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Small Universal Cellular Automata in Hyperbolic Spaces

Part of the Emergence, Complexity and Computation Book Series (volume 4)

DOI: 10.1007/978-3-642-36663-5

ISBN: 978-3-642-36662-8

Publisher: Springer Berlin Heidelberg - Springer Journals

Jan 1, 2013

1 - 9 of 9 Chapters

Available

Why Hyperbolic Geometry?

Margenstern, Maurice

Pages: 1 –26

[In this first chapter, I shall introduce the unfamiliar reader to hyperbolic geometry. Section 1.1 is a very short survey about the history of this geometry. This history is a very important step in the general history of sciences. But this would require almost a whole book. Another summary can...
Available

Cellular Automata and the Railway Model

Margenstern, Maurice

Pages: 27 –56

[As suggested by the title of this book, here we are interested in universality questions only for cellular automata in hyperbolic spaces. In Section 2.1, we have a self-contained presentation of the implementation of cellular automata in hyperbolic spaces. In Section 2.2, we explain the model we...
Available

In the Pentagrid

Margenstern, Maurice

Pages: 57 –78

[In this chapter, we present the best result, up to now, of a planar cellular automaton in the pentagrid which is weakly universal. In Section 3.1 the scenario of the implementation is thoroughly described. In Section 3.2, we translate this description in a full set of rules.]
Available

In the Heptagrid

Margenstern, Maurice

Pages: 79 –114

[This chapter deals with the same problem in the heptagrid. This time we present two results, in Section 4.1 and in Section 4.2. The first result is an adaptation to the heptagrid of what we performed in Chapter 3. The second result requires new ideas detailed in Section 4.2.]
Available

In the Tilings {p, 3}

Margenstern, Maurice

Pages: 115 –189

[In this chapter, in Section 5.1, we introduce cellular automata in a grid {p, 3}. In Section 5.2 we describe the new scenario for a weakly universal planar cellular automaton on a grid {13, 3}, still within our railway model. In Section 5.3, we transform the precise description of the previous...
Available

In the Dodecagrid

Margenstern, Maurice

Pages: 191 –248

[This chapter deals with the same problem in the dodecagrid, in the hyperbolic 3D space. Section 6.1 reminds the first improvement of the first result established in this frame. Section 6.2 describes the best solution while Section 6.3 establishes the corresponding rules with a few words about...
Available

Strongly Universal Hyperbolic Cellular Automata

Margenstern, Maurice

Pages: 249 –280

[In this chapter, we deal with a simple way to prove universality theorems: it consists in embedding results of one-dimensional cellular automata in the desired grid of the hyperbolic plane or of the hyperbolic 3D space. In Section 7.1, we present a general statement which will entail results...
Available

The Help of Programming

Margenstern, Maurice

Pages: 281 –314

[In this chapter, we deal with the programming tools used for this book. Section 8.1 gives an account of the program used to check the results of the previous chapters. Section 8.2 deals with the drawing of the figures used in the book.]
Available

Further Work

Margenstern, Maurice

Pages: 315 –315

[Paradoxically, this chapter will be very short.]

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