quasi
Posts:
11,062
Registered:
7/15/05


Re: rational ngon inscribed in a unit circle
Posted:
Dec 15, 2013 5:11 AM


quasi wrote: >quasi wrote: >>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 numerica >>>>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. > >The above conjecture wasn't the claim I intended  I posted >it too quickly, Not that I have a counterexample yet, just >that I intended a weaker conclusion. > >But I'm pretty sure I _can_ find a counterexample to the above >conjecture using another method, not brute force search. It >will still need some programming, and I don't have time to do >the coding right now, but I'll give it a try tomorrow.
Here's a counterexample to the above conjecture, rescaled to integral radius and integral edge lengths:
radius 1105, edges 528, 544, 936, 1040, 2184
I've found many such counterexamples.
As a tease, who can find another one?
The revised conjecture below is currently still alive (but probably not for long).
>Here's the conjecture I intended to pose ... > >Conjecture [revised]: > >There does not exist a PRUC pentagon. > >Equivalently: > >If a pentagon inscribed in a unit circle has rational edge >lengths, then for some reordering of the edges within the >circle, some diagonal has rational length. > >Remarks: > >This revised conjecture may fail, but it should be a lot >harder to find a counterexample.
I'm now pretty sure the revised conjecture will also fail, and while I don't have a counterexample, I do have a plan of action that has a reasonable chance of finding such a counterexample. When I get a chance, unless someone else finds a counterexample first, I'll try to implement the plan.
quasi

