Constructing $\pi^2$ and e with straightedge and compass Is there a way to construct two curves with lengthratios $\pi^2$, or two areas of ratio $\pi^2$, on a plane surface, with a straightedge and compass?
And is e=2.71... possible?
 A: No. These cannot be constructed.
First, $\pi^2$ and $e$ cannot be constructed because they are transcendental numbers. [We say $t$ is a transcendental number if there is no non-zero polynomial $p(x) = a_nx^n+\cdots+a_1x+a_0$ with integer coefficients $a_i$ such that $p(t)=0$.] Any constructable number is algebraic (i.e. not transcendental). [Note: All constructable numbers are algebraic, but not all algebraic numbers are constructable.]
As for a ratio of two lengths, if we construct a circle of radius $r$ and line of length $\ell$ (this can only be done if $r$ and $\ell$ are constructable numbers). Then the length (i.e. circumference) of the circle is $2\pi r$. So the ratio of these lengths is $2 \pi r /\ell$. Next, $2r/\ell$ is constructable (since $r$ and $\ell$ are). Thus we can construct curves whose length's ratios are of the form $c\pi$ or $c/\pi$ where $c$ is a constructable number.
The ratio of the lengths of 2 lines or 2 circles will just be a constructable number.
Since $\pi^2$ is not of the form $c$, $c\pi$, or $c/\pi$ for any constructable number $c$ (since $\pi$ itself is not constructable), we cannot construct curves whose length's ratio is $\pi^2$.
A similar argument will rule out $\pi^2$ appearing as an area (areas of constructable rectangles are constructable numbers and areas of constructable circles are $\pi r^2$ for some constructable number $r$).
Edit: Ok. There are some gaps. I hadn't considered more complicated regions. I'm not sure exactly what one could cook up in terms of areas. But I don't think it'll be anything more complicated than $a+b\pi$ for constructable numbers $a$ and $b$ (which would preclude $\pi^2$). 
A: As stated before, $e$ cannot be constructed with straightedge and compass. 
It can however be constructed by using other tools, as a sliding object (model train), two telescopes and a wheel.
Place a model train of unit length on the $x$-axis, at the interval $[-1,0]$ (see image below). 

Mount a telescope on the left end of the train, such that this telescope can rotate in the $xy$-plane.
Mount on the end of the telescope a wheel, of which the riding direction is parallel to this telescope. 
Mount a second telescope on the front of the train, which has a fixed, orthogonal  orientation (parallel along the y-axis). Connect that telescope to the axis of the wheel as well. Together these telescopes will assure that the axis of this wheel always has the same $x$-coordinate as the front of the train.    
Now by sliding the train over the $x$-axis, the path of the wheel forms the function $f(x) = e^x$. 

Its distance to the $x$-axis when $x = 1$ is equal to $e$.
