Abstract
In this paper, we give an account of the algorithmic approach developed by the author to study tilings of hyperbolic spaces. We sketchily remember the results which were obtained by this approach and we conclude by possible applications, indicating a few ones already performed and proposing three others.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Ben-Jacob, E., Becker, I., Shapira, Y.: Bacterial Linguistic Communication and Social Intelligence. Trends in Microbiology 12(8), 366–372 (2004)
Berger, R.: The undecidability of the domino problem. Memoirs of the American Mathematical Society 66, 1–72 (1966)
Chelghoum, K., Margenstern, M., Martin, B., Pecci, I.: Palette hyperbolique: un outil pour interagir avec des ensembles de données. In: IHM 2004, Namur (2004) (Hyperbolic chooser: a tool to interact with data, (French))
Chelghoum, K., Margenstern, M., Martin, B., Pecci, I.: Tools for implementing cellular automata in grid {7,3} of the hyperbolic plane. In: DMCS 2004 (2004)
Cook, M.: Universality in Elementary Cellular Automata. Complex Systems 15(1), 1–40 (2004)
Hedlund, G.: Endomorphisms and automorphisms of shift dynamical systems. Math. Systems Theory 3, 320–375 (1969)
Henderson, D.W., Taimina, D.: Crocheting the Hyperbolic Plane. Mathematical Intelligencer 23(2), 17–28 (2001)
Herrmann, F., Margenstern, M.: A universal cellular automaton in the hyperbolic plane. Theoretical Computer Science 296, 327–364 (2003)
Iwamoto, C., Margenstern, M.: Time and Space Complexity Classes of Hyperbolic Cellular Automata. IEICE Transactions on Information and Systems 387-D(3), 700–707 (2004)
Iwamoto, C., Margenstern, M., Morita, K., Worsch, T.: Polynomial Time Cellular Automata in the Hyperbolic Plane Accept Exactly the PSPACE Languages. In: SCI 2002 (2002)
Kari, J.: Reversibility and Surjectivity Problems of Cellular Automata. Journal of Computer and System Sciences 48(1), 149–182 (1994)
Margenstern, M.: New Tools for Cellular Automata of the Hyperbolic Plane. Journal of Universal Computer Science 6(12), 1226–1252 (2000)
Margenstern, M.: A contribution of computer science to the combinatorial approach to hyperbolic geometry. In: SCI 2002 (2002)
Margenstern, M.: Revisiting Poincaré’s theorem with the splitting method. In: Bolyai 200. International Conference on Hyperbolic Geometry, Cluj-Napoca, Romania, October 1-4 (2002)
Margenstern, M.: Implementing Cellular Automata on the Triangular Grids of the Hyperbolic Plane for New Simulation Tools. In: ASTC 2003 (2003)
Margenstern, M.: Can Hyperbolic Geometry Be of Help for P Systems? In: Martín-Vide, C., Mauri, G., Păun, G., Rozenberg, G., Salomaa, A. (eds.) WMC 2003. LNCS, vol. 2933, pp. 240–249. Springer, Heidelberg (2004)
Margenstern, M.: The tiling of the hyperbolic 4D space by the 120-cell is combinatoric. Journal of Universal Computer Science 10(9), 1212–1238 (2004)
Margenstern, M.: A new way to implement cellular automata on the penta- and heptagrids. Journal of Cellular Automata 1(1), 1–24 (2006)
Margenstern, M.: On the communication between cells of a cellular automaton on the penta- and heptagrids of the hyperbolic plane. Journal of Cellular Automata 1(3), 213–232 (2006)
Margenstern, M.: A universal cellular automaton with five states in the 3D hyperbolic space. Journal of Cellular Automata 1(4), 315–351 (2006)
Margenstern, M.: Cellular Automata in Hyperbolic Spaces. Theory, vol. 1, p. 422. Old City Publishing, Philadelphia (2007)
Margenstern, M.: The Domino Problem of the Hyperbolic Plane is Undecidable. Bulletin of the EATCS 93, 220–237 (2007)
Margenstern, M.: The domino problem of the hyperbolic plane is undecidable. Theoretical Computer Science 407, 29–84 (2008)
