The snub dodecahedron is an Archimedean solid consisting of 92 faces (80 triangular, 12 pentagonal), 150 edges, and 60 vertices. It is sometimes called the dodecahedron simum (Kepler 1619, Weissbach and Martini 2002) or snub icosidodecahedron. It is a chiral solid, and therefore exists in two enantiomorphous forms, commonly called laevo (left-handed) and dextro (right-handed). The laevo snub dodecahedron is illustrated above together with a wireframe version and a net that can be used for its construction.
In the snubification construction, the laevo form is obtained by twisting every pentagonal face clockwise as viewed from outside the original dodecahedron. Counterclockwise twists produce the dextro form (McCooey).
It is also the uniform polyhedron with Maeder index 29 (Maeder 1997), Wenninger index 18 (Wenninger 1989), Coxeter index 32 (Coxeter
et al. 1954), and Har'El index 34 (Har'El 1993). It has Schläfli
symbol s
and Wythoff symbol 
.
Some symmetric projections of the snub dodecahedron are illustrated above.
It is implemented in the Wolfram Language as PolyhedronData["SnubDodecahedron"].
An attractive dual of the two enantiomers superposed on one another is illustrated above.
The dual polyhedron of the snub dodecahedron is the pentagonal hexecontahedron, with which it is illustrated above.
It can be constructed by snubification of a dodecahedron of unit edge length. The outward displacement of each face is
 |
(1)
|
where 
is the circumradius given below. The magnitude of
the twist angle is
![theta=cos^(-1)((5-sqrt(5))/2[phi/2sqrt(3R^2-1)-h/(20)sqrt(250+110sqrt(5))])=13.1064033769... degrees,](/images/equations/SnubDodecahedron/NumberedEquation2.svg) |
(2)
|
where 
is the golden ratio and 
is the distance from the center to
a pentagonal face of the resulting
solid. The displacement 
is the increase in this distance from its value in the original
dodecahedron. Here, the notation 
used below indicates a polynomial
root.
The inradius 
of the dual, midradius 
of the solid and dual, and circumradius 
of the solid for 
are given
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(3)
|
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(4)
|
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(5)
|
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(6)
|
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(7)
|
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(8)
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A simpler expression for the circumradius is
 |
(9)
|
where 
is the sole real root of
 |
(10)
|
with 
(T. Piezas III, pers. comm., Mar. 3, 2006).
For unit edge length, the radius 
of the sphere
concentric with its circumsphere and tangent to
the triangular faces satisfies
 |
(11)
|
It is also the largest positive root of the polynomial equation
 |
(12)
|
The surface area is given by
![S=sqrt(15[95+6sqrt(5)+8sqrt(15(5+2sqrt(5)))]),](/images/equations/SnubDodecahedron/NumberedEquation7.svg) |
(13)
|
and the volume is given by the polynomial root
 |
(14)
|
