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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

  • 20
    Draw a circle with the compass. Label it a $2^{63}$-gon ;)2012-08-03
  • 1
    A $2^{63}$-gon is a polygon with $2^{63}$ equal sides.2012-08-03
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    @PeterTamaroff: and a regular one has equal angles as well.2012-08-03
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    And that is possible to repeat 62 times with a straightedge and paper? Isn't that akin to halving a sheet of paper and halving it again etc.?2012-08-03
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    You should probably try it with a heptagon first, before moving on to the hard stuff.2012-08-03
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    See also: [Can a circle's circumference be divided into arbitrary number of equal parts using straight edge and compass only?](http://math.stackexchange.com/q/57529)2012-08-03
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    This is doable if you have a big enough piece of paper. At 1 inch per side, it'll only take a sheet 7.9 light years wide. Of course, the time to do the construction would be a bit daunting.2012-08-03
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    @Silicon, that was a cruel joke... :)2012-08-03

1 Answers 1

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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)

  • 0
    This will probably be indistinguishable from the way that I usually draw regular $2^{63}$-gons: place the point of the compass and draw a circle.2012-08-03
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    I strongly doubt that you'll have $2^{63}$ equally spaced points after this procedure. Even if you just make an error of only 0.1% per step, you'll have a total error of more than 6% in the last constructed points.2012-08-03
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    @PeterTamaroff: speak for yourself.2012-08-03
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    @robjohn Fair enough.2012-08-03
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    @PeterTamaroff: But an idealist will also be able to distinguish between a circle and a $2^{63}$-gon.2012-08-03
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    @celtschk Maybe some nanometrics should be involved here? You're right.2012-08-03
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    A few beers and a mighty fine compass would improve my accuracy too.2012-08-03
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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