A regular polygon with 17,476 sides. Back in the day, Euler proved that such a polygon can be constructed, as can any polygon with number of sides equal to 2^k times the product of distinct prime Fermat numbers. Since 17476 = 2^2*17*257 and 17 and 257 are Fermat primes, it is possible to make one using a straight-edge and compass.

To construct such a thing, you first need to have regular polygons with 17 and 257 sides each. Our stragety is to eventually stick vertices on the unit circle at every polar coordinate of the form (1,k*2π/17476), where k is an integer between 0 and 17475. But first we have to be able to construct an angle of 2*π/17476 radians. All we have at our disposal are the 17-gon and 257-gon. But, by connecting two adjacent vertices of the 17-gon to its center, we have an angle of 2π/17. Similarly, we can get an angle of 2π/257 from the 257-gon. Now we have to start from the 0-degree line of the polar coordinate system, and add and subtract angles of 2π/17 and 2π/257 radians until we get an angle of 2π/17476.

To figure out how many of each angle we need, we solve the Diophantine equation

a*(2π/257) + b*(2π/17) = 2π/17476.

Multiplying through by 17476/2π, we get

a*(4*17) + b*(4*257) = 1

Unforunately, this equation has no solution in the integers, because you can't add an integral number of multiples of four together and get a number that isn't a multiple of four. So, we instead settle for finding the solution to 17a + 257b = 1, which will show us how many of our angles we need to add to get the angle 2π/(257*17). We use Euclid's Algorithm:


 qk   rk    ak   bk  
-------------------
     257     0   1
15    17     1   0
 8     2   -15   1 
       1   121  -8
Note that rk+2 = rk mod rk+1, qk+1 = floor(qk/qk+1), and ak+2 = ak-rk+1*ak+1, and the equationbk+2 is similar.
Also, for each k, rk = 257ak + 17bk... the whole reason we did this.

(upon re-reading, it occurs to me that that notation might not make an ounce of sense to most people)

257 =   0*17 + 1*257
 17 =   1*17 + 0*257   floor(257/17) = 15,  257-15*17 = 2
  2 = -15*17 + 1*257   floor(17/2) = 8,     17-2*8 = 1
  1 = 121*17 - 8*257
So, now we know that we can construct an angle of 2π/(257*17) radians by adding 121 angles of 2π/257 radians each together, and then adding 8 angles of 2π/17 radians each in the opposite direction. And by bisecting this angle twice, we have the angle 2π/(4*17*257) = 2π/17476 rad. Now all that has to be done is to plot out 17,476 vertices, and connect each one to its two neighbors.

All that, and it'll probably just look like a circle.

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