A Mathematical Theory of Origami Constructions and Numbers
Source: www.emis.de
Topic: Origami
Sort Desciption: origami and numbers. The axioms are hierarchically structured so that the ... Pythagorean, Euclidean and Origami numbers are thus obtained using this set ...
Content Inside: New York Journal of Mathematics New York J. Math. 6 (2000) 119–133. A Mathematical Theory of Origami Constructions and Numbers Roger C. Alperin Abstract. In this article we give a simplified set of axioms for mathematical origami and numbers. The axioms are hierarchically structured so that the addition of each axiom, allowing new geometrical complications, is mirrored in the field theory of the possible constructible numbers. The fields of Thalian, Pythagorean, Euclidean and Origami numbers are thus obtained using this set of axioms. The other new ingredient here relates the last axiom to the algebraic geometry of pencils of conics. It is hoped that the elementary nature of this article will also be useful for advanced algebra students in understanding more of the relations of field theory with elementary geometry. Contents 1. Introduction 119 2. Geometrical Axioms and Algebraic Consequences 121 2.1. Thalian Constructions 121 2.2. Thalian Numbers 125 3. Pythagorean Constructions and Numbers 126 4. Euclidean Constructions and Numbers 127 5. Conic Constructions and Origami Numbers 129 5.1. Simultaneous Tangents 129 5.2. Higher Geometry 130 References 133 1. Introduction About twelve years ago, I learned that paper folding or elementary origami could be used to demonstrate all the Euclidean constructions; the booklet, [J57], gives postulates and detailed the methods for high school teachers. Since then, I have noticed a number of papers on origami and variations, [G95], [EMN94] and even websites [H96]. What are a good set of axioms and what should be constructible Received February 28, 2000. Mathematics Subject Classification. 11R04, 12F05, 51M15, 51N20. Key words and phrases. origami, algebraic numbers, pencil of conics, Pythagorean numbers. ISSN 1076-9803/00 119 120 Roger C. Alperin all came into focus for me when I saw the article [V97] on constructions with conics in the Mathematical Intelligencer. The constructions described ...
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