Article

Glider Dynamics on the Sphere: Exploring Cellular Automata on Geodesic Grids

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Abstract

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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... 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. ...
Thesis
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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.
... 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. ...
Chapter
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.
... 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/ ...
Book
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. ...
Article
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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. ...
Article
Full-text available
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. ...
Article
Full-text available
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. ...
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