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This paper describes the dynamics of mobile structures (gliders) in 2D cellular automata (CA) on geodesic grids. 2D CA are typically arranged on regular grids with periodic boundary conditions - equivalent to the topology of a torus. This paper describes an alternative topology - the sphere, with an underlying agenda to better understand natural closed systems. The positive curvature of the sphere as manifested in geodesic grids is described as a rich environment for CA. The necessary grid discontinuities are accepted as integral components of the environment. They are not considered as defects but rather as environmental features to be exploited. To explore the potential for a uniquely spherical computational space, a novel XOR gate built on Conway's Game of Life is demonstrated, utilizing the double-crossing of glider paths following geodesic great circles.

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... Since hexagonal lattice is free from spurious symmetries of the square grid it has been implemented for predicting the spreading of wildfire in [43]. GOL on the surface of geodesic sphere, with all triangular facets, has been studied in [46]. For a corresponding animation see [45]; and for a corresponding interactive demonstration of ST CA on icosahedral geodesic sphere see [56]. ...

... This type of CAs is especially interesting for CASS, due to its relatively straightforward applicability also in hexagonal and triangular tessellations, as described in Chap. 3: "Polarized Film Shading System in regular grids" in Sect. 3 46) GFE has been calculated once for each CA rule at the same sequence of randomly generated initial conditions. Therefore, the results are rather illustrative than conclusive. ...

... 46 The patterns of selected T 2-CAs: 56, 84, 88 and 106. The convention as inFig. ...

This chapter collects the findings of the research on the cellular automaton-based shading systems (CASS) for building envelopes. CASS is based on congruent modular units, thus it has the potential of being inexpensive and robust. Two approaches for the realization of CASS are presented: based on the liquid crystal technology, and based on the rotation of polarized film elements. Several optimization methods of CASS are presented. The optimization criteria include: the “grayness” monotonicity, and cellular automaton (CA) pattern distribution error which represent: the level of control over the CA pattern, and its uniformity over entire array of cells, respectively. The robustness of CASS for various types of failure is discussed.

... By the introduction of random wirings in an otherwise regular lattice, a complex behaviour may completely degrade to a simple fixed pattern [16,42]. Subtle variations in the dynamics of a complex cellular automaton defined on a sphere have also been shown to expose novel logic implementations [57]. ...

... The variations in topology were based on the topological genus of the mesh. Thus our study can be considered as a broad quantitative study similar to that of Marr et al. [32,33] but looking at topologies more in line with those considered by Ventrella [57]. ...

... The triangular tessellation is sufficient to represent any surface, which reduces the complexity of topology construction. Very few studies have investigate the triangular tessellation [7], rather the square tessellation is the most common two dimensional representation [57,60]. Binary-state cellular automata in the triangular tessellation can be constructed with a neighbourhood size of 4, which is only one greater than the one dimensional elementary cellular automata; thus these simple triangular neighbourhoods may be considered to be the two dimensional equivalent of elementary cellular automata. ...

This thesis presents an empirical study of the effects of topology on cellular automata rule spaces. The classical definition of a cellular automaton is restricted to that of a regular lattice, often with periodic boundary conditions. This definition is extended to allow for arbitrary topologies. The dynamics of cellular automata within the triangular tessellation were analysed when transformed to 2-manifolds of topological genus 0, genus 1 and genus 2.
Cellular automata dynamics were analysed from a statistical mechanics perspective. The sample sizes required to obtain accurate entropy calculations were determined by an entropy error analysis which observed the error in the computed entropy against increasing sample sizes. Each cellular automata rule space was sampled repeatedly and the selected cellular automata were simulated over many thousands of trials for each topology. This resulted in an entropy distribution for each rule space.
The computed entropy distributions are indicative of the cellular automata dynamical class distribution. Through the comparison of these dynamical class distributions using the E-statistic, it was identified that such topological changes cause these distributions to alter. This is a significant result which implies that both global structure and local dynamics play a important role in defining long term behaviour of cellular automata.

... In our problem statement, features are inputs referred to as different welllog data. We developed a standard supervised SOM, referred herein as SSOM, based on a torus geometry (Ventrella, 2011), as exhibited in Fig. 5(a). A torus cross-section provides a two-dimensional domain composed by a rectangular grid of neurons, also known as nodes Carreira V.R. et al. ...

