Is the group of symmetries of the rectangle-not-square related to the Klein bottle mathematically?
The reason I am asking is because I want to put a Klein bottle coffee cup in a joke about V_4 and wonder if this would be a good idea.
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$\begingroup$ Is it an obvious yes? Am I missing something? Or is it an obvious no it's just the guy? $\endgroup$– Erin CarmodyCommented Mar 6, 2023 at 0:53
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$\begingroup$ Ok sorry I got the answer from Twitter that these are two different people. Still... $\endgroup$– Erin CarmodyCommented Mar 6, 2023 at 0:55
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$\begingroup$ Asking whether two mathematical objects "are connected" is not a mathematical question. $\endgroup$– Daniel AsimovCommented Mar 6, 2023 at 17:13
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$\begingroup$ Daniel, I admit I could have solved my question with a Google search, but it is not unmathematical to ask if two mathematical objects are connected. A lot of math is about connecting mathematics. $\endgroup$– Erin CarmodyCommented Mar 8, 2023 at 0:54
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$\begingroup$ I don't this is a particularly great question, since it is relatively easy to answer with some online searching and "asking Twitter", but I admit there is precedent: mathoverflow.net/questions/227995/… $\endgroup$– Yemon ChoiCommented Mar 8, 2023 at 10:49
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1 Answer
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No, according to wikipedia (bottle, group) they're both named after Felix Klein, but appeared in different papers on completely different topics. The bottle comes from his notes from 1882 and the group in Lectures on the icosahedron and the solution of equations of the fifth degree.
Klein group is also a bit of a pun, since Klein means small in German.
answered Mar 6, 2023 at 1:04
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$\begingroup$ Since they are in different papers does that mean they are not connected? $\endgroup$ Commented Mar 6, 2023 at 2:47
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1$\begingroup$ You can see from the topics of the linked papers. One is about Galois theory, the other about topology. $\endgroup$ Commented Mar 6, 2023 at 2:53
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$\begingroup$ Ah nice and I heard it was Vierergruppe for 4. But still this only gives evidence that they might be connected. $\endgroup$ Commented Mar 6, 2023 at 2:54
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$\begingroup$ Oh yes one must never mix topology and algebra. $\endgroup$ Commented Mar 6, 2023 at 2:55
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2$\begingroup$ I mean I linked the sources you can see for yourself whether either of these has such a mix... $\endgroup$ Commented Mar 6, 2023 at 3:11