Here is a graphic I drew with an earlier version of Mathematic (probably version 6) which I no longer have available:

Sphere embedded in a truncated icosahedron

At the time Polyhedra was an external package, and I produced the cut-away by truncating a list of component polyhedra of the truncated icosahedron before drawing.

Recent versions of Mathematica have an entirely different (and now integrated) set of polyhedron functions, that look nothing like what I used before. Can anyone suggest a strategy for redrawing this figure now?

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    $\begingroup$ I guess you're referring to PolyhedronData["TruncatedIcosahedron"]? $\endgroup$ – Michael E2 Aug 9 '16 at 0:42
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    $\begingroup$ Do you still have the old code, at least? Can you elaborate on how you "truncat(ed) a list of component polyhedra of the truncated icosahedron"? $\endgroup$ – J. M.'s technical difficulties Aug 9 '16 at 4:42

An adaptation of my answer here:

SeedRandom[3]; (* for reproducibility *)
With[{poly = PolyhedronData["TruncatedIcosahedron"] /. 
    Polygon[pp_] :> Polygon[RandomSample[pp, 24]]}, 
 With[{r0 = PolyhedronData["TruncatedIcosahedron", "Circumradius"],
       r1 = 1,                   (* input: r1 = inner boundary vertex distance *)
       r2 = 2},                  (* input: r2 = outer boundary vertex distance *)
  With[{pts = First@Cases[poly, 
       GraphicsComplex[p_, e__] :> Flatten[{p *(r1/r0), p*(r2/r0)}, 1],
     {EdgeForm[], Lighter@ColorData[97, 2],
      Cases[poly, Polygon[p_] :> Polygon@Join[p, p + Length[pts]/2], Infinity], 
       Polygon[p_] :> Polygon[
           Join[#, Reverse@# + Length[pts]/2] & /@ Partition[#, 2, 1, 1] & /@ p,
     ], PlotRange -> All, Options[poly]]]]]

Mathematica graphics

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