... ( Fig. 5b). This particular configuration guarantees that all nodes are connected and avoids boundary problems (Ventrella, 2011;Kohonen, 2013). More about the model structure of SOMs are detailed in Hoan (2016) and Han et al. (2019). ...

... Hexagonal DGGS as a geospatial data model can offer appropriate basis for spatial analysis and simulation at the global level (Kiester & Sahr, 2008). Spherical CA also provide an suitable grid topology for modelling and understanding naturally closed systems at the global scale (Ventrella, 2011). In a hexagonal grid, the relative location and neighbourhood of cells can be defined using the hexagonal coordinate system which has three coordinate axes that are evenly spaced apart and are usually denoted as . ...

Existing geosimulation land-use change models are predominantly designed to operate at local or regional spatial scales. When these models are applied on data at the global level, they do not consider the effects of spatial distortions caused by the curvature of the Earth’s surface and often lack some refinements related to land suitability analysis. Therefore, the main objective of this study is to integrate multi-criteria evaluation with spherical cellular automata, to develop a novel modelling approach for simulating global urban land-use change. The world region is clustered into sub-regions to improve the suitability analysis. The obtained results reveal differences in urban growth rate and the size of the extent across the four clusters. The 64% of the total global urbanization are occurring in urban region clusters characterized by high gross domestic product and population density, while urban regions in isolated locations have the lowest urban growth rate.

... This is not appropriate because of the spherical nature and curvature of the Earth which distorts the shape and size of grids. Such distortions are absent at the local scale but are however prominent when implemented on a larger scale (Ventrella, 2011). ...

Urbanization process is one of the drivers of environmental and social changes across the globe entailing many environmental problems. Long term land-use change geosimulation models are useful tools to represent the complex human–environment interactions and evaluate the impacts of urbanization on the environment. However, many modelling approaches are not always fit to fully address this process at global scale and the issues of distortions related to Earths’ curvature. Thus, the goal of this research study is to model and examine the long term global urban land-use change using spherical cellular automata approach. The developed model is implemented to simulate urban land-use change across 235 world countries and using two scenarios considering zero-migration and constant-fertility. The simulation results indicate that, between 2015 and 2095, the total global urban extent will double in size with the most extensive change in urban areas occurring in Africa and Asia. The proposed spherical model can be used to assist global urbanization policy making and environmental impact assessments.

... The transition rule takes into account the number of neighbors for multiple states using a set of sub-rules that are applied in sequence at time t to determine the final state at time t + 1. Transition rules are evolved using an interactive tool for breeding gliders. The technique is described in [163]. The web site http://www.ventrella.com/earthday/ ...

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.

... To avoid grid discretization effects on the simulations owing to non-uniformities of the icosahedral mesh, the interaction radius between cells on the sphere was set to be greater than the length scale of the typical internode spacing (for complications, see Ventrella [24]). The speed at which an excitation wavefront could propagate (conduction velocity) was determined by two free parameters: the search radius and the threshold of number of active neighbours. ...

Mathematical models of cardiac electrical excitation are increasingly complex, with multiscale models seeking to represent and bridge physiological behaviours across temporal and spatial scales. The increasing complexity of these models makes it computationally expensive to both evaluate long term (more than 60 s) behaviour and determine sensitivity of model outputs to inputs. This is particularly relevant in models of atrial fibrillation (AF), where individual episodes last from seconds to days, and interepisode waiting times can be minutes to months. Potential mechanisms of transition between sinus rhythm and AF have been identified but are not well understood, and it is difficult to simulate AF for long periods of time using state-of-the-art models. In this study, we implemented a Moe-type cellular automaton on a novel, topologically equivalent surface geometry of the left atrium. We used the model to simulate stochastic initiation and spontaneous termination of AF, arising from bursts of spontaneous activation near pulmonary veins. The simplified representation of atrial electrical activity reduced computational cost, and so permitted us to investigate AF mechanisms in a probabilistic setting. We computed large numbers (approx. 10⁵) of sample paths of the model, to infer stochastic initiation and termination rates of AF episodes using different model parameters. By generating statistical distributions of model outputs, we demonstrated how to propagate uncertainties of inputs within our microscopic level model up to a macroscopic level. Lastly, we investigated spontaneous termination in the model and found a complex dependence on its past AF trajectory, the mechanism of which merits future investigation.