Margenstern, M.: The Finite Tiling Problem Is Undecidable in the Hyperbolic Plane. International Journal of Foundations of Computer Science 19(4), 971–982 (2008)
Margenstern, M.: On a Characteriztion of Cellular Automata in Tilings of the Hyperbolic Plane. International Journal of Foundations of Computer Science 19(5), 1235–1257 (2008)
Margenstern, M.: Cellular Automata in Hyperbolic Spaces. Implementation and computations, vol. 2, p. 360. Old City Publishing, Philadelphia (2008)
Margenstern, M.: The Injectivity of the Global Function of a Cellular Automaton in the Hyperbolic Plane is Undecidable. Fundamenta Informaticae 94(1), 63–99 (2009)
Margenstern, M.: About the Garden of Eden theorems for cellular automata in the hyperbolic plane. Electronic Notes in Theoretical Computer Science 252, 93–102 (2009)
Margenstern, M.: The periodic domino problem is undecidable in the hyperbolic plane. In: Bournez, O., Potapov, I. (eds.) RP 2009. LNCS, vol. 5797, pp. 154–165. Springer, Heidelberg (2009)
Margenstern, M.: A universal cellular automaton on the heptagrid of the hyperbolic plane with four states. Theoretical Computer Science (to appear, 2010)
Margenstern, M.: A weakly universal cellular automaton in the hyperbolic 3D space with three states. arXiv:1002.4290v1[cs,DS], p. 54
Margenstern, M.: A weakly universal cellular automaton in the hyperbolic 3D space with two states. arXiv:1005.4826v1[cs,FL], p. 38
Margenstern, M., Martin, B., Umeo, H., Yamano, S., Nishioka, K.: A Proposal for a Japanese Keyboard on Cellular Phones. In: Umeo, H., Morishita, S., Nishinari, K., Komatsuzaki, T., Bandini, S. (eds.) ACRI 2008. LNCS, vol. 5191, pp. 299–306. Springer, Heidelberg (2008)
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)
Margenstern, M., Skordev, G.: Fibonacci Type Coding for the Regular Rectangular Tilings of the Hyperbolic Plane. Journal of Universal Computer Science 9(5), 398–422 (2003)
Margenstern, M., Skordev, G.: Tools for devising cellular automata in the hyperbolic 3D space. Fundamenta Informaticae 58(2), 369–398 (2003)
Margenstern, M., Song, Y.: A universal cellular automaton on the ternary heptagrid. Electronic Notes in Theoretical Computer Science 223, 167–185 (2008)
Margenstern, M., Song, Y.: A new universal cellular automaton on the pentagrid. Parallel Processing Letters 19(2), 227–246 (2009)
Martin, B.: VirHKey: a VIRtual Hyperbolic KEYboard with gesture interaction and visual feedback for mobile devices. In: Mobile HCI 2005, pp. 99–106 (2005)
Morgenstein, D., Kreinovich, V.: Which Algorithms are Feasible and Which are not Depends on the Geometry of Space-Time. Geocombinatorics 4(3), 80–97 (1995)
Moore, E.F.: Machine Models of Self-reproduction. In: Proceedings of the Symposium in Applied Mathematics, vol. 14, pp. 17–33 (1962)
Myhill, J.: The Converse to Moore’s Garden-of-Eden Theorem. In: Proceedings of the American Mathematical Society, vol. 14, pp. 685–686 (1963)
Robinson, R.M.: Undecidability and nonperiodicity for tilings of the plane. Inventiones Mathematicae 12, 177–209 (1971)
Robinson, R.M.: Undecidable tiling problems in the hyperbolic plane. Inventiones Mathematicae 44, 259–264 (1978)
Stewart, I.: A Subway Named Turing, Mathematical Recreations in Scientific American, pp. 90–92 (1994)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Margenstern, M. (2010). An Algorithmic Approach to Tilings of Hyperbolic Spaces: 10 Years Later. In: Gheorghe, M., Hinze, T., Păun, G., Rozenberg, G., Salomaa, A. (eds) Membrane Computing. CMC 2010. Lecture Notes in Computer Science, vol 6501. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18123-8_6
Download citation
DOI: https://doi.org/10.1007/978-3-642-18123-8_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-18122-1
Online ISBN: 978-3-642-18123-8
eBook Packages: Computer ScienceComputer Science (R0)