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Half-Silvered Mirrors and Wythoff's Game

Published online by Cambridge University Press:  20 November 2018

Horacio Porta  and
Kenneth B. Stolarsky
Horacio Porta
Affiliation:
Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, Illinois, 61801
Kenneth B. Stolarsky
Affiliation:
Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, Illinois, 61801
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Abstract

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We propose the following problem. Given an array of vertical mirrors that simultaneously transmit and reflect, and a single incoming ray of light, describe the configuration of all light rays that are generated. We solve the problem here for a certain infinite configuration of mirrors; the solution involves the winning positions (a(n), b(n)) of Wythoff's game.

Keywords

10A3210A3510F2078A05Secondary 90D42

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 33 , Issue 1 , 01 March 1990 , pp. 119 - 125
Copyright
Copyright © Canadian Mathematical Society 1990

References

1. Adair, R. K., The Great Design: Particles, Fields, and Creation, Oxford University Press, New York, 1987.Google Scholar
2. Arnold, V. I., Mathematical Methods of Classical Mechanics, Springer Verlag, New York, 1978.Google Scholar
3. Bellman, R., Introduction to Matrix Analysis, McGraw-Hill, New York, 1960.Google Scholar
4. Berlekamp, E. R., Conway, J. H. and Guy, R. K., Winning Ways, Academic Press, London, 1982.Google Scholar
5. Bombieri, E. and Taylor, J. E., Quasicrystals, Tilings, and Algebraic Number Theory: some preliminary connections, Contemporary Mathematics, 64 (1987), 241–264.Google Scholar
6. de Bruijn, N. G., Sequences of zeros and ones generated by special production rules, Kon. Nederl. Akad. Wetensch. Proc. Ser. A 84 (=Indag. Math. 43) (1981), 27-37.Google Scholar
7. de Bruijn, N. G., Algebraic theory of Penrose tilings of the plane, Kon. Nederl. Akad. Wetensch. Proc. Ser. A 84 (=Indag. Math. 43) (1981), 38-66.Google Scholar
8. Christoffel, E. B., Observatio arithmetica, Annali di Matematica, Ser. 2, 6 (1875), 145–152.Google Scholar
9. Connell, I. G., A generalization ofWythoff's game, Canadian Math. Bull., 2 (1959), 181–190.Google Scholar
10. Connell, I. G., Some properties of Beatty sequences I, Canadian Math. Bull., 2 (1959), 190–197.Google Scholar
11. Connell, I. G., Some properties of Beatty sequences II, Canadian Math. Bull., 3 (1960), 17–22.Google Scholar
12. Cornfeld, I. P., Fomin, S. V. and Sinai, Ya. G., Ergodic Theory, Springer Verlag, Berlin, 1980.Google Scholar
13. Coxeter, H. S. M., The golden section, phyllotaxis and Wythojf's game, Scripta Mathematica 19 (1953), 135–143.Google Scholar
14. Fraenkel, A. S., Levitt, J., and Shimshoni, M., Characterization of the set of values f(n) = [na], « = 1 , 2 , … , Discrete Mathematics, 2 (1972), 335–345.Google Scholar
15. Fraenkel, A. S., Mushkin, M., and Tassa, U., Determination of [nO] by its sequence of differences, Canadian Math. Bull., 21 (1978), 441–446.Google Scholar
16. Gratias, D. and Michel, L., éd., International Workshop in Aperiodic Crystals, J. de Physique Colloq., c3 (47) (1986), 1–503.Google Scholar
17. Grunbaum, B. and Shephard, G. C., Tilings and Patterns, Freeman, San Francisco, 1986.Google Scholar
18. Hoggatt, V. E. and Bicknell-Johnson, M., Reflections across two and three glass plates, The Fibonacci Quarterly 17 (1979), 118–142.Google Scholar
19. Ikebe, Y., Inagaki, T., and Miyamoto, S., The monotonicity theorem, Cauchy's interlace theorem and the Cour ant-Fisher theorem, Amer. Math. Monthly 94 (1987), 352–354.Google Scholar
20. Lipson, S. G. and Lipson, H., Optical Physics, Cambridge University Press, Cambridge, 1969.Google Scholar
21. Lunnon, W. F. and P. Pleasants, A. B., Quasicrystallographic tilings, J. Math, pures et appl., 66 (1987), 217–253.Google Scholar
22. Moser, L. and Wyman, M., Problem B-6, The Fibonacci Quarterly l(n.l) (1963), 74.Google Scholar
23. Niven, I., Diophantine Approximations, Interscience Tracts in Pure and Applied Mathematics #14, Interscience Publishers, John Wiley and Sons, New York, 2nd. éd., 1966.Google Scholar
24. Niven, I. and Zuckerman, , An Introduction to the Theory of Numbers, 2nd. éd., John Wiley and Sons, New York, 1966.Google Scholar
25. Pleasants, P. A. B., Quasicrystallography: some interesting new patterns, in “Elementary and Analytical Theory of Numbers”, Banach Center Publications 17, PWN Polish Scientific Publ., Warsaw, 1984, 439–461.Google Scholar
26. Porta, H. and Stolarsky, K. B., The edge of a golden semigroup, Proc. 1987 Jânos Bolyai Math. Soc. Conf. Number Theory, to appear.Google Scholar
27. Porta, H. and Stolarsky, K. B., A number system related to iterated maps whose ultimately periodic set in Q(\/5), to appear.Google Scholar
28. Porta, H. and Stolarsky, K. B., Wythoff pairs as semigroup invariants, to appear in Advances in Mathematics.Google Scholar
29. Porta, H. and Stolarsky, K. B., Certain arithmetic semigroups, to appear in Proceedings of International Conference on Analytic Number Theory, Urbana, 1989.Google Scholar
30. Rayleigh, J. W. S., The Theory of Sound, Vol 1, 2nd. éd., Dover Publications, New York, 1945.Google Scholar
31. Series, C., The geometry of Markojf numbers, Math. Intellig., 7 (3) (1985), 20–29.Google Scholar
32. Smith, H. J. S., Note on continued fractions, Messenger of Math., Ser. 2, 6 (1876), 1–14.Google Scholar
33. Stolarsky, K. B., Beatty sequences, continued fractions, and certain shift operators, Canadian Math. Bull., 19 (1976), 472–482.Google Scholar
34. Schoenberg, I. J., Mathematical Time Exposures, Math. Assoc. Amer., 1982.Google Scholar
35. Shallit, J., A generalization of automatic sequences, Theoretical Comp. Sci., to appear.Google Scholar