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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: 1,216
Registered: 2/15/09
Re: rational n-gon inscribed in a unit circle
Posted: Dec 14, 2013 10:05 PM
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On Saturday, December 14, 2013 1:39:26 AM UTC-8, quasi wrote:
> A revised definition ...
>
>
>
> Call an n-gon "rational" if all edge lengths are rational.
>
>
>
> Call an n-gon "rational-unit-cyclic" if it's rational and
>
> can be inscribed in a unit circle.
>
>
>
> 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.
>
>
>
> Note: In my previously posted tentative definition, there were
>
> additional requirements, namely: n > 3, pairwise distinct edge
>
> lengths, and no edge length equal to 2. Those additional
>
> requirements have now been dropped.
>
>
>
> Any rational-unit-cyclic triangle is PRUC, and the class of all
>
> such triangles can be represented parametrically.
>
>
>
> There exist PRUC quadrilaterals. For example, the quadrilateral
>
> found by Richard Tobin with sides 8,36,57,62 and radius 32 is
>
> a PRUC quadrilateral scaled by a factor of 32.
>
>
>
> There are other PRUC quadrilaterals as well, but there's no
>
> clear pattern that I can see relating the numerical values of
>
> the edge lengths. It's not clear how to generate them except by
>
> brute force search.
>
>
>
> So that's the question.
>
>
>
> For some n > 3, either general or specific, is there some
>
> subclass of the class of PRUC n-gons which can be generated
>
> by a method other than brute force search? Perhaps a parametric
>
> representation or a recursion?
>
>
>
> In the meantime, the current search methods might help by
>
> revealing a pattern for some subclass of PRUC n-gons.
>
>
>
> quasi




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 rational 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
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
a third point of rational distance to each, compared 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
Read rational n-gon inscribed in a unit circle
quasi
12/10/13
Read Re: rational n-gon inscribed in a unit circle
ross.finlayson@gmail.com
12/10/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/11/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/11/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/11/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/12/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/12/13
Read Re: rational n-gon inscribed in a unit circle
Helmut Richter
12/12/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/11/13
Read Re: rational n-gon inscribed in a unit circle
scattered
12/11/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/11/13
Read Re: rational n-gon inscribed in a unit circle
fom
12/11/13
Read Re: rational n-gon inscribed in a unit circle
fom
12/11/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/11/13
Read Re: rational n-gon inscribed in a unit circle
fom
12/11/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/11/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/12/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/12/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/12/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/12/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/12/13
Read Re: rational n-gon inscribed in a unit circle
Brian Q. Hutchings
12/13/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/13/13
Read Re: rational n-gon inscribed in a unit circle
Brian Q. Hutchings
12/12/13
Read Re: rational n-gon inscribed in a unit circle
Thomas Nordhaus
12/12/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/12/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/12/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/12/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/12/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/13/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/13/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/13/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/13/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/13/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/13/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/13/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/13/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/12/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/13/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/13/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/13/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/14/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/14/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/14/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/14/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/14/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/15/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/15/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/15/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/15/13
Read Re: rational n-gon inscribed in a unit circle
David Bernier
12/15/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/18/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/18/13
Read Re: rational n-gon inscribed in a unit circle
ross.finlayson@gmail.com
12/19/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/14/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/14/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/14/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/14/13
Read Re: rational n-gon inscribed in a unit circle
ross.finlayson@gmail.com
12/15/13
Read Re: rational n-gon inscribed in a unit circle
Brian Q. Hutchings

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