... To avoid grid discretisation effects on the simulations due to non-uniformities of the icosahedral mesh, the interaction radius between cells on the sphere was set to be greater than the length scale of the typical inter-node spacing (for complications, see Ventrella [20]). The speed at which an excitation wavefront could propagate (conduction velocity) was determined by two free parameters: the search radius and the threshold of number of active neighbours. ...

Mathematical models of cardiac electrical excitation are increasingly complex, with multiscale models seeking to represent and bridge physiological behaviours across temporal and spatial scales. The increasing complexity of these models makes it computationally expensive to both evaluate long term (>60 seconds) behaviour and determine sensitivity of model outputs to inputs. This is particularly relevant in models of atrial fibrillation (AF), where individual episodes last from seconds to days, and inter-episode waiting times can be minutes to months. Potential mechanisms of transition between sinus rhythm and AF have been identified but are not well understood, and it is difficult to simulate AF for long periods of time using state-of-the-art models. In this study, we implemented a Moe-type cellular automaton on a novel, topologically correct surface geometry of the left atrium. We used the model to simulate stochastic initiation and spontaneous termination of AF, arising from bursts of spontaneous activation near pulmonary veins. The simplified representation of atrial electrical activity reduced computational cost, and so permitted us to investigate AF mechanisms in a probabilistic setting. We computed large numbers (~10^5) of sample paths of the model, to infer stochastic initiation and termination rates of AF episodes using different model parameters. By generating statistical distributions of model outputs, we demonstrated how to propagate uncertainties of inputs within our microscopic level model up to a macroscopic level. Lastly, we investigated spontaneous termination in the model and found a complex dependence on its past AF trajectory, the mechanism of which merits future investigation.

... Often cellular automata in two dimensions are defined on a regular lattice with periodic boundary conditions, this is topologically a genus-1 2-manifold (i.e., a torus). Ventrella identified that unique glider patterns where possible when a complex 2-d cellular automaton was defined on a sphere [21]. This interesting result invokes many questions relating to topology and dynamics. ...

In this paper, we demonstrate that the distribution of Wolfram classes within a cellular automata rule space in the triangular tessellation is not consistent across different topological general. Using a statistical mechanics approach, cellular automata dynamical classes were approximated for cellular automata defined on genus-0, genus-1 and genus-2 2-manifolds. A distribution-free equality test for empirical distributions was applied to identify cases in whichWolfram classes were distributed differently across topologies. This result implies that global structure and local dynamics contribute to the long term evolution of cellular automata.

... Because hexagonal grid does not present spurious symmetries of the square lattice it has been implemented for wildfire spreading prediction in [52]. Ref. [53] explores GL on geodesic sphere, whose all facets are triangular. ...

The original prototype of the cellular automaton (CA) shading system (CASS) for building facades was based on rectangular array of cells and used liquid crystal technology. This paper introduces polarized film shading system (PFSS) – an alternative approach based on opto-mechanical modules whose opacity is a function of the rotation of polarized film elements. PFSS in regular tessellations: triangular, square and hexagonal are discussed. Simulations for each type of tessellation are presented and visualized. Visual attractiveness of emergent CA patterns manifested by “particles” and “solitons” is discussed.

Several complex dynamic spatial systems are operating on the global scale. Their representation with existing geosimulation models is limited to planar level and do not consider the curvature of the Earth's surface. Thus, the objective of this study is to propose and develop a spherical geographic automata (SGA) modeling approach to represent and simulate dynamic spatial processes at the global level. The proposed SGA model is implemented for three case studies including simulations of: (1) Game of Life as population dynamics; (2) urban land‐use growth; and (3) deforestation all operating on the spherical Earth's surface. Simulation results indicate that the proposed SGA modeling approach can represent spatial processes such as expansion and shrinkage dynamics on the Earth's surface. The proposed approach has the potential to be adopted to represent different complex systems such as ecological, epidemiological, socioeconomic, and Earth systems processes to support environmental management and policymaking at the global level.

As mentioned in the previous chapter, it seems substantially easier to control the state of a regular-polygon surface with one- than two-dimensional CAs. However, 2D CAs also offer certain advantages which are investigated here.

