quasi
Posts:
12,018
Registered:
7/15/05


Re: rational ngon inscribed in a unit circle
Posted:
Dec 12, 2013 5:38 PM


Richard Tobin wrote: >quasi wrote: > >>> n=8: 1/4 1/4 1/4 1 1 1 5/4 5/4 > >>An 8gon with those sides can't be inscribed in a unit circle. > >Correction already posted.
When I replied, your correction wasn't yet visible on my news server.
But yes, your corrected version works.
Very nice. >I divided 9 by 8 and got 5/4.
Hehe.
>Here's a cyclic 35gon with radius 13, maybe: > > 1 1 1 1 1 1 1 1 1 1 > 1 1 1 1 1 1 1 1 1 1 > 2 2 2 2 > 3 3 3 3 3 3 3 3 > 5 6 17
Close, but I don't think the above example works for radius 13.
But nice work dispatching conjecture (1).
I'll try a revision ...
Conjecture (1) [revised]:
If n > 6, there does not exist a rational ngon with pairwise distinct edge lengths and no two vertices diametrically opposite which can be inscribed in a unit circle.
Remark:
I think I've made it harder to beat, but I suspect it will still fail. In any case, it makes for a nice challenge.
quasi

