
Re: rational ngon inscribed in a unit circle
Posted:
Dec 10, 2013 10:47 PM


On Tuesday, December 10, 2013 4:49:10 PM UTC8, quasi wrote: > Call an ngon rational if all edge lengths are rational. > > > > For n > 6, does there exist a rational ngon 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 ngons ("_rational_ ngons") 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 ngon. The angles are as well rational for the regular ngon. Using that as an approximation, into thirds and regular polygons from n = 3, that at n = 7 is a known problem with constructing regular 7sided polygon with compass and rule. Here for example it would be an infinite process to that a regular seven sided polygon or obviously any ngon 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 rationalpoint (or rational) ngons. 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 rationalratio triangles aren't rationalpoint. With the two other points of the triangle, they can be rational and the to ratio and point (rationalpoint). Then as the continuous rational coordinate points, they are the onepoint 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 rationalratio triangles and rational rationalpoint 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

