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Given a fixed volume and fixed surface area I would like to construct polyhedra that minimize the total length of the edges. This seems like a straight-forward problem to solve by brute force for reasonably small number of vertices, but I imagine this has already been done, or at least considered.

Can anybody think of a source for such structures?

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This has been considered, see, for example, this report.

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    $\begingroup$ From Igor's link: [B01] S. Berger, Edge Length Minimizing Polyhedra, Thesis, Rice University, (2001) $\endgroup$ – Joseph O'Rourke Dec 7 '13 at 21:54
  • $\begingroup$ Thanks for the citation, but the thesis looks at either fixed area or fixed volume. A more concrete example of the question would be: find a shape that encloses 1m^3 of volume, using 5m^2 of flat material, s.t. the total length of the seams is minimized. $\endgroup$ – Rodrigo Dec 8 '13 at 10:47

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