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There is, I believe, a sequence of polyhedra whose shape approaches that of the icosahedron (they all have twelve pentagonal faces and the rest hexagons), and starts:

regular dodecahedron (C$_{12}$, dodecahedrane)

truncated icosahedron (C$_{60}$, buckminsterfullerene)

?: (C$_{240}$, buckminsterfullerene)

?: (C$_{540}$, ?)

I am interested in the polyhedra that these molecules are associated with, namely:

  • What is the correct name for this sequence of polyhedra? (oeis.org doesn't have any sequences that match "12,60,240,540" at present)
  • Is there a formula in the literature for the volume of these polyhedra? (the surface area is not a difficult computation)
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1 Answer 1

The volume of a regular dodecahedron, having 12 congruent regular pentagonal faces each with edge length $a$, is given by the generalized expression $$\bbox[4pt, border: 1px solid blue;]{\color{blue}{V_{\text{dodecahedron}}}=\color{red}{\frac{(15+7\sqrt{5})a^3}{4}}\color{purple}{\approx 7.663118961 \space a^3}}$$ (Note: For derivation & detailed explanation, kindly go through HCR's formula for platonic solids )

The volume of a truncated icosahedron, having 12 congruent regular pentagonal faces & 20 congruent regular hexagonal faces each with edge length $a$, is given by the generalized expression $$\bbox[4pt, border: 1px solid blue;]{\color{blue}{V_{\text{truncated icosahedron}}}=\color{red}{\frac{(125+43\sqrt{5})a^3}{4}}\color{purple}{\approx55.28773076\space a^3}}$$ (Note: For derivation & detailed explanation, kindly go through Mathematical analysis of truncated icosahedron by HCR )

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