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#1
06-06-2016, 05:49 PM
 panache45 Member Join Date: Oct 2000 Location: NE Ohio (the 'burbs) Posts: 39,370
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?
#2
06-06-2016, 05:52 PM
 Nancarrow Guest Join Date: Oct 2004 Posts: 1,697
A heptagon (seven sides).
#3
06-06-2016, 05:59 PM
 panache45 Member Join Date: Oct 2000 Location: NE Ohio (the 'burbs) Posts: 39,370
Yes, of course. I knew you could get close, but not precisely.
#4
06-06-2016, 06:20 PM
 Giles Charter Member Join Date: Apr 2004 Location: Newcastle NSW Posts: 12,834
7 sides
9 sides
11 sides
13 sides
14 sides
15 sides
18 sides
19 sides
21 sides
#5
06-06-2016, 06:46 PM
 markn+ Guest Join Date: Feb 2015 Posts: 1,120
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
06-06-2016, 08:04 PM
 Chronos Charter Member Moderator Join Date: Jan 2000 Location: The Land of Cleves Posts: 73,145
15 sides is easily doable, since both 5 and 3 are constructible, and are relatively prime.
#7
06-06-2016, 08:57 PM
 Giles Charter Member Join Date: Apr 2004 Location: Newcastle NSW Posts: 12,834
Quote:
 Originally Posted by Chronos 15 sides is easily doable, since both 5 and 3 are constructible, and are relatively prime.
Of course you are right -- my mistake.
#8
06-06-2016, 09:00 PM
 Lumpy Charter Member Join Date: Aug 1999 Location: Minneapolis, Minnesota US Posts: 15,629
Of course you can construct n-sided polygons with a ruled straightedge; I wonder why the Greek geometrists considered that cheating.
#9
06-06-2016, 09:15 PM
 Exapno Mapcase Charter Member Join Date: Mar 2002 Location: NY but not NYC Posts: 29,361
Quote:
 Originally Posted by Lumpy Of course you can construct n-sided polygons with a ruled straightedge; I wonder why the Greek geometrists considered that cheating.
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
06-06-2016, 10:38 PM
 Chronos Charter Member Moderator Join Date: Jan 2000 Location: The Land of Cleves Posts: 73,145
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
06-06-2016, 10:53 PM
 MikeS Charter Member Join Date: Oct 2001 Location: New London, CT Posts: 3,752
Quote:
 Originally Posted by Exapno Mapcase 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.
If Wikipedia is to be believed, you can trisect an angle or double a cube using a marked straightedge. You cannot, however, square the circle (which is a problem the Greeks knew of and cared about.) Roughly speaking, doubling the cube becomes possible because the cube root of two is an algebraic number, while π is a transcendental (i.e., non-algebraic) number and so squaring the circle is still impossible.

There's also a list of the regular n-gons that can and cannot be constructed using a marked straightedge here. In particular, it is still impossible to construct a regular hendecagon.
#12
06-07-2016, 12:28 AM
 Nava Guest Join Date: Nov 2004 Location: Hey! I'm located! WOOOOW! Posts: 36,823
Quote:
 Originally Posted by markn+ 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
I don't know a Fermat prime from a hole in the wall, but would that "power of 2" include 20? Because otherwise I don't know how you come up with 3, a number whose regular polygon can be drawn with only a compass.
#13
06-07-2016, 06:41 AM
 Chronos Charter Member Moderator Join Date: Jan 2000 Location: The Land of Cleves Posts: 73,145
Yup, 1 is a power of 2.
#14
06-07-2016, 11:37 AM
 markn+ Guest Join Date: Feb 2015 Posts: 1,120
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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