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.
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.
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.
> 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.)
> 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.
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.
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!
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.
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.
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.
> 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.)
> 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.
Yep, you're right on both counts; I've updated the page with those corrections. Thanks!
> 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.
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.
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.
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!
They're dealing with planar graphs, graphs embedded in planes. Faces are not edges.
https://en.wikipedia.org/wiki/Planar_graph#Euler's_formula
Pretty sure they meant the word face, that would be the generic term for edge. (An edge being a 1 dimensional face)
Taaaake it to the limit: N=∞, area=0, job done