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Topic: rational n-gon inscribed in a unit circle
Replies: 59   Last Post: Dec 19, 2013 12:56 AM

 Messages: [ Previous | Next ]
 ross.finlayson@gmail.com Posts: 2,720 Registered: 2/15/09
Re: rational n-gon inscribed in a unit circle
Posted: Dec 10, 2013 10:47 PM

On Tuesday, December 10, 2013 4:49:10 PM UTC-8, quasi wrote:
> Call an n-gon rational if all edge lengths are rational.
>
>
>
> For n > 6, does there exist a rational n-gon which can be
>
> inscribed in a unit circle?
>
>
>
> quasi

Does there exist a point besides the arc and zero on the circle, that has that
the shortest side of the triangle is an integer, besides x=0 and y = 0? Then
for the inscription to have a rational side, otherwise it is the usual of half
the square root of two or the radius of the unit circle (unit radius <-> unit
diameter). The points would be rational on the unit circle, then as that the
values go through irrationals, they go through rationals. The chords or
outside the polygon inside the circle, these have a rational size chord and
they are all the same for the regular polygon, for there to be rational n-gons
("_rational_ n-gons") inscribed, they are of the rational points here that both
coordinates are rational, whether either could be or each is.

The known sections of the regular, they are for example square root of three,
over two, to one half, here again these aren't rational points.

Then the regular polygons approximate or go through near systems of rational
points that go to them. This is where, for all the points with one rational
and one irrational coordinate, rationals are clearly enough nearest to the
direct progression of n of the n-gon. The angles are as well rational for the
regular n-gon. Using that as an approximation, into thirds and regular
polygons from n = 3, that at n = 7 is a known problem with constructing
regular 7-sided polygon with compass and rule. Here for example it would be an
infinite process to that a regular seven sided polygon or obviously any n-gon
can be circumscribed, and that the the radius and chord from circumference
would go through points, that either are or aren't rational points.

(Then it seems as to having the triangle in the circle and building the squares
around that.)

https://en.wikipedia.org/wiki/Compass_and_straightedge_constructions#Constructing_regular_polygons
https://en.wikipedia.org/wiki/Constructible_polygon

I think these conditions on rational polygons would then be as to
rational-point (or rational) n-gons. For example they are the ratios of all
the polygons that go regular constructions here, with the ratio of the
triangles that re just one point in the quadrant and underneath them, and two
points. With the one point then the rational-ratio triangles aren't
rational-point. With the two other points of the triangle, they can be
rational and the to ratio and point (rational-point). Then as the continuous
rational coordinate points, they are the one-point triangles on the circle with
rational points, as they are the same in ratio, to rational point triangles as
they are for all rational point triangles, here then is most seriously
underdefined in terms of rational-ratio triangles and rational rational-point
triangles.

It is the ratio from one the one side to be measuring out, and the other being
measuring over, that the points on any arc, to the center of the chord, are
measured from traversing the arc or the chord. Then, usual compensating
patterns would suffice. Like linearisation for sine, here over the triangle's
point being the end of the chord, or the midpoint of the chord, it is into
approximations among the values of what values would seem more likely to be
rational points.

Is it to use the cyclotomic fields with integer inputs?

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