
Re: rational ngon inscribed in a unit circle
Posted:
Dec 18, 2013 10:52 PM


On Wednesday, December 18, 2013 3:21:52 PM UTC8, 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 ngons, irrational but formulaic.
I think the point detection has to be exact, because in the ngons, 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 3D 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 trilaterals. 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 ngon 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 ngons fill all these ngons, 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 ngons 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 ngons, then is as to the integer sequences of those, for example of 3, 4, 5 and 5, 12, 13 (Pythagorean triples).
Regards, Ross Finlayson

