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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: 12,052 Registered: 7/15/05
Re: rational n-gon inscribed in a unit circle
Posted: Dec 14, 2013 4:39 AM

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.

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