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ORIGAMI RINGS

Published online by Cambridge University Press:  02 November 2012

JOE BUHLER
Affiliation:
Center for Communications Research, La Jolla, CA 92121, USA (email: buhler@ccrwest.org)
STEVE BUTLER*
Affiliation:
Department of Mathematics, Iowa State University, Ames, IA, 50011, USA (email: butler@iastate.edu)
WARWICK DE LAUNEY
Affiliation:
Center for Communications Research, La Jolla, CA 92121, USA
RON GRAHAM
Affiliation:
Department of Computer Science and Engineering, University of California, San Diego, La Jolla, CA 92093, USA (email: graham@ucsd.edu)
*
For correspondence; e-mail: butler@iastate.edu
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Abstract

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Motivated by mathematical aspects of origami, Erik Demaine asked which points in the plane can be constructed by using lines whose angles are multiples of $\pi /n$ for some fixed $n$. This has been answered for some specific small values of $n$ including $n=3,4,5,6,8,10,12,24$. We answer this question for arbitrary $n$. The set of points is a subring of the complex plane $\mathbf {C}$, lying inside the cyclotomic field of $n$th roots of unity; the precise description of the ring depends on whether $n$is prime or composite. The techniques apply in more general situations, for example, infinite sets of angles, or more general constructions of subsets of the plane.

Type
Research Article
Copyright
Copyright © 2012 Australian Mathematical Publishing Association Inc.

References

[1]Alperin, R. C., ‘A mathematical theory of origami constructions and numbers’, New York J. Math. 6 (2000), 119133.Google Scholar
[2]Alperin, R. C. and Lang, R. J., ‘One-, two-, and multi-fold origami axioms’, in: Origami4 (A K Peters, Natick, MA, 2009), pp. 371393.Google Scholar
[3]Demaine, E. D. and O’Rourke, J., Geometric Folding Algorithms: Linkages, Origami, Polyhedra (Cambridge University Press, Cambridge, 2007).CrossRefGoogle Scholar
[4]Origami5: Fifth International Meeting of Origami Science, Mathematics, and Education (A K Peters/CRC Press, Natick, MA, 2011).Google Scholar
[5]Lang, R. J., Origami Design Secrets (A K Peters, Natick, MA, 2003).CrossRefGoogle Scholar
[6]Lang, R. J., ReferenceFinder, available online at http://www.langorigami.com/science/reffinder/reffinder.php4.Google Scholar
[7]Tachi, T. and Demaine, E., ‘Degenerative coordinates in $22.5^\circ $ grid system’, in: Origami5: Fifth International Meeting of Origami Science, Mathematics, and Education (A K Peters/CRC Press, Natick, MA, 2011).Google Scholar
[8]Washington, L., Introduction to Cyclotomic Fields, 2nd edn (Springer, New York, 1997).CrossRefGoogle Scholar
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