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

 Messages: [ Previous | Next ]
 quasi Posts: 12,067 Registered: 7/15/05
Re: rational n-gon inscribed in a unit circle
Posted: Dec 14, 2013 3:53 PM

quasi wrote:
>quasi wrote:
>>quasi wrote:
>>>
>>>A revised definition ...
>>>
>>>Call an n-gon "rational" if all edge lengths are rational.
>>>
>>>Call an n-gon "rational-unit-cyclic" if it's rational and
>>>can be inscribed in a unit circle.
>>>
>>>Call an n-gon "primitive-rational-unit-cyclic" (PRUC) if it's
>>>rational-unit-cyclic 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 rational-unit-cyclic triangle is PRUC, and the class of all
>>>such triangles can be represented parametrically.
>>>
>>>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 numerical 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 n-gons 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 n-gons.

>>
>>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 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.

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