# Does Alexander-Whitney formula imply Pythagoras theorem? [closed]

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 StricklandApr 3 '17 at 10:41

• This question does not appear to be about research level mathematics within the scope defined in the help center.
If this question can be reworded to fit the rules in the help center, please edit the question.

• 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 $\sqrt{2}$. – Yemon Choi Apr 3 '17 at 3:32
• I'm voting to close this question because I cannot see anything more than speculation based on verbal coincidence – Yemon Choi Apr 3 '17 at 3:33
• Can I consider this comment as your answer to my question? – Frank Apr 4 '17 at 16:52
• Sure, if you wish. – Yemon Choi Apr 4 '17 at 21:03