**compass-and-straightedge**4

[1608.08087] Equisectional equivalence of triangles

october 2017 by Vaguery

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

may 2017 by Vaguery

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
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.

may 2017 by Vaguery

[1503.00566] $N$-Division Points of Hypocycloids

april 2017 by Vaguery

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

march 2017 by Vaguery

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
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.

march 2017 by Vaguery

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