I understand that a straight line of length $\pi$ can't be drawn with a compass and straight edge without neusis.
I'm looking for a nice way to draw a line of length $\pi$ using compass and straight edge, where neusis is allowed.
I saw that you can draw a circle of radius 1, draw a line through its diameter, wrap a string around your circle, mark the string where it crosses the diameter line, and then straighten the string.
However, I'd like to avoid that kind of wrapping/unwrapping if possible, as the materials I'm working with are rigid, hypothetically.
Cheers!
Edit: It's become clear from the great comments that neusis doesn't get you the transcendentals. So I'd also be interested in the following:
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Other techniques which do give the transcendentals, in particular $\pi$, like the rope stretching technique mentioned above.
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Compass and straight edge processes, which when repeated, rapidly approach a length of $\pi$.
Actually I think the comments are enough to set me on the right path, but answers are still welcome. Thanks again!
Best Answer
What I am demonstrating is a construction that gives us pi, approximately, with an error of 0.0046%.
Draw a circle of a known radius, preferably a power of 2 and draw a vertical diameter. I am drawing the circle of radius 2 units.
Now, draw a perpendicular line at point B and cut 3 times of the diameter. I name that point to be F
Now, construct an angle $30^o$ at A as shown and complete the triangle. I am not showing the arcs.
Join F and G and divide the line segments by the diameter. In this case it is 4. So I will bisect the line twice.
That construction does not yield $\pi$. The result is $\sqrt{9+(\frac12+\sqrt{\frac34})^2}$ From this geometric solution you can get $3.141737211$ (closer to $\pi$) but not actually $\pi$. This is to say, you get $0.0046$ % error which is acceptable.