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  • Given a genus-zero polyhedron (e.g., PolyhedronData["Dodecahedron"]) and a point $P$ anywhere on its surface, color all points on the polyhedron by the length of the minimum-length path from $P$.
  • For two specified points, $P_1$ and $P_2$, display that path on the polyhedron.

Optimizing over all possible paths between two points has proven both mathematically and Mathematically difficult--even for a cube! (I cannot find scholarly papers in the mathematical literature that solve this problem, in general.)

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    $\begingroup$ Presumably, $P$ does not have to be a vertex? Is the polyhedron always genus 0, or can one expect holes (e.g. the Szilassi or Császár polyhedra)? $\endgroup$ Commented Apr 18, 2020 at 16:16
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    $\begingroup$ I wonder if the methods in this question become applicable after a suitable triangularization of the given polyhedron? $\endgroup$ Commented Apr 18, 2020 at 16:22
  • $\begingroup$ Excellent suggestion! Before posting I searched but didn't find anything, including that valuable page. $\endgroup$ Commented Apr 18, 2020 at 16:56
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    $\begingroup$ also related: Minimum path on a rectangular prism $\endgroup$
    – kglr
    Commented Apr 18, 2020 at 18:05

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