Timeline for Defining Euler's number via elementary euclidean geometry (and a dimension limit)
Current License: CC BY-SA 3.0
16 events
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| Jun 4, 2019 at 21:35 | answer | added | Atton Rand | timeline score: 1 | |
| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
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| Dec 26, 2016 at 12:39 | comment | added | B K | I just wrote down the formulas for $vol\; Q_n$ and $vol\; C_n$ and then evaluated the limit of their quotient using one of the standard definitions of $e$. If I have more time at some point I can include the computations in the question. | |
| Dec 26, 2016 at 1:28 | comment | added | user78249 | Just out of curiousity, how did you prove the limit was $e^4$? Quite frankly this seems like the best answer to the question you asked. | |
| Dec 25, 2016 at 5:37 | answer | added | Job Bouwman | timeline score: 5 | |
| Oct 17, 2015 at 16:12 | answer | added | Manfred Weis | timeline score: 4 | |
| Oct 17, 2015 at 6:34 | comment | added | Dan Romik | ... the infinite series definition, the limit $\lim_{n\to\infty} (1+1/n)^n$, and Stirling's formula. The one place I can think of where $e$ appears in a truly exotic and nontrivial way is in the number theoretic formula $e=\lim_{n\to\infty} (\textrm{lcm}(1,2,\ldots,n))^{1/n}$ that is one of the equivalent forms of the prime number theorem. So, I completely agree with @BK's motivation for the question: it would be extremely interesting to see some less familiar characterizations of $e$, if there even are any. | |
| Oct 17, 2015 at 6:34 | comment | added | Dan Romik | This question is a lot more interesting than I realized at first. It reminded me of an insight I had many years ago but had completely forgotten about, which is that $e$ is a lot less versatile of a mathematical constant than $\pi$. That is, $e$ doesn't actually appear naturally in mathematics in many different ways. When it does come up (e.g. the Poisson distribution in probability theory, and the various formulas one finds on Wikipedia and in Finch's book Mathematical Constant) this can almost always be traced back to two or three fundamental properties: ... | |
| Oct 16, 2015 at 9:52 | comment | added | B K | The difference between this question and the related ones at Math.SE is that their authors were more or less satisfied with an answer equivalent to yours, but I am not ;-) | |
| Oct 16, 2015 at 9:50 | history | edited | B K | CC BY-SA 3.0 |
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| Oct 16, 2015 at 7:45 | comment | added | Dan Romik | Ah, history repeats itself. Possible duplicate on Math.SE: "Is there any geometric way to characterize $e$?". But I dare say my figure is nicer than the one in that question... | |
| Oct 15, 2015 at 23:03 | answer | added | Todd Trimble | timeline score: 6 | |
| Oct 15, 2015 at 21:57 | comment | added | Gerry Myerson | More simply, the volume of the unit ball in $n$-space is asymptotic to $${1\over\sqrt{n\pi}}\left({2\pi e\over n}\right)^{n/2}$$ | |
| Oct 15, 2015 at 21:45 | history | edited | B K | CC BY-SA 3.0 |
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| Oct 15, 2015 at 20:50 | answer | added | Dan Romik | timeline score: 13 | |
| Oct 15, 2015 at 20:19 | history | asked | B K | CC BY-SA 3.0 |