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Monsky's Theorem

by hyperbrainer on 4/19/25, 9:12 PM with 13 comments

• by ogogmad on 4/19/25, 10:45 PM

  Haven't read the article. But something about this reminds me of Arnold's topological proof of the unsolvability of the quintic (YouTube form: https://www.youtube.com/watch?v=BSHv9Elk1MU ; PDF: https://web.williams.edu/Mathematics/lg5/394/ArnoldQuintic.p...).

  It seems a lot of impossibility theorems - the type that the ancient Greeks would have understood - can be proven using algebraic topology. Perhaps Sperner's lemma can be seen as an algebraic topology theorem? I don't personally know.

• by akoboldfrying on 4/20/25, 1:44 AM

  > To show that detM is non-zero, we can show that its 2-adic valuation is nonzero.

  I think the last word in that sentence should be "finite"?

  Also do I understand correctly that "face" means "maximal line segment"? (I see some other comments discussing this and concluding that "face" means "edge", but to me, an "edge" doesn't permit "intermediate" vertices.)

• by prof-dr-ir on 4/19/25, 10:40 PM

  > no face of P, nor any face of one of the Ti, contains vertices of all three colors

  That should be 'edge', not 'face', no? Otherwise I do not understand what is happening at all with the examples.

• by bobmcnamara on 4/20/25, 12:00 AM

  Taaaake it to the limit: N=∞, area=0, job done