[Tex/LaTex] Truncated Icosahedron in TikZ

pst-solides3dtikz-pgf

I would like to draw the Archimedean solid Truncated Icosahedron (soccer ball, Bucky ball), so one can use the resources on transparency, position and light source to manipulate the images.

enter image description here

The image on the left is executed in POVRay and sources are available at Wikipedia, the one on the right is from Mathematica and data on vertices is available with the command PolyhedronData["TruncatedIcosahedron"].

Listing Vertices, Edges and Faces

The vertices are all even permutations of:

(0, ±1, ±3ϕ)
(±2, ±(1+2ϕ), ±ϕ)
(±1, ±(2+ϕ), ±2ϕ)

where $\phi = (1 + \sqrt{5})/2$ is the golden mean, for an explicit list see Wikipedia here or here.

The number of vertices (60), edges (90) and faces (32) seem a bit too large to be dealt by hand as in other examples like drawing an Octahedron. All edges in this configuration have length 2 so an easy algorithm can determine the edges (the 3 neighbours at distance < 2.1). Given two edges at a vertex it is also (algorithmically) easy to find the third that compose a face — it is the one that makes a zero-volume with the previous two vectors. How easy it is to implement all of that in TikZ? May be getting a ready set of coordinates edges and faces would be much easier?

Cutting the Vertices of a pre-defined Icosahedron

Another possible construction would be cutting off the vertices of an (pre-defined) icosahedron in a way, so that every edge has the same length. Several packages (pst-solids3d, pst-platon) have the Icosahedron pre-define with one-command — the issue here will the execution of the cut-offs.

In this particular case the solid is centred at the origin and all planes (that cut-off) are perpendicular to the vectors defining the original icosahedron, and equidistant to the origin.

Best Answer

Let's start with the skeleton. Coordinates are copy from polyhedron_js.asy.

\documentclass[border=9,tikz]{standalone}

\usepackage{tikz-3dplot}

\begin{document}
\foreach\s in{2,4,...,360}{
    \tdplotsetmaincoords{2.71828+\s}{2.71828+\s*2}
    \tikz[tdplot_main_coords,scale=.1]{
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        \draw
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    }
}

\end{document}
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