Comment by yarg
Seven sided never seemed that problematic to me?
You can't do it exactly, but you can do it to arbitrary degrees of accuracy; at least about as far as you can go without bumping into the precision limits of a compass and straightedge.
1/7 = 1/8 + 1/64 + 1/512 + 1/4096 + 1/32768... as you can see this will hit the limits of human precision in short order.
In general any fraction 1/(2^n - 1) can be expressed as an infinite sum (or a series that comes infinitesimally close)
1/(2^n - 1) = the sum from x equals one to infinity of 1/(2 ^ (x * n)). And we all know how to section any arch-length into fractions over powers of 2.
So starting with a complete loop, segment take the first piece, then take the second piece, segment and take its first piece... keep on adding all the little pieces together until it's so close enough to 1/7 that you can take a compass measure and use that to resegment the rest of the pie - making sure that you recurse enough that after you've market out 6 additional ones, you get near enough a collision to the first that you're not really worried.
But yeah, I'd be surprised if you could compass and straightedge even to a precision of one part in 4096 - and there's no way in hell that anyone's ever pulling off one part in 32768.
This actually reminds me of another claim that I think is wrong for the opposite reason;
That that the Hilbert Curve covers the totality of the square; but the square contains all bound points of the form [real, real], and you can see from the rational construction of the recursive vertix generator that one of the two values for each co-ordinate pair must necessarily be a rational number (albeit one denominated by an infinite integer exponent of two).
Even if you covered all of [real, rational] + [rational, real] (which you don't), you'd still never reach all of [real, real].
Effectively 100% of the plane is not on the curve and 100% of the plane is within an infinitesimal distance of the curve.
Which I actually think is more interesting than saying that the whole damned thing is in there, which it isn't.