compass-and-straightedge   4

[1608.08087] Equisectional equivalence of triangles
We study equivalence relation of the set of triangles generated by similarity and operation on a triangle to get a new one by joining division points of three edges with the same ratio. Using the moduli space of similarity classes of triangles introduced by Nakamura and Oguiso, we give characterization of equivalent triangles in terms of circles of Apollonius (or hyperbolic pencil of circles) and properties of special equivalent triangles. We also study rationality problem and constructibility problem.
plane-geometry  compass-and-straightedge  looking-to-see  rather-interesting  algebra  nudge-targets  consider:feature-discovery
october 2017 by Vaguery
[1507.07970v2] Dividing the circle
There are known constructions for some regular polygons, usually inscribed in a circle, but not for all polygons - the Gauss-Wantzel Theorem states precisely which ones can be constructed.
The constructions differ greatly from one polygon to the other. There are, however, general processes for determining the side of the n-gon (approximately, but sometimes with great precision), which we describe in this paper. We present a joint mathematical analysis of the so-called Bion and Tempier approximation methods, comparing the errors and trying to explain why these constructions would work at all.
compass-and-straightedge  plane-geometry  construction  approximation  nudge-targets  consider:rediscovery
may 2017 by Vaguery
[1503.00566] \$N\$-Division Points of Hypocycloids
We show that the \$n\$-division points of all rational hypocycloids are constructible with an unmarked straightedge and compass for all integers \$n\$, given a pre-drawn hypocycloid. We also consider the question of constructibility of \$n\$-division points of hypocycloids without a pre-drawn hypocycloid in the case of a tricuspoid, concluding that only the \$1\$, \$2\$, \$3\$, and \$6\$-division points of a tricuspoid are constructible in this manner.
compass-and-straightedge  construction  proof  computational-geometry  nudge-targets  consider:looking-to-see
april 2017 by Vaguery
[1507.07970] Dividing the circle
There are known constructions for some regular polygons, usually inscribed in a circle, but not for all polygons - the Gauss-Wantzel Theorem states precisely which ones can be constructed.
The constructions differ greatly from one polygon to the other. There are, however, general processes for determining the side of the \$n\$-gon (approximately, but sometimes with great precision), which we describe in this paper. We present a joint mathematical analysis of the so-called Bion and Tempier approximation methods, comparing the errors and trying to explain why these constructions would work at all.
compass-and-straightedge  constructible-numbers  rather-interesting  nudge-targets  consider:benchmarks  plane-geometry
march 2017 by Vaguery

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