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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 13, 2013 4:24 PM

Richard Tobin wrote:
>
>Radius 32, 4 sides: 8 36 57 62

Nice find!

I had done some searching, but apparently not enough.

I suspect 32 is the smallest positive integer radius for which
there is an integer quadrilateral counterexample (where the
restrictions must be satisfied by all reorderings of the edges).

Some background ...

The problem arose as a question in my mind based on my
resolution of a question by José Carlos Santos in the thread
"Non-constructible numbers of degree 4".

In one of my solutions to the problem, I analyzed a cyclic
decagon with two edges of length 2 and the remaining 8 edges of
length 1. I was able to show that the radius of the
circumscribed circle was a non-constructible (by straight edge
and compass) number of algebraic degree 4 over the rationals.

I began to wonder if there were non-trivial examples of a cyclic
polygon with integer sides for which the radius of the
circumscribing circle was an integer.

I was able to fully resolve the question for n = 3.

For n = 4, I saw some trivial ways, and wondered if those were
the only ways.

The conjectures I posed were intended to promote an exploration
of those types of questions.

With your latest counterexample, all the conjectures previously

I'm not sure if there is any sensible revision that maintains
the spirit of those conjectures. I'll think about it.

In any case, thank for playing -- I hope it was fun.

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