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### Constructible Angles and Regular Polygons

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Date: 04/17/98 at 22:20:26
From: Danny Dawson
Subject: Euclidean Construction of Polygons

I've found that you can create angles (using Euclidean Geometry) that
are multiples of 9 and/or 15 degrees. I've used the construction of
regular polygons to help me, but that's all I can construct. Are there
any other angles that are constructible, and what regular polygons are
constructible? I'd appreciate a prompt respose because I've been
collaborating with a reporter on an article for National Mathematics
week. Thank you.
```

```
Date: 04/20/98 at 11:56:10
From: Doctor Wilkinson
Subject: Re: Euclidean Construction of Polygons

It was very clever of you to notice the connection between
construction of regular polygons and construction of angles.

The problem of which regular polygons can be constructed was solved
completely by Gauss around 1800, although there are still some
mysteries connected with it. The angles you mention correspond to the
regular hexagon and the regular pentagon. But there are other
constructible regular polygons, notably the ones with 15 and 17 sides.

The rule Gauss discovered was that a regular polygon with n sides can
be constructed if and only if n is the product of a power of 2 and of
distinct primes of the form 2^2^k + 1.  For k = 0, 1, 2, 3, 4, this
gives the primes 3, 5, 17, 257, and 65537.  2^2^5 + 1 is not prime,
however, and nobody has ever discovered another prime of this form,
but nobody has ever proved that there aren't any more either.

Looking just at the numbers from 3 to 20, we can build the following
table of which regular polygons can and can't be constructed:

3    yes    3 = 2^2^0 + 1
4    yes    4 = 2^2
5    yes    5 = 2^2^1 + 1
6    yes    6 = 2 * 3
7    no
8    yes    8 = 2^3
9    no
10    yes    10 = 2 * 5
11    no
12    yes    12 = 2^2 * 3
13    no
14    no
15    yes    15 = 3 * 5
16    yes    16 = 2^4
17    yes    17 = 2^2^2 + 1
18    no
19    no
20    yes    20 = 2^2 * 5

Gauss was the first to discover that a regular polygon with 17 sides
could be constructed. The construction of a regular polygon with 15
sides was known to the ancient Greeks.

For more about constructable regular polygons, see the Dr. Math FAQ:

http://mathforum.org/dr.math/faq/formulas/faq.regpoly.html

-Doctors Wilkinson and Sarah,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/
```
Associated Topics:
High School Constructions
High School Geometry
High School Triangles and Other Polygons

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