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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: 12,067 Registered: 7/15/05
Re: rational n-gon inscribed in a unit circle
Posted: Dec 15, 2013 5:11 AM

quasi wrote:
>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.
>>>>
>>>>There exist PRUC quadrilaterals. For example, the
>>>>quadrilateral 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 numerica
>>>>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 a counterexample to the above conjecture, rescaled to
integral radius and integral edge lengths:

radius 1105, edges 528, 544, 936, 1040, 2184

I've found many such counterexamples.

As a tease, who can find another one?

The revised conjecture below is currently still alive (but
probably not for long).

>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.

I'm now pretty sure the revised conjecture will also fail, and
while I don't have a counterexample, I do have a plan of action
that has a reasonable chance of finding such a counterexample.
When I get a chance, unless someone else finds a counterexample
first, I'll try to implement the plan.

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