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Another point to consider is whether "Über den Rauminhalt" is in fact Dehn's first solution to Hilbert's 3rd Problem. I believe his first solution was in the paper "Über raumgleiche Polyeder" in the N …

By Nielsen-Schreier, the subgroup $F$ of $F_2$ generated by $x$ and $y$ is free. Since $x$ and $y$ do not commute, $F$ is not the free group of rank 1, so it must contain a free group of rank 2

I think the question is also answered positively by the main result in a paper of mine -- Efficient computation in groups and simplicial complexes. Trans. Amer. Math. Soc. 276 (1983), no. 2, 715–727 …

OK, the title is opinionated and contentious, but I have a definite question. I know that the title refers to the Bourbaki volume Groupes et Algèbres de Lie (Chapters 4-6), published in 1968, but …

For a really gentle introduction I would recommend Kolmogorov and Fomin's Introductory Real Analysis, available as a Dover paperback. They have a nice introduction to distributions as "generalized fu …

As Andrew L has mentioned, Hartshorne's book Geometry: Euclid and Beyond is very good. If you really want a comprehensive book (for yourself, not for the children you teach) then Hartshorne is as com …

It's interesting that the coprimality of Fermat numbers was already known in Goldbach's time. The reason for attributing the proof to Polya is presumably that such a proof is indicated as an exercise …

I can't improve on the list in Joseph O'Rourke's answer, but I'd like to mention that Winning Ways gets on the list because of its discussion of Conway's "Life" cellular automaton. In particular Winni …

Here are a few: Girolamo Cardano: The Book of My Life. (trans. by Jean Stoner. New York: New York Review of Books, 2002) Norbert Wiener's two volumes Ex-Prodigy: My Childhood and Youth. (MIT Press 195 …

I've hesitated to attempt an answer to this question because I do not know about shape parameters. However, in the hope that what is really wanted is a history of the cross-ratio, here goes. The cro …

The technique for constructing groups with unsolvable word problems applies more generally to construct groups that "simulate'' Turing machines. So, if a Turing machine halts for a recursive set of in …

I'm reluctant to advertise, but since no one else has answered yet, I'll mention the proof on pp. 142--144 of my book Classical Topology and Combinatorial Group Theory.

I'm pleased to hear that some MOers like my book, but I have to say that I think it has too much math for a class of non-science majors. At best, you might mine it for some homework problems because o …

I think it unlikely that anyone ever proposed a weaker Church's thesis, because, as Tim Chow points out, diagonalization was known (and known to be constructive) before anyone ever contemplated a def …