The following might be helpful to determine which permutations of the roots of unity give a polygon with zero area. Let A be the area, then 2 * A = = sum(i=0, n-1, x[i]*y[i+1] - x[i+1]*y[i] ) = sum(i=0, n-1, (x[i] + x[i+1]) * (y[i+1] - y[i]) ) = sum(i=1, n, x[i] * (y[i+1] - y[i-1]) ) where indices are taken modulo n This is from http://geomalgorithms.com/a01-_area.html which gives further references. I found this URL at http://stackoverflow.com/questions/451426/how-do-i-calculate-the-area-of-a-2... Best regards, jj * James Propp <jamespropp@gmail.com> [Jul 30. 2016 08:11]:
For which n>2 does there exist a reentrant n-gon whose vertices are evenly spaced on a circle and whose signed area (aka algebraic area) is zero?
When n is even, there is an easy solution: draw a circuit from 1 to 2 to 3 to ... to n/2-1 to n/2 to n to n-1 to n-2 to ... to n/2+1 to 1 (here I'm numbering the n vertices cyclically).
For n=3, it's trivially impossible to find such an n-gon, and I've checked by brute force (using Mathematica) that there's no such n-gon when n=5 or n=7.
For n=9, there is such an n-gon (which I found through brute-force search): http://mathenchant.org/nonagon.pdf Mathematica assures me that 1/2 (-1/2 Sqrt[3] sin(\[Pi]/18)+1/2 sin(\[Pi]/9)-sin((2 \[Pi])/9)-2 sin(\[Pi]/18) sin((2 \[Pi])/9)-1/2 cos(\[Pi]/18)-1/2 Sqrt[3] cos(\[Pi]/9)+2 cos(\[Pi]/18) cos((2 \[Pi])/9)+2 sin(\[Pi]/9) cos(\[Pi]/9)) simplifies to 0, which implies the claim. Is there a nice (preferably trig-free) way to prove it? That is: is there a nice way to show that the total area of the three scalene black triangles equals the total area of the four nonscalene black triangles, maybe using dissection and area-preserving linear transformations, or maybe using algebraic tricks involving roots of unity?
I suspect that when n is prime, there's no such polygon. Can any of you find a proof?
Also, can any of you construct such a polygon with n=15? It'd be nice if the answer to my original question were "Precisely when n is composite".
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun