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Über dieses Buch

This fascinating, colourful book offers in-depth insights and first-hand working experiences in the production of art works, using simple computational models with rich morphological behaviour, at the edge of mathematics, computer science, physics and biology. It organically combines ground breaking scientific discoveries in the theory of computation and complex systems with artistic representations of the research results. In this appealing book mathematicians, computer scientists, physicists, and engineers brought together marvelous and esoteric patterns generated by cellular automata, which are arrays of simple machines with complex behavior. Configurations produced by cellular automata uncover mechanics of dynamic patterns formation, their propagation and interaction in natural systems: heart pacemaker, bacterial membrane proteins, chemical rectors, water permeation in soil, compressed gas, cell division, population dynamics, reaction-diffusion media and self-organisation.

The book inspires artists to take on cellular automata as a tool of creativity and it persuades scientists to convert their research results into the works of art.

The book is lavishly illustrated with visually attractive examples, presented in a lively and easily accessible manner.

Inhaltsverzeichnis

Self-Organizing Two-Dimensional Cellular Automata: 10 Still Frames

A favorite topic of my research in the 1980s and 1990s was pattern formation of interacting particle systems and cellular automata (CA) started from random initial states, especially in two dimensions. Together with my students, and later colleagues, Bob Fisch, Janko Gravner, and Kellie Evans, I studied a wide variety of local lattice evolutions started from noise. With the advent of personal computers it became possible to interact with simulations of these algorithms in order to observe system behavior and sometimes gain insight leading to rigorous mathematical results.

David Griffeath

Is it Art or Science?

Is the art of cellular automata (CA) a legitimate subject of preoccupation? — yes, of course! Although the study of CA is an exercise in experimental dynamics by computer algorithms on top of which mathematical theories and conjectures are superimposed, the graphic representations themselves confer intuitive subjective impressions that are inescapably art. To the simple art lover these are intriguing immediate images which imagination may strive to interpret or merely accept. To the CA theorist and practitioner the “art” is imbued with layers of deeper meaning, just as Zen art can be experienced either on the surface or by the Zen master.

Andrew Wuensche

Larger Than Life

Images in this section provide a sample of the rich dynamics and complex selforganization exhibited by Larger than Life (LtL), a four-parameter family of twodimensional cellular automata that generalizes John Horton Conway’s celebrated Game of Life (Life) to large neighborhoods and general birth and survival thresholds. LtL was proposed by David Griffeath in the early 1990s to explore whether Life might be a clue to a critical phase point in the threshold-range scaling limit [80].

Kellie Michele Evans

Three Favorite Cellular Automata

Cellular automata (CA) are attractive visually and offer strikingly simple definitions for well motivated models that can produce complex behavior. My study and research with CA began in earnest the 1990’s and since then I have explored many interesting automata. My research on CA has included models with the symmetry of snowflakes, self-organizing waves produced by prey-predator models, and cyclic CA on various nonrectangluar lattices and in three dimensions. In addition, I have explored many local image processing techniques that are essentially CA for applications to smoothing and enhancing images, and creating Vorinoi tilings and other patterns.

Clifford Reiter

Cellular Automata: Dying to Live Again, Architecture, Art, Design

As in our own lives, outcomes are not predictable. We start in a simple unknown state and through multiple growth and death cycles of individual cells, we become something that could never have been predicted. Can design be explored using a similar process? Can design and art be driven by forces that are unrelated, unattached to preconceptions, unpredictable, and able to explore possibilities not foreseen? What to do you when you don’t know what to do? Let the process of life and death take over?

Robert J. Krawczyk

In Search of Movement and Life on a Static Grid

The search for life-like processes that can be simulated on computer has led to many types of proposed systems. Of these, cellular automata are one of the most venerable and one of the most visually appealing.

