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Thread: Sides of a dodecahedron

  1. #1
    Aug 2018

    Sides of a dodecahedron


    I'm having a little construction problem:

    Is it possible to trace all edges of a dodecahedron (or alternatively an icosahedron) with one continuing line?
    So imagine a dodecahedron made up of 12 wooden pentagons. To smoothly cover up the cracks between them, I would now
    like to attach a single rope along all the edges, so every edge is covered without cutting the rope. (Not preferrable: but it may
    overlap once or twice if it has to.)

    (Simplification of the problem)

    There has to be a way to solve this problem mathematically, but I can't think of it.

    Thank you very much in advance!
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  2. #2
    MHF Contributor

    Apr 2005

    Re: Sides of a dodecahedron

    This is basically Euler's famous "bridges of Konigsberg" problem. The surface of a dodecahedron consists of 12 faces, each face a pentagon. Three edges meet at each vertex. If you want a single continuous line through all edges then a line through a vertex would take two of those but that leaves one edge either leading in to that vertex without any way to leave or an edge out of a vertex where your line must have started.
    Thanks from topsquark and DanFromBavaria
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