V3. V3. V3. V3.3.3.3.∞

Snub dodecahedral graph[edit]

Snub dodecahedral graph
Snub dodecahedral graph.png
5-fold symmetry Schlegel diagram
PropertiesHamiltonian, regular
Table of graphs and parameters

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.[2]

See also[edit]

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


  1. ^ How Spherical Are the Archimedean Solids and Their Duals? P. K. Aravind, The College Mathematics Journal, Vol. 42, No. 2 (March 2011), pp. 98-107
  2. ^ Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269

External links[edit]