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A symmetrical pursuit problem on the sphere and thehyperbolic plane

Published online by Cambridge University Press:  01 August 2016

P.K. Aravind
P.K. Aravind*
Affiliation:
Physics Department, Worcester Polytechnic Institute, Worcester, MA 01609, U.S.A.
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The following is a well known pursuit problem: n bugs are at rest at the vertices of a regular polygon. At a certain instant each bug starts crawling towards its neighbour on the right with a constant speed, always altering its course so as to be headed directly towards its neighbour. How long does it take for the bugs to meet at the center of the polygon and what paths do they trace out in getting there? The earliest reference I could find to this problem was in a collection of mathematical puzzles edited by L.A.Graham. A more recent reference to it occurs in the amusing and informative collection of geometrical curiosities by David Wells. The “n-bug” problem was also discussed briefly in two of Martin Gardner’s columns in Scientific American. The cover of the July 65 issue of that magazine shows the equiangular spirals traced out by four bugs as they “square dance” their way towards their tryst at the centre.

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Type
Research Article
Information
The Mathematical Gazette , Volume 78 , Issue 481 , March 1994, pp. 30 - 36
Copyright
Copyright © The Mathematical Association 1994

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References

1. Graham, L.A., Ingenious mathematical problems and methods (Dover, New York, 1959), Problem 66.Google Scholar
2. David, Wells, The Penguin Dictionary of Curious and Interesting Geometry (Penguin Books, New York, 1991), pp. 201–202.Google Scholar
3. Martin, Gardner in “Mathematical Games”, Scientific American, November 1957 and July 1965.Google Scholar
4. For a concise account of spherical trigonometry, together with a listing of all the relevant formulas, see The VNR Encyclopedia of Mathematics Eds. Gellert, W., Kustner, H., Hellwich, M. and Kastner, H. (Van Norstrand, New York, 1977), pp.261–272.Google Scholar
5. For an elementary account of hyperbolic (or Lobachevskian) geometry see Toth, L.Fejes Regular figures (Pergamon Press, New York, 1964),Ch.III, or Coxeter, H.S.M. Non-Euclidean Geometry (5th ed.) (University of Toronto Press, Toronto, 1968), among the many books available on the subject.Google Scholar
6. Figure 1(a), for the spherical case, is obtained as follows: assuming that the bugs are on a circle of latitude in the northern hemisphere, one performs a stereographic projection through the south pole onto a plane passing through the equator; then the geodesic arcs connecting the bugs project onto the circular arcs shown in figure 1(a). Figure 1(b) for the hyperbolic case is constructed using the Poincaré (disk) model of the hyperbolic plane; in this model (see Fejes Toth, ref.5), points on the hyperbolic plane are represented by points within a circle and lines on the hyperbolic plane by circular arcs orthogonal to this circle.Google Scholar
7. Pierce, and Foster, , A short table of integrals (Blaisdell, New York, 1966), p.43, No.298.Google Scholar
8. Gradshteyn, I.S. and Ryzhik, L.M., Tables of Integrals, Series and Products (Academic Press, New York, 1980), Formulas 2.261 and 2.264; the integral (4) has to be recast slightly before these formulas can be used.Google Scholar
9. Coxeter, H.S.M.. “Angels and Devils” in The Mathematical Gardner, edited by Klarner, David A. (Wadsworth International, Belmont, California, 1981).Google Scholar