what figures that are common knowledge have perimeter equal to $e$ in a similar manner like circle have $\pi$? they can be smooth im not interested in answers like the triangle of $\frac{e}{3}$ because i want to define $e$.
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@DietrichBurde first it is not about a perimeter. bu see that the $1$ circles area equals $\frac{\pi}{4}$. what are the names of that figures? – test Sep 20 '21 at 11:33
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1You can also take the perimeter instead the area for an geometric figure under a function like $f(x)=\frac{1}{x}$. But basically the duplicate is saying that this is not "defining $e$", but rather "recognizing" it. I think the answer to your request is negative. The usual definitions for $e$ are much better. (Or at least, you should use hyperbolic geometry instead of Euclidean one.) – Dietrich Burde Sep 20 '21 at 11:38
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1It is conjectured that $e$ is not a "period". That is, $e$ is not an integral of an algebraic function over an algebraic domain. So your figure (if it exists) must be transcendental. – GEdgar Sep 20 '21 at 12:20