Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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

 Search Thread: Advanced Search

 Messages: [ Previous | Next ]
 quasi Posts: 12,067 Registered: 7/15/05
Re: rational n-gon inscribed in a unit circle
Posted: Dec 13, 2013 5:29 PM
 Plain Text Reply

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 quasi
12/10/13 ross.finlayson@gmail.com
12/10/13 quasi
12/11/13 quasi
12/11/13 quasi
12/11/13 quasi
12/12/13 quasi
12/12/13 Helmut Richter
12/12/13 quasi
12/11/13 scattered
12/11/13 quasi
12/11/13 fom
12/11/13 fom
12/11/13 quasi
12/11/13 fom
12/11/13 Richard Tobin
12/11/13 Richard Tobin
12/12/13 quasi
12/12/13 Richard Tobin
12/12/13 Richard Tobin
12/12/13 quasi
12/12/13 Brian Q. Hutchings
12/13/13 quasi
12/13/13 Brian Q. Hutchings
12/12/13 Thomas Nordhaus
12/12/13 Richard Tobin
12/12/13 quasi
12/12/13 Richard Tobin
12/12/13 quasi
12/12/13 quasi
12/13/13 Richard Tobin
12/13/13 Richard Tobin
12/13/13 quasi
12/13/13 Richard Tobin
12/13/13 quasi
12/13/13 Richard Tobin
12/13/13 quasi
12/13/13 quasi
12/12/13 Richard Tobin
12/13/13 quasi
12/13/13 Richard Tobin
12/13/13 quasi
12/14/13 quasi
12/14/13 quasi
12/14/13 quasi
12/14/13 quasi
12/14/13 Richard Tobin
12/15/13 quasi
12/15/13 quasi
12/15/13 Richard Tobin
12/15/13 David Bernier
12/15/13 quasi
12/18/13 Richard Tobin
12/18/13 ross.finlayson@gmail.com
12/19/13 quasi
12/14/13 Richard Tobin
12/14/13 quasi
12/14/13 Richard Tobin
12/14/13 ross.finlayson@gmail.com
12/15/13 Brian Q. Hutchings

© The Math Forum at NCTM 1994-2018. All Rights Reserved.