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Old 06-06-2016, 05:49 PM
panache45 panache45 is offline
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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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Old 06-06-2016, 05:52 PM
Nancarrow Nancarrow is offline
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A heptagon (seven sides).
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Old 06-06-2016, 05:59 PM
panache45 panache45 is offline
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Yes, of course. I knew you could get close, but not precisely.
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Old 06-06-2016, 06:20 PM
Giles Giles is offline
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The complete list would start with:
7 sides
9 sides
11 sides
13 sides
14 sides
15 sides
18 sides
19 sides
21 sides
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Old 06-06-2016, 06:46 PM
markn+ markn+ is offline
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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
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Old 06-06-2016, 08:04 PM
Chronos Chronos is offline
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15 sides is easily doable, since both 5 and 3 are constructible, and are relatively prime.
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Old 06-06-2016, 08:57 PM
Giles Giles is offline
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Originally Posted by Chronos View Post
15 sides is easily doable, since both 5 and 3 are constructible, and are relatively prime.
Of course you are right -- my mistake.
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Old 06-06-2016, 09:00 PM
Lumpy Lumpy is offline
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Of course you can construct n-sided polygons with a ruled straightedge; I wonder why the Greek geometrists considered that cheating.
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Old 06-06-2016, 09:15 PM
Exapno Mapcase Exapno Mapcase is offline
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Originally Posted by Lumpy View Post
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.
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Old 06-06-2016, 10:38 PM
Chronos Chronos is offline
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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.
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Old 06-06-2016, 10:53 PM
MikeS MikeS is offline
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Originally Posted by Exapno Mapcase View Post
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.
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Old 06-07-2016, 12:28 AM
Nava Nava is offline
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Originally Posted by markn+ View Post
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.
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Old 06-07-2016, 06:41 AM
Chronos Chronos is offline
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Yup, 1 is a power of 2.
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Old 06-07-2016, 11:37 AM
markn+ markn+ is offline
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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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