Monsky's Theorem

https://mathmondays.com/monskys-theorem

49•hyperbrainer•9mo ago

Comments

prof-dr-ir•9mo 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•9mo ago

Pretty sure they meant the word face, that would be the generic term for edge. (An edge being a 1 dimensional face)

dmurray•9mo ago

Yes, 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•9mo ago

Maybe 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•9mo ago

Yeah, 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•9mo ago

They're dealing with planar graphs, graphs embedded in planes. Faces are not edges.

https://en.wikipedia.org/wiki/Planar_graph#Euler's_formula

ogogmad•9mo ago

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.

PollardsRho•9mo ago

Thanks 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•9mo ago

Sperner 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.

bobmcnamara•9mo ago

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

akoboldfrying•9mo 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•9mo 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•9mo ago

Yep, you're right on both counts; I've updated the page with those corrections. Thanks!

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