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

• Main Author: Daval Santos, Pocholo Domingo A.
• Format: text
• Language: English
• Published: Animo Repository 1999
• Subjects: Triangle; Geometry; Plane; Vertex operator algebras; Algebraic Geometry
• Online Access: https://animorepository.dlsu.edu.ph/etd_masteral/1971
• Institution: De La Salle University

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.