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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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    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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    Oh , sorry for not clarifying that , but however it does not matter cause the question is closed now.2012-10-17