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.