Kempe's universality theorem

In algebraic geometry, Kempe's universality theorem states that any bounded subset of an algebraic curve may be traced out by the motion of one of the joints in a suitably chosen linkage.[1] It is named for British mathematician Alfred B. Kempe, who in 1876 published his article On a General Method of describing Plane Curves of the nth degree by Linkwork,[2] which showed that for any arbitrary algebraic plane curve, a linkage can be constructed that draws the curve. However, Kempe's proof was flawed and the first complete proof was provided in 2002 based on his ideas.[3][4]

This theorem has been popularized by describing it as saying, "One can design a linkage which will sign your name!"[5]

Kempe recognized that his results demonstrate the existence of a drawing linkage but it would not be practical. He states

It is hardly necessary to add, that this method would not be practically useful on account of the complexity of the linkwork employed, a necessary consequence of the perfect generality of the demonstration.[2]

He then calls for the "mathematical artist" to find simpler ways to achieve this result:

The method has, however, an interest, as showing that there is a way of drawing any given case; and the variety of methods of expressing particular functions that have already been discovered renders it in the highest degree probable that in every case a simpler method can be found. There is still, however, a wide field open to the mathematical artist to discover the simplest linkworks that will describe particular curves.[2]

A series of animations demonstrating the linkwork that results from Kempe's universality theorem are available for the parabola, self-intersecting cubic, smooth elliptic cubic and the trifolium curves.[6]

  1. ^ A. Saxena (2011) Kempe’s Linkages and the Universality Theorem Archived 2016-12-07 at the Wayback Machine, RESONANCE
  2. ^ a b c Kempe, A. B. (1875). "On a General Method of describing Plane Curves of the nth degree by Linkwork". Proceedings of the London Mathematical Society. s1-7: 213–216. doi:10.1112/plms/s1-7.1.213.
  3. ^ M. Kapovich and J. J. Millson (2002), Universality theorems for configguration spaces of planar linkages Topology, Pergamon Press.
  4. ^ Demaine, Erik; O'Rourke, Joseph (2007), "3.2 Kempe's Universality Theorem", Geometric Folding Algorithms, Cambridge University Press, pp. 31–40, ISBN 978-0-521-71522-5.
  5. ^ J. Malkevich, Feature Column, American Mathematical Society.
  6. ^ A. Kobel, (2008) Automated Generation of Kempe Linkages for Algebraic Curves in a Dynamic Geometry System. Saarland University, Saarbrucken, Germany, Faculty of Natural Sciences and Technology I, Department of Computer Science.