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The Trigonometry of Hyperbolic Tessellations

Published online by Cambridge University Press:  20 November 2018

H. S. M. Coxeter
H. S. M. Coxeter*
Affiliation:
Department of Mathematics University of Toronto Toronto, Ontario M5S 2G3
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Abstract

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For positive integers p and q with (p − 2)(q − 2) > 4 there is, in the hyperbolic plane, a group [p, q] generated by reflections in the three sides of a triangle ABC with angles π/p, π/q, π/2. Hyperbolic trigonometry shows that the side AC has length ψ, where cosh ψ = c/s, c = cos π/q, s = sin π/p. For a conformal drawing inside the unit circle with centre A, we may take the sides AB and AC to run straight along radiiwhile BC appears as an arc of a circle orthogonal to the unit circle. The circle containing this arc is found to have radius 1/ sinh ψ = s/z, where z = , while its centre is at distance 1/ tanh ψ = c/z from A. In the hyperbolic triangle ABC, the altitude from AB to the right-angled vertex C is ζ, where sinh ζ = z.

Keywords

51F1551N3052A55
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 40 , Issue 2 , 01 June 1997 , pp. 158 - 168
Copyright
Copyright © Canadian Mathematical Society 1997

References

1. Bonola, R., Non-Euclidean Geometry, Dover, New York, 1955.Google Scholar
2. Bourbaki, N., Groupes et algèbres de Lie, Chapitres IV, V, VI, Hermann, Paris, 1968.Google Scholar
3. Coxeter, H. S. M., The abstract groups, Gm, n, p, Trans. Amer. Soc. 45 (1939), 73–150.Google Scholar
4. Coxeter, H. S. M., Regular compound tessellations of the hyperbolic plane, Proc. Roy. Soc. A 278 (1964), 147–167.Google Scholar
5. Coxeter, H. S. M., Non-Euclidean Geometry, 5th edition, University of Toronto Press, 1965.Google Scholar
6. Coxeter, H. S. M., Inversive distance, Annali di Mat. (4) 71 (1966), 73–83.Google Scholar
7. Coxeter, H. S. M., Twelve Geometric Essays, Southern Illinois University Press, Carbondale, IL., 1968.Google Scholar
8. Coxeter, H. S. M., Twisted honeycombs, Regional Conference Series in Math. 4 (Amer. Math. Soc. 1970).Google Scholar
9. Coxeter, H. S. M., Regular Polytopes, 2nd edition, Dover, New York, 1973.Google Scholar
10. Coxeter, H. S. M., Parallel lines, Canad. Math. Bull. 21 (1978), 385–397.Google Scholar
11. Coxeter, H. S. M., Unvergängliche Geometrie (2. Auflage), Birkhäuser Verlag, Basel, 1981.Google Scholar
12. Coxeter, H. S. M., Introduction to Geometry, 2nd edition, Wiley, New York, 1969.Google Scholar
13. Coxeter, H. S. M., Projective Geometry, 2nd edition, Springer Verlag, New York, 1987.Google Scholar
14. Coxeter, H. S. M., Regular Complex Polytopes, 2nd edition, Cambridge University Press, 1991.Google Scholar
15. Coxeter, H. S. M. and de Craats, Jan van, Philon lines in non-Euclidean planes, J. Geom. 48 (1993), 26–55.Google Scholar
16. Coxeter, H. S. M. and Greitzer, S. L., Geometry Revisited, Math. Assoc. of America, Washington, DC, 1967.Google Scholar
17. Ernst, B., The Magic Mirror of M.C. Escher, Random House, New York, 1976.Google Scholar
18. Tóth, L. Fejes, Regular Figures, Pergamon, New York, 1964.Google Scholar
19. Lobachevsky, N. I., Geometrical researches on the theory of parallels, pp. 1-48 at the end of Bonola [1].Google Scholar
20. Magnus, W., Noneuclidean Tesselations and their Groups, Academic Press, New York, 1974.Google Scholar
21. Sherk, F. A., P.McMullen, Thompson, A. C. and Weiss, A. I., Kaleidoscopes, Wiley-Interscience, New York, 1995.Google Scholar
22. Vinberg, E. B., Reflection groups, Soviet Encyclopaedia of Mathematics, 20, p. 33, Kluwer, Dordrecht, 1992.Google Scholar