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Topic: rational n-gon inscribed in a unit circle
Replies: 59   Last Post: Dec 19, 2013 12:56 AM

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quasi

Posts: 10,317
Registered: 7/15/05
Re: rational n-gon inscribed in a unit circle
Posted: Dec 13, 2013 5:29 PM
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Richard Tobin) wrote:
>quasi wrote:
>>
>>Call an n-gon rational if all edge lengths are rational.
>>
>>For n > 6, does there exist a rational n-gon which can be
>>inscribed in a unit circle?

>
>Lowest denominator heptagon:
>
> radius 40, 7 sides: 10 10 10 45 45 48 64


However it violates two of the restrictions imposed in later
revisions:

(1) For any reordering of the edges, no two vertices are
allowed to be diametrically opposite.

(2) The edges lengths are requied to be pairwise distinct.

A tentative definition ...

For this discussion, for n > 3, call a rational n-gon inscribed
in a unit circle "primitive-rational-unit-cyclic" (PRUC) if

(1) No edge has length 2.

(2) The edge lengths are pairwise distinct.

(3) For any reordering of the edges, no diagonal has rational
length.

An example of a PRUC quadrilateral (rescaled so that the radius
and all edge lengths are integers) is the cyclic quadilateral
found by Richard Tobin with sides 8,36,57,62 and radius 32.

I haven't yet seen an example of a PRUC n-gon with n > 4,
although I'm pretty sure such examples exist.

quasi


Date Subject Author
12/10/13
Read rational n-gon inscribed in a unit circle
quasi
12/10/13
Read Re: rational n-gon inscribed in a unit circle
ross.finlayson@gmail.com
12/10/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/11/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/11/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/11/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/12/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/12/13
Read Re: rational n-gon inscribed in a unit circle
Helmut Richter
12/12/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/11/13
Read Re: rational n-gon inscribed in a unit circle
scattered
12/11/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/11/13
Read Re: rational n-gon inscribed in a unit circle
fom
12/11/13
Read Re: rational n-gon inscribed in a unit circle
fom
12/11/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/11/13
Read Re: rational n-gon inscribed in a unit circle
fom
12/11/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/11/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/12/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/12/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/12/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/12/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/12/13
Read Re: rational n-gon inscribed in a unit circle
Brian Q. Hutchings
12/13/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/13/13
Read Re: rational n-gon inscribed in a unit circle
Brian Q. Hutchings
12/12/13
Read Re: rational n-gon inscribed in a unit circle
Thomas Nordhaus
12/12/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/12/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/12/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/12/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/12/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/13/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/13/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/13/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/13/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/13/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/13/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/13/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/13/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/12/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/13/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/13/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/13/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/14/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/14/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/14/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/14/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/14/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/15/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/15/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/15/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/15/13
Read Re: rational n-gon inscribed in a unit circle
David Bernier
12/15/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/18/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/18/13
Read Re: rational n-gon inscribed in a unit circle
ross.finlayson@gmail.com
12/19/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/14/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/14/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/14/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/14/13
Read Re: rational n-gon inscribed in a unit circle
ross.finlayson@gmail.com
12/15/13
Read Re: rational n-gon inscribed in a unit circle
Brian Q. Hutchings

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