

Regular Heptadecagon Inscribed in a Circle - tokenadult
http://en.wikipedia.org/wiki/File:Regular_Heptadecagon_Inscribed_in_a_Circle.gif

======
gjm11
That's beautiful.

For anyone who's wondering: why a 17-gon? the answer is an astonishing
theorem: with the traditional Euclidean tools of ruler and compass, you can
construct an n-sided polygon if and only if n is the product of a power of 2
(1,2,4,8,...) and some set of "Fermat primes": prime numbers of the form
2^2^n+1 (2,3,5,17,65537, and no one knows whether there are any more but there
probably aren't). You aren't allowed to use any of the Fermat primes more than
once.

So you can construct a regular heptadecagon but not a regular nonagon or a
regular heptagon.

Why? Here's a super-handwavy sketch of the ideas involved. Points in the plane
are complex numbers. Constructing a regular n-gon is like constructing z =
exp(2 pi i / n). That satisfies the equation z^n=1. We only need to consider
prime-power values of n, because if m,n have no common factor then you can do
mn if and only if you can do m and n. (Proof left as an exercise for the
reader.) For such n, it's not hard to figure out the _minimal_ polynomial in z
whose value is 0; when n is prime, e.g., it's z^(n-1) + z^(n-2) + ... + z + 1.

Now, imagine doing any ruler-and-compass construction you like. Start with two
points, which we'll call 0 and 1. Then every individual construction you can
do involves solving either a linear or a quadratic equation. Conversely, you
can solve any quadratic equation by doing ruler-and-compass constructions.
(Details again left as an exercise; or look it up.)

Now think about those "minimal polynomials" for all the numbers you construct
along the way. The degree of such a polynomial (i.e., the highest exponent it
contains) is an important quantity. It turns out (super-handwavy, again) that
when you solve a quadratic equation with coefficients whose minimal
polynomials have degree d, you get something whose minimal polynomial has
degree d or 2d. So at any stage in your ruler-and-compass construction, all
the degrees are powers of 2. In particular, you can never construct anything
whose minimal polynomial has degree that isn't a power of 2. And _that_ is
enough to tell you that when n is a power of a prime number, you can't
construct a regular n-gon unless n is a power of 2 or a Fermat prime.

The other direction is conceptually easier but fiddlier, and I shan't try to
explain it here.

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leif
The one that trisects an angle is so much cooler but I can't remember where to
find it.

~~~
splat
I figured out a way to do it once, but this text box is too narrow to contain
it.

~~~
bantic
nice fermat reference

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nopassrecover
I've never wanted a fast-forward button on an animated gif before, nor seen
one at nearly 2mb. Pretty cool though.

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bprater
Is there software that does what we are seeing in that image?

~~~
skermes
When I was in school we had a program (I think it was this:
<http://www.dynamicgeometry.com/>) that let you do pretty much exactly that.
We were never that ambitious, but it was pretty cool. I remember using it's
animation functions to create some limited spirographs by tracing out lines
defined by pairs of points moving around circles and things like that.

------
jaekwon
theoretically, but go ahead and try that. ;)

~~~
tokenadult
It takes a steady hand.

I just did some looking up, and confirmed that Gauss's tombstone does NOT have
the 17-gon on it,

[http://sunsite.utk.edu/math_archives/.http/hypermail/histori...](http://sunsite.utk.edu/math_archives/.http/hypermail/historia/dec98/0102.html)

although a monument to him is said to have one, which I can't find in one of
the better available online photos.

[http://upload.wikimedia.org/wikipedia/commons/4/4f/Braunschw...](http://upload.wikimedia.org/wikipedia/commons/4/4f/Braunschweig_Brunswick_Gauss-
Denkmal_komplett_%282006%29.JPG)

<http://www.jimloy.com/geometry/17-gon.htm>

[http://archive.ncsa.illinois.edu/Classes/MATH198/whubbard/GR...](http://archive.ncsa.illinois.edu/Classes/MATH198/whubbard/GRUMC/geometryExplorer/help/turtle/17-gon.html)

<http://www.mathpages.com/home/kmath487.htm>

~~~
jcl
The 17-pointed star is on the left side of the Braunschweiger memorial (an
unpopular side to photograph, it seems):

[http://de.wikipedia.org/wiki/Datei:Braunschweig_Gauss-
Denkma...](http://de.wikipedia.org/wiki/Datei:Braunschweig_Gauss-
Denkmal_Stern-Seite.JPG)

[http://de.wikipedia.org/wiki/Datei:Braunschweig_Gauss-
Denkma...](http://de.wikipedia.org/wiki/Datei:Braunschweig_Gauss-
Denkmal_Detail_mit_Stern.JPG)

