quasi
Posts:
10,385
Registered:
7/15/05


Re: rational ngon inscribed in a unit circle
Posted:
Dec 14, 2013 5:28 AM


quasi wrote: > >A revised definition ... > >Call an ngon "rational" if all edge lengths are rational. > >Call an ngon "rationalunitcyclic" if it's rational and >can be inscribed in a unit circle. > >Call an ngon "primitiverationalunitcyclic" (PRUC) if it's >rationalunitcyclic and, for any reordering of the edges, no >diagonal has rational length. > >Note: In my previously posted tentative definition, there were >additional requirements, namely: n > 3, pairwise distinct edge >lengths, and no edge length equal to 2. Those additional >requirements have now been dropped. > >Any rationalunitcyclic triangle is PRUC, and the class of all >such triangles can be represented parametrically. > >There exist PRUC quadrilaterals. For example, the quadrilateral >found by Richard Tobin with sides 8,36,57,62 and radius 32 is >a PRUC quadrilateral scaled by a factor of 32. > >There are other PRUC quadrilaterals as well, but there's no >clear pattern that I can see relating the numerical values of >the edge lengths. It's not clear how to generate them except by >brute force search. > >So that's the question. > >For some n > 3, either general or specific, is there some >subclass of the class of PRUC ngons which can be generated >by a method other than brute force search? Perhaps a parametric >representation or a recursion? > >In the meantime, the current search methods might help by >revealing a pattern for some subclass of PRUC ngons.
I haven't yet found a PRUC pentagon.
I have my search program running, but so far, nothing.
quasi

