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I Why can't a (5,5,4) solid exist?

  1. May 17, 2016 #1
    So i was reading my geometry textbook and it said "and obviously there can be no solid where every vertex has a (5,5,4) arrangement"; unfortunately, this is not an obvious fact to me. Can somebody explain to me what makes this so obvious? This statement was the end of a proof by contradiction so i feel I should really understand why it's true.
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  3. May 17, 2016 #2


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  4. May 19, 2016 #3


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    This statement from the wikipedia article sums it up:

    Specifically p.q.r implies that a p-gon is surrounded by alternating a q-gons and r-gons, so either p is even orq equals r.

    If something is uniform 5.5.4 then focus on a pentagon. Each edge alternates being a pentagon or a square, but when you go all the way around the pentagon you will get two squares (or two pentagons) abutting each other.
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