Editing Truncated dodecahedron (section)

== Construction ==
The truncated dodecahedron is constructed from a [[regular dodecahedron]] by cutting all of its vertices off, a process known as [[Truncation (geometry)|truncation]].{{r|ziya}} Alternatively, the truncated dodecahedron can be constructed by [[expansion (geometry)|expansion]]: pushing away the edges of a regular dodecahedron, forming the [[Pentagon|pentagonal]] faces into [[Decagon|decagonal]] faces, as well as the vertices into [[triangle]]s.{{r|vxac}} Therefore, it has 32 faces, 90 edges, and 60 vertices.{{r|berman}}

The truncated dodecahedron may also be constructed by using [[Cartesian coordinate]]s. With an edge length <math> 2\varphi - 2 </math> centered at the origin, they are all even permutations of
<math display="block">
\left(0, \pm \frac{1}{\varphi}, \pm (2 + \varphi) \right), \qquad
\left(\pm \frac{1}{\varphi}, \pm \varphi, \pm 2 \varphi \right), \qquad
\left(\pm \varphi, \pm 2, \pm (\varphi + 1) \right),
</math>
where <math display="inline"> \varphi = \frac{1 + \sqrt{5}}{2} </math> is the [[golden ratio]].<ref>{{mathworld |title=Icosahedral group |urlname=IcosahedralGroup}}</ref>