Tim J. Hutton

Some Beautiful and Difficult Questions About Cellular Automata

One of the most interesting things about cellular automata (CA) is that such simple sets of rules can give rise not only to such strange and unpredictable behavior, but also that it can create seemingly predictable behavior that is nonetheless extremely difficult to actually pin down. One example of this apparently simple behavior comes from the “Life without Death” cellular automaton, which is the exact same as Conway’s Game of Life with the exception that cells never die. Just by playing around with random starting patterns, it immediately becomes “clear” that the majority of sufficiently large patterns explode quadratically in this rule and fill most of the 2D plane. However, it was not even known if there is a single quadraticallygrowing pattern in this rule until Dean Hickerson found one in 2009, over 20 years after this CA was first studied.

Nathaniel Johnston

Hyperbolic Gallery

We give five illustrations of cellular automata (CA) in hyperbolic spaces. The text under each picture indicates the paper from which it is taken together with some indications on the meaning of the picture. In order to help the reader to grasp something about what the meaning of each picture is, here we give a short introduction to hyperbolic spaces and then the general context of my research. The below figure gives a schematic representation of the hyperbolic plane in what is called the Poincaré’s disc.

Maurice Margenstern

Evolved Gliders and Waves on a Geodesic Grid

Cellular automata (CA) are typically modeled on 1D, 2D, or 3D grids. To avoid boundary artifacts, it is common to create periodic boundary conditions so that opposing boundarieswrap around. In the case of a 2D cellular automaton, this periodic boundary creates the topological equivalent of a torus. A 2D grid can be mapped onto a 3D torus without introducing cell neighborhood discontinuities. This mapping introduces some geometrical distortion, but no topological artifacts — no discontinuities in grid connectivity. This is not the case when mapping a grid onto a sphere, which has positive curvature everywhere.

Jeffrey Ventrella

Constructing Counters Through Evolution

In the one-dimensional synchronisation task, discussed in [150], the final pattern consists of an oscillation between all 0s and all 1s. From an engineering point of view, this period-2 cycle may be considered a 1-bit counter. Building upon such an evolved CA, using a small number of different cellular clock rates, 2- and 3-bit counters can be constructed.

Moshe Sipper

Biological Lattice-Gas Cellular Automata

Biological lattice-gas cellular automata (LGCA) can be viewed as models for collective behaviour emerging frommicroscopicmigration and interaction processes of biological cells. Such LGCA are used to model the interplay of cells with each other and with their heterogeneous environment by describing interactions at a cell-based (microscopic) scale and facilitating both efficient simulation and theoretical analysis of emergent, tissue-scale (macroscopic) parameters [47].

Andreas Deutsch

The Enlightened Game of Life

The interaction of light with complex matter is one of the fundamental subjects in modern physics and biology. This topic is relevant from a variety of different perspectives, including, among others, modelling of animal behaviour (as the moonlight driven coral reef spawning), evolution (the development of the eye), and light activated matter (as, e.g., opto-genetics and laser-driven micro-motors). From a very abstract point of view, if one accepts cellular automata (CA) as the simplest mathematical model for life (as originally suggested by Conway), one can try to include the interaction with light and electromagnetic radiation by enlarging the set of evolution rules to obtain a photo-sensible CA.

Claudio Conti

Small Synchronizers and Prime Generators

We illustrate a prime sequence generation problem and a firing squad synchronisation problem on cellular automata (CA) with different communication models. For a long time there was little use of prime numbers in practical applications. But nowadays, it has been known that large scale prime numbers play a very important role in encryption in computer security networks. The question is “How can we generate prime numbers in real-time on a small-state CA?”.

Hiroshi Umeo

Ecological Patterns of Self-Replicators

The aesthetic sense of humans has part of its origins in human evolution. Our ancestors must have evolved the abilities to feel various aesthetic values on other living creatures they shared their habitats with, sometimes with joy, sometimes with fear, sometimes with awe. The same principle might apply to the art of cellular automata (CA) too; if you see living creatures flourishing or struggling in CA, that might trigger a certain kind of aesthetic feeling that is rather different from the one you feel for purely mathematical, geometrical patterns.

