quasi
Posts:
12,067
Registered:
7/15/05


Re: rational ngon inscribed in a unit circle
Posted:
Dec 14, 2013 2:29 PM


quasi wrote: >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.
Still nothing, so based on that, I'll revive one of my previous conjectures for the special case n = 5 ...
Conjecture:
If a pentagon inscribed in a unit circle has rational edge lengths, then for some reordering of the edges within the circle, two vertices are diametrically opposite.
quasi

