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I was wondering whether, if I add string and thumbtacks to my geometry kit, I am able to do any new constructions. The idea being, with string, I can draw ellipses for instance and the intersection of these with lines, circles and other ellipses might produce points which aren't constructable from compass and straightedge alone. But then again, maybe not? This is why I'm asking. Thanks!

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up vote 8 down vote accepted

Even if we stick to using our string to draw ellipses, we get beyond Euclidean constructibility. Almost anything sensible that one tries yields intersection points that are not Euclidean constructible. For fun, let's use the string to solve a famous problem, the Duplication of the Cube. The use of conic sections, at least the parabola and hyperbola, for duplicating the cube goes back to Greek times.

We are assuming that for Euclidean constructions we are initially given two points $A$ and $B$. We identify $A$ with the origin, $B$ with $(1,0)$, and draw axes as usual. Then the two ellipses with equations $$2x^2-2x+y^2+2\sqrt{2}y+1=0\qquad\text{and}\qquad 3x^2-2x+y^2 +(2\sqrt{2}+1)y+\sqrt{2}=0$$ are string constructible. Subtract. We obtain $y=-(x^2+\sqrt{2}-1)$. Substitute for $y$ in the first equation, and simplify. We get $x^4-2x=0$. So one of the intersection points has $x$-coordinate $\sqrt[3]{2}$, and we have duplicated the cube.

Remark: I think I can make $4x^3-3x-\cos\theta=0$ pop out of the intersection of two string constructible ellipses, where $\cos\theta$ is a constructible number, through the kind of fooling around that produced $x^3-2=0$.

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Thanks, this is very neat. –  Mike Mar 17 '12 at 6:17
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Cartesian ovals, which were the object of Maxwell's first paper at 14, are quartic plane curves definable by string and thumbtacks. They do not seem to be Euclidean curves, but I'm not sure. See a picture and another one belowenter image description here.

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