Origami and Geometric Constructions
Source: www.langorigami.com
Topic: Origami
Sort Desciption: The goal of origami geometric constructions is to define one or more points or ...... While many geometric constructions are possible with origami and many ...
Content Inside: Lang, Origami and Geometric Constructions 1 Origami and Geometric Constructions By Robert J. Lang Copyright ©1996–2003. All rights reserved. Introduction.2 Preliminaries and Definitions .2 Binary Divisions.4 Binary Folding Algorithm.5 Binary Approximations.9 Rational Fractions.11 Crossing Diagonals.12 Fujimoto’s Construction.15 Noma’s Method.18 Haga’s Construction.20 Irrational Proportions.22 Continued Fractions .22 Quadratic Surds.26 Angle Divisions.31 Axiomatic Origami.37 Preliminaries.39 Folding.41 Alignments.41 Bringing a point to a point P ↔ P .42 Bringing a point onto a line (P ↔ L).42 Bringing one line to another line ( L ↔ L).42 Alignments by folding.43 Multiple Alignments .44 Constructability.44 Axiom 6 and Cubic Curves .45 Approximation by Computer.49 References.53 Lang, Origami and Geometric Constructions 2 Introduction Compass-and-straightedge geometric constructions are familiar to most students from high- school geometry. Nowadays, they are viewed by most as a quaint curiosity of no more than academic interest. To the ancient Greeks and Egyptians, however, geometric constructions were useful tools, and for some, everyday tools, used for construction and surveying, among other activities. The classical rules of compass-and-straightedge allow a single compass to strike arcs and transfer distances, and a single unmarked straightedge to draw straight lines; the two may not be used in combination (for example, holding the compass against the straightedge to effectively mark the latter). However, there are many variations on the general theme of geometric constructions that include use of marked rules and tools other than compasses for the construction of geometric figures. One of the more interesting variations is the use of a folded sheet of paper for geometric construction. Like compass-and-straightedge constructions, folded-paper constructions are both academically interesting and practically useful ...
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