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I entered $2^{63}$ as a stand alone value at WolframAlpha. Among the responses was a factoid that 'A regular 9223372036854775808-gon is constructible with a straightedge and compass.'

What is such a shape and how can I construct one?

2^63-gon notice

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    @Silicon, that was a cruel joke... :)2012-08-03

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Start with a circle and a diameter. Bisect the diameter and extend the bisector to the circle to make a square. Bisect each right angle at the center of the circle $61$ times, extending the bisector to the circle. You have $2^{63}$ points, equally spaced, around the circle. As Dario says, you won't be able to tell it from the original circle.

The regular polygons constructible with straightedge and compass have the number of sides of the form $2^n$ times a product of zero or more of the Fermat primes: $3, 5, 17, 257, 65537$ (each to only the first power)

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    @celtschk: my errors are random, so they only build up as $\sqrt n$, making about $0.8\%$ at the end. As $2^{63} \approx 9.2\cdot 10^{18}$ this would be $7.4$ parts in $10^{20}$ of the original circle diameter. I challenge you to find it.2012-08-03