Editing Snub dodecahedron (section)

==Geometric relations==
{{multiple image
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 | image1 = Polyhedron 12 max.png |width1=3652 |height1=3960
 | image2 = Polyhedron small rhombi 12-20 max.png |width2=3973 |height2=4000
 | image3 = Polyhedron snub 12-20 left max.png |width3=3966 |height3=4000
 | footer = Dodecahedron, rhombicosidodecahedron and snub dodecahedron <small>(animated [[:File:P4-A11-P5.gif|expansion]] and [[:File:A11-A13.gif|twisting]])</small>
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{{multiple image
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 | image1 = Polyhedron great rhombi 12-20 subsolid snub left big.png
 | image2 = Polyhedron great rhombi 12-20 big.png
 | image3 = Polyhedron great rhombi 12-20 subsolid snub right big.png
 | footer = Uniform alternation of a truncated icosidodecahedron
}}

The ''snub dodecahedron'' can be generated by taking the twelve [[pentagon]]al faces of the [[dodecahedron]] and [[Expansion (geometry)|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.)

The snub dodecahedron can also be derived from the [[truncated icosidodecahedron]] by the process of [[alternation (geometry)|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. 

Alternatively, combining the vertices of the snub dodecahedron given by the [[Snub_dodecahedron#Cartesian_coordinates|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 permutation]]s contained in the five sets of [[Truncated_icosidodecahedron#Cartesian_coordinates|truncated icosidodecahedron Cartesian coordinates]]). The alternations are those with an odd number of minus signs in these three sets: 

{{multiple image
 | align = right  | total_width = 350
 | image1 = Truncated_icosidodecahedron_comparing_regular_vs_quasi-regular_overlay_of_regular_and_quasi-regular_constructions.svg
 | image2 = Snub_decahedron_comparing_regular_vs_quasi-regular.svg
 | footer = Overlay of  regular and semiregular truncated icosidodecahedra and snub dodecahedra
}}
<math display=block>\begin{array}{ccccccc}
  \Bigl(& \pm \tfrac{1}{\varphi} &,& \pm \tfrac{1}{\varphi} &,& \pm [3 + \varphi] & \Bigr), \\[2pt]
  \Bigl(& \pm \tfrac{1}{\varphi} &,& \pm\,\varphi^2 &,& \pm [3\varphi - 1] & \Bigr),  \\[2pt]
  \Bigl(& \pm [2\varphi - 1] &,& \pm\,2 &,& \pm [2 + \varphi] & \Bigr),
\end{array}</math>
and an even number of minus signs in these two sets:
<math display=block>\begin{array}{ccccccc}
  \Bigl(& \pm \tfrac{2}{\varphi} &,& \pm\,\varphi &,& \pm [1 + 2\varphi] & \Bigr), \\[2pt]
  \Bigl(& \pm\,\varphi &,& \pm\,3 &,& \pm\,2\varphi & \Bigr),
\end{array}</math>

where <math>\varphi = \tfrac{1 + \sqrt 5}{2}</math> is the [[golden ratio]]. 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.