A Result About the Density of Iterated Line Intersections in the Plane

Let $S$ be a finite set of points in the plane and let $\mathcal{T}(S)$ be the set of intersection points between pairs of lines passing through any two points in $S$. We characterize all configurations of points $S$ such that iteration of the above operation produces a dense set. We also discuss partial results on the characterization of those finite point-sets with rational coordinates that generate all of $\mathbb Q^2$ through iteration of $\mathcal{T}$.Comment: 10 pages, 8 figures (low-res for the arXiv), Computational Geometry: Theory and Application

Reflecting a triangle in the plane

Abstract. We prove that if the three angles of a triangle T in the plane are different from (60~176176 (30 ~ 30 ~ 120 ~ (45~176176 ~ 60~176 then the set of vertices of those triangles which are obtained from T by repeating 'edge-reflection ' is everywhere dense in the plane

Reflecting triangles with a dense set of vertex points in the plane

This thesis is an expository work based on the main part of the paper Reflecting a Triangle in the Plane by Peter Frankl, Imre Barany, and Hiroshi Maechara published in Graphs and Combinatories (1993) 9: 97-104. The article states that if the three angles of a triangle (delta) in the plane are different from (60 degrees, 60 degrees, 60 degrees), (30 degrees, 30 degrees, 120 degrees), 45 degrees, 45 degrees, 90 degrees), (30 degrees, 60 degrees, 90 degrees), then the set of vertices of those triangles which are obtained from (triangle) by repeating edge-reflection (denoted by Omega ABC) is dense in the plane. This thesis will prove the following main results: Theorem 0.0.1 Let delta ABC be a rational triangle with angles alpha less than or equal to beta less than or equal to y. If (alpha, beta, y) is not equal to (60 degrees, 60 degrees, 60 degrees), (30 degrees, 30 degrees, 120 degrees),(45 degrees, 45 degrees, 90 degrees), (30 degrees, 60 degrees, 90 degrees),then Omega ABC is dense in the plane. Theorem 0.0.2 Let ABC be an irrational triangle, then Omega ABC is dense in the plane