This chapter collects the key findings of the pioneering concept of a CAbased shading system for building envelopes, CASS for short. CASS can be considered as a realistic approach to realization of the architect’s dream of an adaptive skin of a building which could dynamically change its appearance according to varying external conditions or the inhabitants’ will. Since CASS is based on identical modular units it has potential of being relatively robust and inexpensive.Most importantly it takes visual advantage of CA’s emergent properties which result in new, intriguing, organic aesthetics. The practical approach focused on tangible and “buildable” results make this material particularly suitable for designers and building engineers — groups that usually are not well addressed in CA research. Additionally, this paper presents an alternative approach to CA which may contribute to the mainstream research in the field.

Classical results on the surjectivity and injectivity of pa rallel maps are shown to be extendible to the cases with non-Euclidean cell spaces of particular types. Also shown are obstructions to extendibility, which may shed light on the nature of classical results such as the Gard en-of-Eden theorem.

We exploit the similarity between irregular Cellular Automata (CA) and Geometric Proportional Analogies (GPA), as both involve manipulations of geometric objects (points, lines and polygons). We describe how each GPA effectively defines a CA-like transition rule and we adapt an algorithm (called Structure Matching) used for solving GPAs to solving CAs. Irregular CAs improve on regular CAs by allowing an irregular tessellation of the plane, while further extensions support transition rules that lie beyond the scope of traditional CA. We describe three facets of the resulting model; layered inferences, incremental structures and the merge operation. Examples describe how structure matching (Mullally et al, 2005) is used to update and enhance a topographic land-cover map.

The Game of Life cellular automaton is a classical example of a massively parallel collision-based computing device. The automaton exhibits mobile patterns, gliders, and generators of the mobile patterns, glider guns, in its evolution. We show how to construct basic logical perations, AND, OR, NOT in space-time configurations of the cellular automaton. Also decomposition of complicated Boolean functions is discussed. Advantages of our technique are demonstrated on an example of binary adder, realized via collision of glider streams.

SUMMARY Dynamical theory of cellular automata on groups is developed. Main results are non-Euclidean extensions of Sato and Honda's results on the dynamics of Euclidean cellular automata. The notion of the period of a configuration is redefined in a more group theoretical way. The notion of a co-finite configuration substitutes the notion of a periodic configuration, where the new term is given to it to reflect and emphasize the importance of finiteness involved. With these extended or substituted notions, the relations among period preservablity, injectivity, and Poisson stability of parallel maps are established. Residually finite groups are shown to give a nice topological property that co-finite configurations are dense in the configuration space.

In recent years, a number of data structures for global geo-referenced data sets have been proposed based on regular, multi-resolution partitions of polyhedra. We present a survey of the most promising of such systems, which we call Geodesic Discrete Global Grid Systems (Geodesic DGGSs). We show that Geodesic DGGS alternatives can be constructed by specifying five substantially indepen- dent design choices: a base regular polyhedron, a fixed orientation of the base regular polyhedron relative to the Earth, a hierarchical spatial partitioning method defined symmetrically on a face (or set of faces) of the base regular polyhedron, a method for transforming that planar partition to the corresponding spherical/ellipsoidal surface, and a method for assigning point representations to grid cells. The majority of systems surveyed are based on the icosahedron, use an aperture 4 triangle or hexagon partition, and are either created directly on the surface of the sphere or by using an equal- area transformation. An examination of the design choice options leads us to the construction of the Icosahedral Snyder Equal Area aperture 3 Hexagon (ISEA3H) Geodesic DGGS.

Article Outline: Glossary Definition of the Subject Introduction The Locating Problem in Hyperbolic Tilings Implementation of Cellular Automata in Hyperbolic Spaces Complexity of Cellular Automata in Hyperbolic Spaces On Specific Problems of Cellular Automata Universality in Cellular Automata in Hyperbolic Spaces The Connection with Tiling Problems Future Directions Acknowledgments Bibliography © 2012 Springer Science+Business Media, LLC. All rights reserved.

A cell-based wildfire simulator that uses an irregular grid is presented. Cell-based methods are simpler to implement than fire front propagation methods but have traditionally been plagued by fire shape distortion caused by the fire only being able to travel in certain directions. Using an irregular grid randomises the error introduced by the grid, so that the shape of simulated fire spread is independent of the direction of the wind with respect to the underlying grid.The cell-based fire spread simulator is implemented using discrete event simulation, which is a much more efficient computational method than conventional wildfire simulation techniques because computing resources are not used in repeatedly computing small updates to parts of the fire whose dynamics change infrequently, namely those areas of a fire that move slowly. The resulting simulator is comparable in accuracy with traditional fire front propagation schemes but is much faster and can therefore be used as an engine for fire simulation applications that require large numbers of simulations, such as in the role of a risk analysis engine. Additional keyword: discrete event simulation.

