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Cellular Automata in Hyperbolic Spaces

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Abbreviations

Fibonacci sequence:

A sequence of natural integers, denoted by f n and defined by the recurrent equation \( { f_{n+2}=f_{n+1}+f_n } \), for all \( { n\in I\!\!N } \), and by the initial values \( { f_0=f_1=1 } \).

Hyperbolic geometry:

This geometry was discovered independently by both Nikolaj Lobachevsky and Jànos Bolyai around 1830. This geometry satisfies the axioms of Euclidean geometry, the axiom of parallels being excepted and replaced by the following one: through a point out of a line, there are exactly two parallels to the line. In this geometry, there are also lines which never meet: they are called non-secant. They are characterized by the existence, for any couple of such lines, of a unique common perpendicular. Also, in this geometry, the sum of the interior angles of a triangle is always less than π. The difference to π defines the area of the triangle. In hyperbolic geometry, distances are absolute: there is no notion of similarity. See also Poincaré's disc.

Pentagrid:

The tiling \( { \{5,4\} } \), with five sides and four tiles around a vertex. The angles are right angles.

Poincaré's disc:

A model of the hyperbolic plane inside the Euclidean plane. The points are the points which are interior to a fixed disc D. The lines are the trace in D of diameters or circles which are orthogonal to the border of D. The model was first found by Beltrami and then by Poincaré who also devised the half-plane model also called after his name. The half-plane model is a conformal image of the disc model.

Ternary heptagrid:

The tiling \( { \{7,3\} } \) with seven sides and three tiles around a vertex.

Tessellation:

A particular case of a finitely generated tiling. It is defined by a polygon and by its reflections in its sides and, recursively, of the images in their sides.

Tiling:

A partition of a geometric space; the closure of the elements of the partition are called the tiles. An important case is constituted by finitely generated tilings: there is a finite set of tiles G such that any tile is a copy of an element of G.

Tiling \( {\{p,q\}} \) :

This is tessellation based on the regular polygon with p sides and with vertex angle \( { {2\pi} / q } \).

Invariant group of a tiling:

A group of transformations which define a bijection on the set of tiles. Usually, in a geometrical space, they are required to belong to the group of isometries of the space.

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Acknowledgments

The author is particularly in debt to Andrew Adamatzky for giving him the task to write this article.

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Authors and Affiliations

  1. Université Paul Verlaine, Metz, France

    Maurice Margenstern

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  1. Maurice Margenstern
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Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Margenstern, M. (2009). Cellular Automata in Hyperbolic Spaces. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_53

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