Regular Heptadecagon Inscribed in a Circle

Oh believe me, after doing engineering school (mechanical or civil) you would be able to do that easily in half an hour and the end result would be quite pretty.

Either that or you would never get a degree as technical drawing teachers developed the art of nitpicking to incredible levels (or at least that was the case at my university).

bprater• 16 years ago

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

skermes• 16 years ago

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.

leif• 16 years ago

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

tokenadult2• 16 years ago

http://www.jimloy.com/geometry/trisect.htm#tools

But of course these are not the Euclidean tools.

splat• 16 years ago

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

bantic• 16 years ago

nice fermat reference

nopassrecover• 16 years ago

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

gjm11• 16 years ago

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.

Regular Heptadecagon Inscribed in a Circle

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