Hiroki Sayama

The Art of Penrose Life

John Horton Conway’s Game of Life (GoL) [39, 71] is a simple two-dimensional, two state cellular automaton (CA), remarkable for its complex behaviour [39, 136]. The classic GoL is defined on a regular square lattice.

Susan Stepney

Asynchronous Cellular Automata Simulating Complex Phenomena

The concept of asynchronous cellular automaton (ACA) differs from that of classical synchronous cellular automata (CA) in the mode of operation, namely cells transit to next states at random being updated sequentially. ACA are especially suitable for simulation chemical, biological, and physical phenomena, which are dissipative, nonlinear, and stochastic. Usually, such phenomena are represented as a combination of movements and transformations of real or abstract particles.

Olga Bandman

A Multiparticle Lattice-Gas Cellular Automaton Simulating a Piston Motion

The well known lattice-gas cellular automata (CA), [68] simulate incompressible viscous fluid flow. The cell states contains Boolean six component vectors $$s(x) = (s_1, . . . , s_6)$$, where $$s_i = 1$$ is interpreted as a particle with a unit mass, propagating towards the ith cell of the neighborhood T(x), which contains six adjacent to x cells in a hexagonalcellulararray. Global transition rule $$\Theta (X)$$ is a superposition of two global operators: $$\Theta _{C}(X)$$, simulating particles collision, and $$\Theta _{P}(X)$$, simulating the propagation. Transition rules obey mass and momentum conservation laws. Simulation capabilities of the model are limited by the maximal Reynolds number, which does not allow to use the model for simulating inviscid gas flows.

Yuri Medvedev

Two Layer Asynchronous Cellular Automata

Asynchronous cellular automata (ACA) whose transition functions contain activation and inhibition components are capable to simulate a great variety of complex systems behavior. The competition between activators and inhibitors impact on the ACA functioning contributes to regulation of ACA evolution properties [178].

Anastasiya Kireeva

Cellular Automata Simulation of Bacterial Cell Growth and Division

The exact mechanisms of growth and division in bacterial cells (further, bacterial cell is referred to as bacterium to avoid confusion with a cell in cellular automata) is not clear so far. Investigations in the field are focused on studying the process in the bacteria of Escherichia coli (E.coli) [59, 100].

Anton Vitvitsky

Seismic Cellular Automata

The proposed cellular automata driven potential-basedmodel for earthquake simulation is a dynamic system constituted of cells-charges. It is assumed that the system balances through the exercitation of electrostatic Coulomb-forces among charges, without the existence of any other form of interconnection in-between. Such kinds of forces are also responsible for this level to be bonded with a rigid but moving plane below.

Ioakeim G. Georgoudas, Georgios Ch. Sirakoulis, Emmanuel M. Scordilis, Ioannis Andreadis

DNA Cellular Automata

The elementary cellular automata (CA) evolution rule can be extracted from a given number of CA evolution patterns and this can also be applied to the CAs that model DNA sequences. We map DNA to CA in such manner that sugar-phosphate backbone of a DNA molecule corresponds to the CA lattice and the organic bases to the CA cells. At each position of the lattice one of the four bases A (Adenine), C (Cytosine), T (Thymine) and G (Guanine) of the DNA molecule may be allocated, corresponding to the four possible states of the CA cell.

Charilaos Mizas, Georgios Ch. Sirakoulis, Vasilios Mardiris, Ioannis Karafyllidis, Nicholas Glykos, Raphael Sandaltzopoulos

Reversibility, Simulation and Dynamical Behaviour

A cellular automaton (CA) is reversible if it repeats its configuration in a cycle. Reversible one-dimensional CA are studied as automorphisms of the shift dynamical system, and analyses using graph-theoretical approaches and with block permutations. Reversible CA are dynamical systems which conserve their initial information. This is why they pose a particular interest in mathematics, coding and cryptography.

