4 ms·
Monsky's Theorem
- bobmcnamara 1y agoTaaaake it to the limit: N=∞, area=0, job done
- prof-dr-ir 1y ago> 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.
- erooke 1y agoPretty sure they meant the word face, that would be the generic term for edge. (An edge being a 1 dimensional face)
- dmurray 1y agoYes, this would more normally be called "edge". It's not incorrect to call it a face, by analogy with higher-dimensional solids, but confusing.
- FabHK 1y agoMaybe they called it "face" and not "edge" because an edge is normally understood to be what's between two vertices (of a graph; so an edge has two vertices, beginning and end), while here "face" is what's between two corners of a given triangle (so a face can contain more than two vertices, and so multiple edges). See the bottom "face" of the top centre triangle in the 4 examples.
- hswanson 1y agoYeah, that's exactly it! "Face" is the term used in the original paper, so that's what I was using. I've updated the page to make the distinction clearer. Thanks!
- drewcoo 1y agoThey're dealing with planar graphs, graphs embedded in planes. Faces are not edges. https://en.wikipedia.org/wiki/Planar_graph#Euler's_formula https://en.wikipedia.org/wiki/Planar_graph#Euler's_formula
- ogogmad 1y agoHaven'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 https://www.youtube.com/watch?v=BSHv9Elk1MU ; PDF: https://web.williams.edu/Mathematics/lg5/394/ArnoldQuintic.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.
- PollardsRho 1y agoThanks for sharing this proof! As someone who enjoys math but never got myself through enough Galois theory to finish the standard proof, it's fantastic to see a proof that's more elementary while still giving a sense of why the group structure is important.
- xyzzyz 1y agoSperner lemma is very much an algebraic topology theorem. The ideas involved in it form the basis for the theory of simplicial homology, which in turn will lead you to general homology and cohomology theories.
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- akoboldfrying 1y ago> 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.)
- erooke 1y ago> Also do I understand correctly that "face" means "maximal line segment"? In the statement of Sperners lemma this seems to be how he means it. You have a triangle who's faces have been subdivided. The face he is referring to is the face before subdivision I think. This lines up with the usual statement I'm familiar with for Sperners lemma which involves triangulating an n-simplex.
- hswanson 1y agoYep, you're right on both counts; I've updated the page with those corrections. Thanks!