Cellular automaton models have enjoyed popularity in recent years as easily constructed models of many complex spatial processes, particularly in the natural sciences, and more recently in geography also. Most such models adopt a regular lattice (often a grid) as the basis for the spatial relations of adjacency that govern evolution of the model. A number of variations on the cellular automaton formalism have been introduced in geography but the impact of such variations on the likely behavior of the models has not been explored. This paper proposes a method for beginning to explore these issues and suggests that this is a new approach to the investigation of the relationships between spatial structure and dynamics of spatial processes. A framework for this exploration is suggested, and details of the required methods and measures are provided. In particular, a measure of spatial pattern—spatial information—based on entropy concepts is introduced. Initial results from investigation along the proposed lines are reported, which suggest that a distinction can he made between spatially robust and fragile processes. Some implications of this result and the methodology presented are briefly discussed.

Cellular automata are discrete dynamical systems with simple construction but complex self-organizing behaviour. Evidence is presented that all one-dimensional cellular automata fall into four distinct universality classes. Characterizations of the structures generated in these classes are discussed. Three classes exhibit behaviour analogous to limit points, limit cycles and chaotic attractors. The fourth class is probably capable of universal computation, so that properties of its infinite time behaviour are undecidable.

We present a new generalized definition of spatial hierarchy and use it to create a data structure for spatial hierarchies on the plane and the sphere. The data structure is then used as the basis for hierarchical, multi-resolution cellular automata which are topology-independent so that many topologies may be studied. In these systems the dynamics of a focal cell is dependent on its neighbors and also the cell above and below it in the next coarser or finer resolution. Results from a multi-resolution version of the Game of Life show complex and unexpected behavior which is dependent on the topology chosen and initial conditions. These results and the software which produced them provide a proof of concept for the new data structure and algorithms. These may be especially useful for data analysis and simulation at the global scale.

This study proposes an alternative cellular automata (CA) model, which relaxes the traditional CA regular square grid and synchronous growth, and is designed for representations of land-use change in rural – urban fringe settings. The model uses high-resolution spatial data in the form of irregularly sized and shaped land parcels, and incorporates synchronous and asynchronous development in order to model more realistically land-use change at the land parcel scale. The model allows urban planners and other stakeholders to evaluate how different subdivision designs will influence development under varying population growth rates and buyer preferences. A model prototype has been developed in a common desktop GIS and applied to a rapidly developing area of a midsized Canadian city.

Three decades of CA-modelling in the social sciences have shown that the cellular automata framework is a useful tool to explore the relationship between micro assumptions and macro outcomes in social dynamics. However, virtually all CA-applications in the social sciences rely on a potentially highly restrictive assumption, a rectangular grid structure. In this paper, we relax this assumption and introduce irregular grids with variation in the structure and size of neighbourhoods between locations in the grid. We test robustness of two applications from our previous work that are representative for two broad classes of CA models, migration dynamics and influence dynamics. We tentatively conclude that both influence dynamics and migration dynamics have important general properties that are robust to variation in the grid structure. At the same time, we find in both examples substantively interesting implications of the irregular grid that could not be identified with a regular grid structure.

- B Grunbaum
- G C Shephard

Grunbaum, B. and Shephard, G. C. Tilings and Patterns. W. H. Freeman and
Company, 1987 ( " balanced tilings explained in pages 129-134).

Turing Universality of the Game of Life. Collision-Based Computing

- P Rendell

Rendell, P. Turing Universality of the Game of Life. Collision-Based
Computing, ed. Adamatzky, Springer, 2002

Earth Day 2009 – A Spherical Cellular Automaton (web site published online at http://www.ventrella.com

- J Ventrella

Ventrella, J. Earth Day 2009 – A Spherical Cellular Automaton
(web site published online at http://www.ventrella.com/EarthDay/EarthDay.html
2009

Life at the Edge of Chaos. Artificial Life

- C Langon

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