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Origami and Geometric Constructions PDF File size: 1592.42 Kb
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Compass-and-straightedge geometric constructions are familiar to most students from highschool geometry.Thus, within origami, there is a practical interest in devising folding sequences

Lang, Origami and Geometric Constructions

Origami and Geometric Constructions1
By Robert J. Lang Copyright ©1996­2010. All rights reserved. Introduction………………………………………………………………………………………………………………….. 3 Preliminaries and Definitions ………………………………………………………………………………………… 3 Binary Divisions ……………………………………………………………………………………………………………. 5 Binary Folding Algorithm………………………………………………………………………………………….. 6 Binary Approximations……………………………………………………………………………………………. 10 Rational Fractions……………………………………………………………………………………………………….. 13 Crossing Diagonals ………………………………………………………………………………………………….. 13 Fujimoto’s Construction ………………………………………………………………………………………….. 17 Noma’s Method ……………………………………………………………………………………………………….. 19 Haga’s Construction ………………………………………………………………………………………………… 21 Irrational Proportions …………………………………………………………………………………………………. 24 Continued Fractions ………………………………………………………………………………………………… 24 Quadratic Surds………………………………………………………………………………………………………. 28 Angle Divisions………………………………………………………………………………………………………… 34 Axiomatic Origami………………………………………………………………………………………………………. 40 Preliminaries …………………………………………………………………………………………………………… 43 Folding ……………………………………………………………………………………………………………………. 45 Alignments………………………………………………………………………………………………………………. 46 Bringing a point to a point P P ………………………………………………………………………… 46 Bringing a point onto a line ( P L ) …………………………………………………………………….. 46 Bringing one line to another line ( L L ) ……………………………………………………………… 46

This is an article I originally wrote in 1996; in 2003, an abbreviated version appeared in the book, A Tribute to a Mathemagician. The 2003 version appeared on my website, http://www.langorigami.com. This version (2010) corrects some errors that appeared in earlier versions: a few typographical errors, proper credit to Jacques Justin for the 7 "axioms," and a correction of Abe’s name, among others. My thanks go to the various correspondents who have sent me corrections to the 2003 version.

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