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What are the constructible angles ?

Wikipidia sais:

The only angles of finite order that may be constructed starting with two points are those whose order is either a power of two, or a product of a power of two and a set of distinct Fermat primes.

I don't understand the exact meaning of this, does it say that an angle is constructible if and only if it is a power of two or a product of a power and $?$ (this part I didn't understand either)

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That is the theorem, yes, if and only if. A less fancy way of putting it is for $n$ a positive integer, the angle $2\pi/n$ is constructible iff $n$ is a power of $2$ (possibly equal to $1$) times a product (possibly empty) of distinct Fermat primes. The order stuff generalizes this in an unimportant way. –  André Nicolas Apr 20 '12 at 14:09
    
@AndréNicolas not really. The angle 98 degrees = 6π/10 is constructible, but your way doesn't include it. –  Ekuurh Apr 20 '12 at 14:18
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@Ekuurh: $6\pi/10$ is $108^\circ$... –  J. M. Apr 20 '12 at 14:30
    
@Ekuurh: Our disagreement would be over whether going beyond the constructible polygons is an important generalization. –  André Nicolas Apr 20 '12 at 14:44
    
@J.M. Honest mistake :) Yea, I think the set of all constructible angles should be classified. –  Ekuurh Apr 20 '12 at 18:12
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1 Answer

up vote 1 down vote accepted

It means an angle is constructible if and only if its order is either a power of two, or a power of two times a set of Fermat primes. For example, 10 = 2*5, and 2 is a power of two and 5 is a fermat prime, thus you can make an angle of 360/10 = 36 degrees. but 40 degrees = 360/9 cannot be constructed, because 9=3*3, and 3, 3 are not distinct.

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This means that $2^0$ is ok in both cases ? –  Belgi Apr 20 '12 at 14:22
    
Sure, $2^0$ is fine, $0$ is a good number, perhaps a little too unassuming. –  André Nicolas Apr 20 '12 at 14:46
    
@Belgi Yea, it's fine, and by accepting that 1 is a power of 2, we get that the constructible angles are exactly the angles whose order is a power of two times a set of distinct Fermat primes. –  Ekuurh Apr 20 '12 at 18:14
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