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Using a calculator we can easily check that $$\color{Green}{e^{\pi}-\pi}=19.999\cdots\color{Green}{\approx 20}$$ This article and this one provides some details about this almost near identity, but no explanation. My question is :

Can we obtain the above almost equality geometrically?

Add: Let me add more details to address the issues pointed some people in the comments section. As I think both numbers $\pi$ and $e$ are very geometric and their appearance in some unexpected places is due to this geometric nature. Also, here by geometry, I do not mean just Euclidean geometry (or ruler and compass construction), but in the inclusion sense.

Secondly, consider the graphs of functions $y=e^x$ and $y=x+20.$ We can compute their exact intersection points of them using Lambert W function, and approximately they are $x=-19.99999\cdots$ and $x=3.1416\cdots.$ One of these numbers almost indistinguishable from $-20$ and the other is very close to $\pi.$ How can we explain this unreasonable result?

Bumblebee
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    The first source you quoted states but no satisfying explanation as to "why" $e^{\pi}-\pi \approx 20$ is true has yet been discovered". Are you asking there are any progress in this "open" problem or what? – achille hui Oct 06 '15 at 10:13
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    Why on earth should you expect a geometric explanation of the fact that $\sum_{j=2}^{\infty} \frac{\pi^{j}}{j!}$ is close to $19$ (it's not that close in any case)? – Geoff Robinson Oct 06 '15 at 11:27
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    This answer shows a geometric interpretation for the terms of the power series for sine and cosine; consequently, $e^\theta$ is simply the total length of the polygonal spiral $P_0P_1P_2P_3\dots$. I don't see an obvious connection between the $e^\pi$ spiral and $20$, but that could just be a failure of imagination on my part. It might be worth noting that the not-very-good approximation $\sqrt{2}+\sqrt{3}\approx \pi$ has a geometric "explanation", so this question isn't completely unreasonable. – Blue Oct 06 '15 at 15:56
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    The point for that $\sqrt2+\sqrt3$ is that both of them are "constructible" (is that a word?). Personally I don't know a geometrical approximation of $e$, let alone $e^\pi$. – Quang Hoang Oct 07 '15 at 06:19
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    Seems unlikely that a pure geometric solution can be found, as there is no ruler and compass construction for the exponential. The number $e$ itself doesn't seem to mate well with geometry. (And by the way, we have no explanation at all regarding the quasi-equality.) –  Oct 07 '15 at 06:28
  • @YvesDaoust: That's true. But I hope there should be an approach using trigonometry, complex numbers or in another form. – Bumblebee Oct 07 '15 at 06:31
  • @Nilan: you should rephrase your question, focusing on geometry caused its closing. –  Oct 07 '15 at 06:32
  • @YvesDaoust "you should rephrase your question, focusing on geometry caused its closing." I have mixed feelings about your comment. I agree that it caused the query to be closed; and I certainly feel that the question is overbroad. However, the core of the question is its focus on geometry and I (for one) would have trouble identifying a (narrow) sub-topic in geometry for the question to focus on. – user2661923 Oct 29 '20 at 17:49
  • @user2661923: what makes you think that $e$ is tractable by geometry ? (and plays an important role ???) –  Oct 29 '20 at 17:56
  • @YvesDaoust: $e$ is the unique real number such that the tangent to the graph (of the exponential) $y=e^x$ at $x=0$ is $1.$ To me it is a very geometric property. – Bumblebee Oct 29 '20 at 17:58
  • @YvesDaoust Actually, being totally ignorant in the field, it wouldn't surprise me if $e$ was or was not tractable by geometry. However, consider the question from the OP's perspective. He was not asserting that there is a relation between $e$ and geometry. Instead, he was asking if one existed. Seems like a very reasonable and interesting question to ask. – user2661923 Oct 29 '20 at 17:59
  • @user2661923: are you kidding ? "Also we know that both $\pi$ and $e$ plays an important role in geometry." Do you deny that the OP wrote this ? –  Oct 29 '20 at 18:04
  • @YvesDaoust Very good point. I blind-spot glossed over that portion of his query, focusing only on the question in bold print. – user2661923 Oct 29 '20 at 18:08
  • @user2661923: $\pi$ cannot be constructed by compass and ruler. I have no idea of how $e$ could be constructed, and even less $e^\pi$. And showing that a quantity is a good approximation of another "geometrically" does not seem obvious. The OP is throwing a random request with no serious consideration of feasibility and wrong assumptions. I fail to see in what way this question would be interesting. –  Oct 29 '20 at 18:14
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    @YvesDaoust: I don't see why the word "geometry" means just "compass and ruler type constructions" to most people. I add few more lines to my question to make it clear. – Bumblebee Oct 29 '20 at 18:41
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    I'm not an expert in this area but, perhaps you can get some inspiration From Heegner numbers and Ramanujan's constant, which have algebraic reasons for why they're close to integers: https://en.wikipedia.org/wiki/Heegner_number – Alex R. Oct 29 '20 at 18:46
  • @Bumblebee: the Lambert function belongs to calculus. It is a wrong point of view to consider that the graph of a transcendental function is "geometry". –  Oct 29 '20 at 18:46
  • I see no reason to expect any special meaning behind this... strange occurrences happen all the time and it is human nature to focus on those strange occurrences and attempt to assign meaning to them... however there are flukes and coincidences all over the place. Had this one not occurred then you would be asking about whatever other coincidence you had come across was, like some $\varphi^{\sqrt{2}}\approx 2$. If there was strict equality, then perhaps I would be inclined to believe there to be some tangible reason/connection, but merely approximately? Nah... – JMoravitz Oct 29 '20 at 18:49
  • @JMoravitz: take a few mathematical constants, a few naturals, and form expressions with the arithmetic operations. The probability of "interesting" coincidences will increase with the number of constants and length of the expressions. A dumb computer program will find them. All these are indeed pseudocoincidences. You can find as many as you want, and as accurate as you want. –  Oct 29 '20 at 18:54
  • @YvesDaoust: So you don't see things like differential geometry as a part of geometry? – Bumblebee Oct 29 '20 at 18:58
  • @Bumblebee: no, as applied calculus. I am not even sure that it requires the Euclidean axiomatics. –  Oct 29 '20 at 18:59
  • @JMoravitz: Yes, it is true that equalities are more interesting than approximation. But it is wrong to think that almost equalities are not important. There is a whole story and very interesting mathematics behind Ramanujan Constant. See here and here – Bumblebee Oct 29 '20 at 19:03
  • @YvesDaoust: So, what are your boundaries of "Geometry"? BTW according to modern standard there are holes in Euclid's presentation, and peoples now use either Hilbert's axioms or Birkhoff's axioms, though I am not familiar with theses formulations. – Bumblebee Oct 29 '20 at 19:09
  • @Bumblebee: sterile discussion. Next time enter an explicit question. –  Oct 29 '20 at 19:11
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    It looks like this question is a duplicate: https://math.stackexchange.com/questions/724872/why-is-e-pi-pi-so-close-to-20 – Alex R. Oct 29 '20 at 22:10

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