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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 14, 2013 12:55 PM

Richard Tobin wrote:
>quasi wrote:
>>
>>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.

>
>Given a set of rational sides, how in general do you determine
>whether any of the diagonals are rational?

Since the radius and the edge lengths are known, the sine and
cosine of each of the central angles to the edges can be found.
If two edges are placed adjacent, trig sum formulas can be used
to find sine and cosine of the combined angle. The law of
cosines can then be used to get the length of the chord between
two adjacent vertices. For diagonals between vertices more than
two vertices apart, the idea is essentially the same but more
complicated.

Before declaring a given sequence of edge lengths PRUC, the
irrationality of the diagonals must be confirmed for all
circular permutations of the edge lengths.

The idea of a PRUC n-gon is that it's a rational-unit-cyclic
n-gon which can't be obtained, after some circular permutation
of its edges, by gluing together two smaller rational-unit-cyclic
polygons along a common length edge and then erasing the common
edge (now a diagonal).

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