quasi
Posts:
11,467
Registered:
7/15/05


Re: rational ngon inscribed in a unit circle
Posted:
Dec 14, 2013 12:55 PM


Richard Tobin wrote: >quasi wrote: >> >>Call an ngon "primitiverationalunit cyclic" (PRUC) if it's >>rationalunitcyclic and, for any reordering of the edges, no >>diagonal has rational length. > >Given a set of rational sides, how in general do you determine >whether any of the diagonals are rational?
Since the radius and the edge lengths are known, the sine and cosine of each of the central angles to the edges can be found. If two edges are placed adjacent, trig sum formulas can be used to find sine and cosine of the combined angle. The law of cosines can then be used to get the length of the chord between two adjacent vertices. For diagonals between vertices more than two vertices apart, the idea is essentially the same but more complicated.
Before declaring a given sequence of edge lengths PRUC, the irrationality of the diagonals must be confirmed for all circular permutations of the edge lengths.
The idea of a PRUC ngon is that it's a rationalunitcyclic ngon which can't be obtained, after some circular permutation of its edges, by gluing together two smaller rationalunitcyclic polygons along a common length edge and then erasing the common edge (now a diagonal).
quasi

