On the Infinigons of the Hyperbolic Plane, A combinatorial approach

• Autorzy: Margenstern M.
• Języki publikacji: EN

Abstrakty

EN

In this paper, we pay a new visit to an object of hyperbolic geometry which, perhaps, did not draw on itself all the attention it may deserve. The paper gives a simple definition of this object, infinigons, which was implicit in general considerations about tilings of the hyperbolic plane, and which was not definied in its all possible extensions. From the simple construction of the infinigons and using the ideas of the splitting method being introduced by the author in the case of tilings being based on the replication of regular polygons, we give an algorithmic construction of the infinigrids. On the way, we give a simple geometrical characterisation of the infinigons in terms of pencils of lines.

Słowa kluczowe

EN

hyperbolic geometry; tilings; combinatorial geometry

• Wydawca: IOS Press
• Czasopismo: Fundamenta Informaticae
• Rocznik: 2003
• Tom: Vol. 56, nr 3
• Strony: 255--272
• Opis fizyczny: wykr., bibliogr. 21 poz.

Twórcy

autor

Margenstern M.

• Université de Metz, I.U.T. de metz, LITA, EA 3097, Ile du Saulcy, 57045 Metz Cedex, France,

Bibliografia

• [1] Broughton S. A., Constructing Kaleidoscopic Tiling Polygons in the Hyperbolic Plane, American Mathematical Monthly, vol. 107, N 8, 689-709, (2000).
• [2] C. Caratheodory. Theory of functions of a complex variable, vol.II, 177-184, Chelsea, New-York, 1954.
• [3] H.S.M. Coxeter and G.J. Whitrow, World-structure and non-Euclidean honeycombs, Proceesing of the Royal Society of London. ser. A, 201, 417-437, (1950).
• [4] H.S.M. Coxeter, Regular honeycombs in hyperbolic space, International Congress of Mathemaricians, (1954), 3, 155-169.
• [5] D. B. A. Epstein, J. W. Cannon, D. F. Holt, S. V. F. Levi, M. S. Paterse, W. P. Thurston, Word Processing in Groups, Jones and Barlett Publ., Boston, (1992).
• [6] Goodman-Strauss Ch., Compass and Straightedge in the Poincaré Disk, American Mathematical Monthly, vol. 108, N 1, 38-49, (2001).
• [7] C. Goodman-Strauss, A strongly aperiodic set of tiles in the hyperbolic plane, submitted.
• [8] S. Grigorieff, M. Margenstern, Register cellular automata in the hyperbolic plane, invited talk at SCI’2002, Orlando, July, 14-18, (2002).
• [9] F. Herrmann, M. Margenstern, A universal cellular automaton in the hyperbolic plane, Theoretical Computer Science, 296-2, 38p., (2003).
• [10] Ch. Iwamoto, M. Margenstern, K. Morita, T. Worsch, Polynomial-Time Cellular Automata in the Hyperbolic Plane Accept Exactly the PSPACE Languages, Proceedings of SCI’2002, Orlando, July, 14-18, 2002, (2002).
• [11] C. Mann, On Heesch’s problem and other tiling problems, PhD Thesis, University of Arkansas, (2001).
• [12] Margenstern M., New tools for Cellular Automata in the Hyperbolic Plane, Journal of Universal Computer Science, vol 6, issue 12, 1226-1252, (2000).
• [13] M. Margenstern. Cellular automata in the hyperbolic plane. A survey, Romanian Journal of Information Science and Technology, 5, 1-2, 155-179, (2002), (invited paper).
• [14] M. Margenstern. Cellular Automata and Combinatoric Tilings in Hyperbolic Spaces, a Survey, Lecture Notes in Computer Sciences, proceedings of DMTCS’03, Dijon, France, July 7-12, (2003), (invited talk).
• [15] M. Margenstern, A combinatorial approach to infinigons and to infinigrids of the hyperbolic plane, SCI’2002, Orlando, July, 14-18, 2002, (2002).
• [16] Margenstern M., Morita K., NP-problems are tractable in the space of cellular automata in the hyperbolic plane, Theoretical Computer Science, 259, 99-128, (2001).
• [17] G.A. Margulis, S. Mozes, Aperiodic tilings of the hyperbolic plane by convex polygons, Israel Journal of Mathematics, 107, 319-325, (1998).
• [18] H. Meschkowski, Noneuclidean geometry, translated by A. Shenitzer, Academic Press, New-York, 1964.
• [19] Effectve simulations on hyperbolic networks, Fundamenta Informaticae, 53, (2002), 203-231.
• [20] C. Papazian, Recognizers on hyperbolic graphs of finite automata, AUTOMATA’2001, September, 27-29, 2001, Giens, France.
• [21] A. Ramsay, R.D. Richtmeyr, Introduction to hyperbolic geometry. Springer Verlag, 1995.