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.
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.)
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.
prof-dr-ir•9mo ago
That should be 'edge', not 'face', no? Otherwise I do not understand what is happening at all with the examples.
erooke•9mo ago
dmurray•9mo ago
FabHK•9mo ago
See the bottom "face" of the top centre triangle in the 4 examples.
hswanson•9mo ago
drewcoo•9mo ago
https://en.wikipedia.org/wiki/Planar_graph#Euler's_formula