Juan Carlos Seck Tuoh Mora, Norberto Hernandez Romero, Joselito Medina Marin

Aesthetics and Randomness in Cellular Automata

We propose two images obtained with an asynchronous and a stochastic cellular automaton (CA). Deterministic cellular automata are now well-studied models and even if there is still so much to understand, their main properties are now largely explored. By contrast, the universe of asynchronous and stochastic is mainly a terraincognita. Only a few islands of this vast continent have been discovered so far. The two examples below present space-time diagrams of one-dimensional cellular automata with nearest-neighbour interaction.

Nazim Fatès

Cellular Automata with Memory

In conventional discrete dynamical systems, the new configuration depends solely on the configuration at the preceding time step. This contribution considers an extension to the standard framework of dynamical systems by taking into consideration past history in a simple way: the mapping defining the transition rule of the system remains unaltered, but it is applied to a certain summary of past states.

Ramon Alonso-Sanz

Turing Machine in Conway Game of Life

The complexity of the behaviour that some simple cellular automata (CA) can exhibit is clearly shown by a demonstration of universal computation. The ability to perform any calculation that a Turing machine can make. The construction of such a machine necessarily requires areas covered by repeating patterns for storage of data and programs as well as areas of variation encoding information on a more direct way producing an uneven texture when viewed at a distance. These textures together with the inherent symmetry of the design lead naturally to artistically pleasing images.

Paul Rendell

Aperiodicity and Reversibilty

We show cellular automata (CA) configurations obtained in our studies of spacetime dynamics on aperiodic tilings and reversible self-reproducing CA. The families of CA considered have almost nothing in common but their unconventionality. In 2012 Goucher found a glider propagating CA on Penrose tilings [75]. His work was inspired by the study of the variants of the Game of Life on Penrose tilings conducted by Owens and Stepney [130, 131].

Katsunobu Imai

Painting with Cellular Automata

Images created by Nature fascinate us, and always have. People try to reproduce them in different forms, according to their own skills and/or personal feelings. Artists, for example, use sounds, colours, stones to express their fascination with what they see or what they sense. Scientists, on the other hand, demonstrate their fascination through their strong desire to understand natural phenomena. Discoveries being made day by day in physiology labs may revise many of our conceptions, including the concept of computation.

Danuta Makowiec

Patterns in Cellular Automata

The idea of a one dimensional cellular automaton (CA) is quite simple, and its evolution in time is ideal for a two dimensional presentation, as on a video screen. To start with, a cell is a region, even a point, with differing forms, called states. For convenience, these states are usually numbered with small integers beginning with zero, rather than described. For the purposes of automata theory the nature of the states does not matter, only their relation to one another, and the way they change with time according to their environment. Since they are abstract, they can just as well be represented by coloured dots on a video screen, which is what makes them so dramatic when interpreted as an abstract artistic design.

Harold V. McIntosh

Gliders in One-Dimensional Cellular Automata

Gliders are non-trivial complex patterns emerging typically in complex cellular automata (CA). The most famous glider (mobile self-localizations, particles, waves) is the five-live cells glider moving diagonally in five steps in the two-dimensional CA Conway’s Game of Life [71]. Gliders are abstractions of travelling localizations often found in living systems, and used to derive novel properties of objects in artificial life, complex systems, physical systems, and chemical reactions. Examples include, gliders in reaction-diffusion systems [16, 128], Penrose tilings [75], three-dimensional glider gun [15], gliders in hyperbolic spaces [109].

Genaro J. Martínez

Excitable Automata

Cellular automata (CA) are computationally efficient and user-friendly tools for simulation of large-scale spatially extended locally connected systems. CA representation of reaction-diffusion and excitable systems is especially interesting because this allows us to effortlessly map already established massively-parallel architectures onto novel material base of chemical systems, and also design novel non-classical and nature-inspired computing architectures [22].

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