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.

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

Let ξ ≈ 0.94315125924 be the real zero of the cubic polynomial x3 + 2x2 − φ2, where φ is the golden ratio.

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 coordinates of the vertices are integral linear combinations of 1, φ, ξ, φξ, ξ2 and φξ2.

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.

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.

This means that ξ is the ratio between the midradii of a snub dodecahedron and the icosahedron in which it is inscribed.

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

The four positive real roots of the sextic equation in R2

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

[1] 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.

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.)

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.

Alternatively, combining the vertices of the snub dodecahedron given by the Cartesian coordinates (above) and its mirror will form a semiregular truncated icosidodecahedron.

The comparisons between these regular and semiregular polyhedrons is shown in the figure to the right.

Cartesian coordinates for the vertices of this alternative snub dodecahedron are obtained by selecting sets of 12 (of 24 possible even permutations contained in the five sets of truncated icosidodecahedron Cartesian coordinates).

The alternations are those with an odd number of minus signs in these three sets:

The mirrors of both the regular truncated icosidodecahedron and this alternative snub dodecahedron are obtained by switching the even and odd references to both sign and position permutations.

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 .

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.