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#1




Constructing a polygon within a given circle
What is the regular polygon with the least number of sides that cannot be constructed within a given circle, using only a straightedge and a compass?

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#2




A heptagon (seven sides).

#3




Yes, of course. I knew you could get close, but not precisely.

#4




The complete list would start with:
7 sides 9 sides 11 sides 13 sides 14 sides 15 sides 18 sides 19 sides 21 sides 


#5




See Constructable Polygon. The only regular polygons that can be constructed are those where the number of sides is a product of a number of distinct Fermat primes multiplied by a power of 2.
Mark 
#6




15 sides is easily doable, since both 5 and 3 are constructible, and are relatively prime.

#7




Of course you are right  my mistake.

#8




Of course you can construct nsided polygons with a ruled straightedge; I wonder why the Greek geometrists considered that cheating.

#9




Is there anything the Greeks cared about that couldn't be constructed with a compass and ruler? If there isn't, that takes most of the game out of the game.



#10




You can make a ruled straightedge easily, using the Greeks' own techniques (Euclid's Proposition VI.9). The problem isn't in putting the marks on the ruler; it's in interpolating between the marks.

#11




Quote:
There's also a list of the regular ngons that can and cannot be constructed using a marked straightedge here. In particular, it is still impossible to construct a regular hendecagon. 
#12




Quote:

#13




Yup, 1 is a power of 2.

#14




Yes, as Chronos says, 1 is a power of 2.
Also, "a number of Fermat primes" includes the case of zero Fermat primes, which is why you can construct a square (4 sides = 2^2, with no primes involved). A Fermat prime is a prime of the form (2^(2^n))+1. There are only 5 known Fermat primes: 3, 5, 17, 257, and 65537. Mark 
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