A Mathematical Theory of Origami Constructions and Numbers
نویسنده
چکیده
About twelve years ago, I learned that paper folding or elementary origami could be used to demonstrate all the Euclidean constructions; the booklet, [9], gives postulates and detailed the methods for high school teachers. Since then, I have noticed a number of papers on origami and variations, [3], [2] and even websites [7]. What are a good set of axioms and what should be constructible all came into focus for me when I saw the article [13] on constructions with conics in the Mathematical Intelligencer. The constructions described here are for the most part classical, going back to Pythagorus, Euclid, Pappus and concern constructions with ruler, scale, compass, and angle trisections using conics. Klein mentioned the book of Row, [12], while describing geometrical constructions in [10], but went no further with it. Row’s book uses paper folding, as he says, ‘kindergarten tools’, to study geometrical constructions and curve sketching. We shall describe a set of axioms for paper folding which will be used to describe, in a hierarchial fashion, different subfields of the complex numbers, in the familiar way that ruler and compass constructions are used to build fields. The axioms for the origami constructible points of the complex numbers, starting with the constructible points 0 and 1 are that it is the smallest subset of constructible points obtained from the following axioms: (1) The line connecting two constructible points is a constructible line. (2) The point of coincidence of two constructible lines is a constructible point. (3) The perpendicular bisector of the segment connecting two constructible points is a constructible line. (4) The line bisecting any given constructed angle can be constructed.
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تاریخ انتشار 1999