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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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 ross.finlayson@gmail.com Posts: 2,630 Registered: 2/15/09
Re: rational n-gon inscribed in a unit circle
Posted: Dec 18, 2013 10:52 PM

On Wednesday, December 18, 2013 3:21:52 PM UTC-8, Richard Tobin wrote:
> In article <o20ra99n5qlu21u23tod7rsnuqfarmlcmq@4ax.com>,
>
> quasi <quasi@null.set> wrote:
>
>
>

> >>There does not exist a PRUC pentagon.
>
>
>

> >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.
>
>
>
> Have you got anywhere with this? I have searched up to radius 500 and
>
> found none. All the rational pentagons I found have a rational
>
> diagonal, and almost all have a diameter as a diagonal. A few have
>
> *all* the diagonals rational, so you could inscribe a rational
>
> pentagram in a rational pentagon, e.g. 65: 32 50 66 78 126.
>
>
>
> -- Richard

Sounds like your group actions and so on then also special functions that are
rational for 0 and 1 then irrational, but formulaic. While here for the
pentagon, these are rational, in the higher n-gons, irrational but formulaic.

I think the point detection has to be exact, because in the n-gons, it's easier
to find _seeming_ rational constructions, for a given tolerance or precision,
than to prove an enumerative rational polygon connects. Then for this a fixed
point result in the rational, simply has that it would be exact. These are for
enumerating lines as they would form the sides of triangles, for example
reducing all the lines with the same slope to one line [gradient] as it would
form a triangle to congruency. This is a different enumeration than rational
triangles exhaustively as from integer points, here in the integer lattice. So
for the direct construction of rational triangles in enumeration, they are
exact in fixed point and here extended fixed point in rational points.
Otherwise enumerating only rational triangles in the line model where they are
trilaterals, then it is plane to tetrant instead of quadrant. (Here for 3-D
octant.) For the trilateral, the idea is to give up an orthonormal basis e1,
e2, for a basis that is f(x) = x from zero to one and then symmetrically about
1/3 * 2pi and 2/3 * 2pi, here to that the points are rotational instead of so
as rigid. Then the basis has a unit radius in that it forms a circle here in
the vector space and so on it is then for the enumeration of the rational
tri-laterals. Then, their value in the triangle as on the circle, is also that
the basis coordinates would be rational, so it would be a rational triangle
from a projection to that, the elements of the circle that are each of all
slopes of a line on the plane: the circle, here the unit circle. Then the
point would be that the projection would be preserving the rational triangles
for congruency and here ordering of the points then that for example as they
are enumerated in those terms, this builds general space terms that are then
easily extractable here from reducing the problem given the case of the
"circle" and then, though, "rational polygons with points on the circle" as
here n-gon inscribed in the circle, as the circle circumscribes it. Then,
basically giving each polygon the same center as the circle, it is basically
about that there is a uniform rational lattice, in the uniform integer lattice,
that would otherwise admit scale (here in the rigid between the scale of the
integers and the scales of the pos. rationals < 1). So, rational n-gons fill
all these n-gons, and that here as open subsets and as dense. Anyways then a
general idea about reducing the integer or rational trangles or other polygons,
then the transform is linear because it's over all the discrete inputs.
Finding the constructions that are very near and rational, would be along where
they would have approximating terms in that way. This is where the series of
those itselves naturally approximates the nearest element that can be
approximate. Obviously enough, this would be among all the rational n-gons
that are closer than that, as there would be those (infinitely, except as to
sweep). Then ordering the reductions of an approximation, in construction and
finite, but closest in congruence in the rational triangles or n-gons, then is
as to the integer sequences of those, for example of 3, 4, 5 and 5, 12, 13
(Pythagorean triples).

Regards, Ross Finlayson

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