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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 ]
 Brian Q. Hutchings Posts: 6,427 Registered: 12/6/04
Re: rational n-gon inscribed in a unit circle
Posted: Dec 15, 2013 8:14 PM

reformat

> It seems easier to construct rational point polygons than rational side length
>
> polygons, for the regular and convex it is all inside the triangle. (It is
>
> easy to construct the rational divisions of the angles.) Then, for the chord
>
> and line length, it is always the chords of the radius for the side length to
>
> match 1-1 with arc length, for just a rat

ional side length edge. Then for
>
> the rest of the edges it is whether the rest of the circle, worked out from the
>
> arc length, with have these rational edges. The idea is to treat it from 0 to
>
> 2pi, around the circle, but also -pi to pi. 0 to 2pi is counter-clockwise, ccw
>
> (right hand rule), -pi to pi is clockwise, or ccw. This is where the
>

> properties of all the vertices of the polygon are that they are on the circle
>
> for convex vertices (for convex polygons) and if internal then concave. It
>
> seems obvious there are rational side length pairs, connecting any two points,
>
> eg of the square of the line through them as a diagonal. So, concave polygons
>
> with rational side lengths, can be constructed from rational convex polygons,
>
> here as to whether its possible to construct, for an irrational length, two
>
> rational lengths connected by a third point here inside the circle, of a
>
> rational length through the third point between the vertices otherwise at an
>
> irrational distance. Then the idea is that there are those, concave polygons
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> bounded by a circle with only rational side lengths for any collection of
>
> vertices on the circle, then that

from actual rational side lengths, they have
>
> all of regular geometry around them then, the regular n-gons. Then the idea
>
> is to use constructions of arc length, and the convex, but that the concave is
>
> the case for all rational lengths in the geometry, then as to that the
>
> cyclotomic fields are regular with usual reversibility in cyclotomic fields,
>
> here for 0 to 2pi or -pi to pi. Given two points, is it possible to construct
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> a third point of rational distance to each, compare

ed to a third distance fixed
>
> as one? It seems so where it is less than two. Given two points that are
>
> irrational, is it possible to construct a point at rational distances from each
>
> endpoint, here without distance fixed? Here it is so if above, for whatever
>
> rational distance there would be, that the path would be of rational length.
>
> Then though as to the rational division of the angles, that is of the
>
> isosceles, and the base being regular in the isosceles where it is irrational,
>
> has whether it is then of the irrational components in the angle formulae, to
>
> divide those out, then for building the convex polygons on the circle with
>
> those. Then the side length, of the convex, to each be an integer in relation
>
> to the others as some integers, as scaled from 1.0, for them to be rational, it
>
> divides the arc length with the same ratios. The arc length is 2pi radians.
>
> What it is there is that the

polygon on the circle is the outside of the
>
> connected set of all the vertices with the vertices on the circle, all the
>
> edges are inner. Here for the outermost edges as there are, these to be
>
> regular or as to rational would be then that as connected, the points have
>
> usual constructions to all the other points, here up in the cyclomatic fields
>

> of those, the vertices of the polygon on the circle, as connected set of edges,
>
> with that there are outer edges so the polygon is convex, then that it is on
>
> the circle so the connections conserve a maximum or inequality, here of the
>
> angles and thus distances. Then like Gauss is finding in the Mersenne, the
>
> prime and co-prime in the cyclomatic may have tractable forms for rational
>
> (PRUC one point) n-gons.
>
>
>
> For general interest in geometry, there is the regular inscribed n-gon. Is the

>
> line segment the two-sided case? It is always rational. There are a lot of
>
> ways to find many of these. It's nice to type these, I wrote that an hour ago
>
> then just now add a few words. Mostly though it's all one spew. And me typing
>
> it is the spell-check. The polygon's vertices that are each irrational, adding
>
> to those more vertices of rational distance, the result can be a polygon of
>
> only rational lengths. Then obviously the original vertices at at most one not
>
> originally rational, that's though a definition because the original n
>
> irrational points just have one that can surely be made rational as zero by
>
> subtracting it, but also 4-5 them here for the 2n-1, and triangle points, then
>
> also for 4 and 5 and line points.

Date Subject Author
12/10/13 quasi
12/10/13 ross.finlayson@gmail.com
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