There are many diverse proofs of the Pythagorean theorem, which says something non-trivial about the diagonal of the standard square. Its length may be approximated by the convergents $1, \frac{3}{2}, \frac{7}{5}, \frac{17}{12}, \frac{41}{29},\cdots$ of the regular continued fraction $\sqrt{2}=[1;2,2,2,2,2\ldots]$.

In algebraic topology and in homological algebra the diagonal of a square has to be approximated by a natural chain map $$D_X\colon C(X)\to C(X)\otimes C(X).$$ For a singular $n$-cube $f\colon I^n\to X$ it is given by the magic formula of Alexander-Whitney $$D_X(f)=\sum_{H}\rho_{H, K}\; A_H(f) \otimes B_K(f)$$ where $A_H$ and $B_K$ denote generalized face operators. The summation in this formula is over all ordered subsets $H$ of the integers $\{1,2,\ldots, n\}$ and $K$ is the complementary subset to $H$. The sign $\rho_{H,K}$ is the signum of the permutation $H K$. (c.f. Massey: A Basic Course in Algebraic Topology, chapter XI.5, p. 287)

This cubicial diagonal-approximation has a lot of interesting consequences. For example the Eilenberg-Zilber theorem, cup-products, Steenrod-operations etc.

My question: Is it reasonable to expect a relationship between the two types of approximations or a proof of Pythagoras's theorem using methods from algebraic topology or homological algebra?


closed as off-topic by Yemon Choi, Lee Mosher, Jan-Christoph Schlage-Puchta, Stefan Kohl, Neil Strickland Apr 3 '17 at 10:41

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    $\begingroup$ I do not see why "approximation" in the sense of Alexander-Whitney would have any connection to the fact that the length of the diagonal of a square in the Euclidean metric is $\sqrt{2}$. $\endgroup$ – Yemon Choi Apr 3 '17 at 3:32
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    $\begingroup$ I'm voting to close this question because I cannot see anything more than speculation based on verbal coincidence $\endgroup$ – Yemon Choi Apr 3 '17 at 3:33
  • $\begingroup$ Can I consider this comment as your answer to my question? $\endgroup$ – Frank Apr 4 '17 at 16:52
  • $\begingroup$ Sure, if you wish. $\endgroup$ – Yemon Choi Apr 4 '17 at 21:03