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Platonic Solids

  1. Jul 1, 2010 #1
    What is the easiest way to see that, in a regular polyhedron, the sum of the face angles about each vertex is less than 2π? This seems elementary, but I truly have very little background in geometry. Preliminary searches on PF and Google/Wikipedia didn't turn up anything substantial aside from hints that there are more general results regarding angle defects. I am looking for both some intuitive reasoning and some rigor, if possible. Please go slowly, as I am completely lacking in 3D visualization and geometric rigor. Thank you all in advance!
  2. jcsd
  3. Jul 1, 2010 #2
    You can study the http://en.wikipedia.org/wiki/Net_(polyhedron)" [Broken] of the polyhedron and add the angles around each vertex.

    Or you can imagine the tip of an infinitely obtuse/flat otherwise regular tetrahedron. It's so flat you can think of it as an equilateral triangle made up of three smaller equilateral triangles. It's trivial to see that the angles around the middle vertex are (almost) 360°. It is also trivial to extend this observation for whatever base of pyramid. This is the maximum sum of angles we can have around any vertex of a convex polyhedra.

    We also know that a cone's net is two circles that don't overlap and are tangent, with one of the circles missing an arc:


    We can see that this is decidedly less than 360° if the cone has any height to it at all.

    Of course, this isn't true with concave polyhedra.
    Last edited by a moderator: May 4, 2017
  4. Jul 1, 2010 #3
    Thank you for the answer -- I had never encountered nets before. They are an interesting concept. Your second suggestion, about using the limiting case, I understand, except what is the reasoning that lifting the height decreases the combined face angles? My visualization is horrible. This is related to the problem I have with nets; why can we not have a net that has overlap in the plane suddenly not overlap when we fold it? I can see why this is true if we fold all the faces in the same direction, but what is to say there is not some complicated way of folding it that somehow creates more space for the angles?

    (EDIT: This previous question, upon further thought, might be easily explained away, which would help convince me. A follow-up would then be whether there is a proof, without knowing the 5 Platonic solids, that every regular polyhedron has a net.)

    Finally, I do like the idea of the cone; does the face angle stay the same if we project it slightly outward onto the "circumscribing" cone? I'm not sure.

    I like all of these suggestions and will feel a lot more comfortable using them if I can find one explanation, maybe an elaboration of one of these existing three, that can fully convince me. That's never an easy task :). Is there also a connection between the solid angle at a vertex and the sum of the face angles -- maybe one that can mathematically justify this? And what exactly is it about convexity that prevents the above arguments from working?

    One additional piece of curiosity. Coxeter is using this fact to prove that there are only 5 Platonic solids (the same proof is outlined as the classical method on Wikipedia). He mentions that this is a "familiar theorem." Does it have a name?
    Last edited: Jul 1, 2010
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