Date: Dec 12, 2013 5:38 PM
Author: quasi
Subject: Re: rational n-gon inscribed in a unit circle

Richard Tobin wrote:
>quasi wrote:
>

>>> n=8: 1/4 1/4 1/4 1 1 1 5/4 5/4
>
>>An 8-gon 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 35-gon 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 n-gon 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