{ Formally, we can characterize the 2D Moore neighbourhood c i ( t+ - PDF document

� � � � � � � � � � Two-Dimensional Cellular Automata 2 Christian Jacob, University of Calgary Two-dimensional CA Neighbourhoods Totalistic Cellular Automata • Consider the following 1D CA rule: 111 110 101 100 011 010 001 000 0 1 1 0 1 0 0 0 • This automaton changes c i into a 1 only if the sum of the cells c i-1 , c i , and c i+1 is exactly 2: von Neumann Moore Extended Moore r = 1 r = 1 r = 1, 2, 3 1, if c i -1 ( t ) + c i ( t ) + c i +1 ( t ) = 2 { Formally, we can characterize the 2D Moore neighbourhood c i ( t+ 1) = N ij ( r ) of radius r in a cell lattice L for a cell c ij by 0, otherwise N ij ( r ) = {( k , l ) � L | | k - i | � r and | l - j | � r }. This is an example of a totalistic rule. 3 Christian Jacob, University of Calgary 4 Christian Jacob, University of Calgary Encoding of Totalistic Rules Encoding of Totalistic Rules (2) • We first look at one-dimensional, binary CA rules ( V = 2) : • With this encoding all legal rule codes are even numbers. c i ( t +1) = f ( � j � N i(2) c i + j ( t ) ), – All 32 rules for V = 2 and r = 2 (K = 2 r +1) are encoded by all even numbers between 0 and 62. where f : V 2 r +1 � {0, 1}. – Example: mod-2 rule • We can make a table of f in the form of a tuple Sum: � 0 � 1 � 2 � 3 � 4 � 5 ( f (0), f (1), f (2), …, f (2 r + 1) ). f (Sum): � 0 � 1 � 0 � 1 � 0 � 1 • We can encode this tuple by the following formula: C f : � 0 · 2 0 + � 1 · 2 1 + � 0 · 2 2 + � 1 · 2 3 + � 0 · 2 4 + � 1 · 2 5 = 42 C f = � j=0, …, 2 r +1 f ( j ) · 2 j . 5 Christian Jacob, University of Calgary 6 Christian Jacob, University of Calgary

� � � Some Example 2D Cellular Automata: 1022 Encoding of Totalistic Rules (3) 5-neighbourhood, outer totalistic • This encoding for binary, totalistic CAs ( V = 2) C f = � j=0, …, 2 r +1 f ( j ) · 2 j can be generalized to any number of V values per cell: C f = � j=0, …, 2 r +1 f ( j ) · V j , where f maps to any of the k values for a neighbourhood 0-39 of radius r : f : {0, 1, …, 2 r +1} � {0, 1, …, V }. 20 30 40 7 Christian Jacob, University of Calgary 8 Christian Jacob, University of Calgary Some Example 2D Cellular Automata: 510 Some Example 2D Cellular Automata: 374 5-neighbourhood, outer totalistic 5-neighbourhood, outer totalistic 0-39 0-39 20 30 40 20 30 40 9 Christian Jacob, University of Calgary 10 Christian Jacob, University of Calgary Some Example 2D Cellular Automata: 614 Some Example 2D Cellular Automata: 174 5-neighbourhood, outer totalistic 5-neighbourhood, outer totalistic 0-39 0-39 20 30 40 20 30 40 11 Christian Jacob, University of Calgary 12 Christian Jacob, University of Calgary

Some Example 2D Cellular Automata: 494 Patterns from Seed Regions: 143954 5-neighbourhood, outer totalistic 9-neighbourhood, outer totalistic Single-Cell Seed Random Disordered Seed 0-39 20 30 40 13 Christian Jacob, University of Calgary 14 Christian Jacob, University of Calgary Patterns from Seed Regions: 50224 Patterns from Seed Regions: 15822 9-neighbourhood, outer totalistic 9-neighbourhood, outer totalistic Single-Cell Seed Random Disordered Seed Single-Cell Seed Random Disordered Seed 15 Christian Jacob, University of Calgary 16 Christian Jacob, University of Calgary Patterns from Seed Regions: 85507 Patterns from Seed Regions: 191044 9-neighbourhood, outer totalistic 9-neighbourhood, outer totalistic Single-Cell Seed Random Disordered Seed Single-Cell Seed Random Disordered Seed 17 Christian Jacob, University of Calgary 18 Christian Jacob, University of Calgary

Patterns from Seed Regions: 93737 9-neighbourhood, outer totalistic Single-Cell Seed Random Disordered Seed Spreading Phenomena 19 Christian Jacob, University of Calgary 20 Christian Jacob, University of Calgary Slow Diffusive Growth: 256746 Spreading Phenomena 9-neighbourhood, outer totalistic • Spreading is the process in which an object extends itself over an increasingly larger area by incorporating regions adjacent to itself. • The spreading or kinetic growth (KG) models describe a wide variety of natural processes: – Tumor growth – Epidemic spread – Gelation – Rumor-mongering – Fluid flow through porous media – … 21 Christian Jacob, University of Calgary 22 Christian Jacob, University of Calgary Slow Diffusive Growth: 736 Slow Diffusive Growth: 174826 9-neighbourhood, outer totalistic 9-neighbourhood, outer totalistic 23 Christian Jacob, University of Calgary 24 Christian Jacob, University of Calgary

Slow Diffusive Growth: 175850 Eden Model 9-neighbourhood, outer totalistic • The original kinetic growth model was the Eden model , introduced by the biologist M. Eden in the 1960s. • The Eden model simulates accretive growth of a cell cluster (originally for simulating the spread of tumor cells) in a square lattice by sequentially adjoining randomly selected cells from the perimeter. • In more detail: – We start with a cluster list consisting of a seed site. – A site is randomly chosen from the perimeter list, consisting of the nearest sites adjacent to the seed site (v. Neumann neighbourhood). – The cluster and perimeter lists are updated. – Another site is randomly selected from the perimeter list, and so on. 25 Christian Jacob, University of Calgary 26 Christian Jacob, University of Calgary Variations of the Eden Model The Invasion Percolation Model • We look at two variations of the Eden model: • The invasion percolation model describes the flow of fluid through porous media. – The Invasion Percolation Model • Invasion Percolation Algorithm: – Each site in the perimeter list has a random number associated with it. – The Single Percolation Cluster Model – The cluster spreads by incorporating the perimeter site with the lowest associated random number. • This model describes a process of spreading that “follows the path of least resistance.” 27 Christian Jacob, University of Calgary 28 Christian Jacob, University of Calgary Invasion Percolation Model: Spreading Invasion Percolation Model: Spreading (2) 29 Christian Jacob, University of Calgary 30 Christian Jacob, University of Calgary

The Single Percolation Cluster Model Single Percolation Cluster Model: p = 1.0 • The single (or random) percolation cluster model describes the epidemic spread of disease. • Single Percolation Cluster Algorithm: – Each randomly selected perimeter site has a probability p of joining the cluster. – If the selected site is placed in the cluster, it is removed from the perimeter list. – Even if the selected site is not placed in the cluster, it is removed from the perimeter list so that it cannot be chosen again later. • When p = 1, this model reduces to the Eden model. 31 Christian Jacob, University of Calgary 32 Christian Jacob, University of Calgary Single Percolation Cluster Model: p = 1.0 Single Percolation Cluster Model: p = 0.5 33 Christian Jacob, University of Calgary 34 Christian Jacob, University of Calgary Invasion vs. Single Percolation Excitable Media Invasion Percolation p = 1.0 Single Percolation p = 0.5 35 Christian Jacob, University of Calgary 36 Christian Jacob, University of Calgary