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Is there any geometric way to characterize $e$?

We know that the length of perimeter of a circle of unit diameter is $\pi$ ; is there a similar geometric interpretation of $e$ , without invoking complex numbers, in terms of lenth ( and not area )?

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    Does this answer your question? http://math.stackexchange.com/questions/159707/is-there-any-geometric-way-to-characterize-e/159711#1597112012-10-11
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    @ Qiaochu Yuan:- Ah! I get it , it's still an open problem.2012-10-11
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    geometric form of e https://geomathry.wordpress.com/2017-09-14

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Do a step of length $1$, then a step of half the previous one, then one third the previous one, then one forth the previous one,...

How far do you get?

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    That's quite a good way of converting the definition of e into a geometrical construction , but the process does not end after a finite terms.2012-10-15
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    ... which was not requested.2012-10-15
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    Oh , sorry for not clarifying that , but however it does not matter cause the question is closed now.2012-10-17