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1-Dimensional Cellular Automata 1-Dimensional Cellular Automata Math Circles 2018 University of Waterloo At first, Zoink was alone. But Budlings have the special power of friendification! I Every Budling can create Budlings beside him. I But if a Budling gets surrounded, he dies! Planet Friendship This is Zoink. He is a Budling from Planet Friendship. I Every Budling can create Budlings beside him. I But if a Budling gets surrounded, he dies! Planet Friendship This is Zoink. He is a Budling from Planet Friendship. At first, Zoink was alone. But Budlings have the special power of friendification! I But if a Budling gets surrounded, he dies! Planet Friendship This is Zoink. He is a Budling from Planet Friendship. At first, Zoink was alone. But Budlings have the special power of friendification! I Every Budling can create Budlings beside him. Planet Friendship This is Zoink. He is a Budling from Planet Friendship. At first, Zoink was alone. But Budlings have the special power of friendification! I Every Budling can create Budlings beside him. I But if a Budling gets surrounded, he dies! First 16 days on Planet Friendship First 16 days on Planet Friendship First 16 days on Planet Friendship First 16 days on Planet Friendship First 16 days on Planet Friendship First 16 days on Planet Friendship First 16 days on Planet Friendship First 16 days on Planet Friendship First 16 days on Planet Friendship First 16 days on Planet Friendship First 16 days on Planet Friendship First 16 days on Planet Friendship First 16 days on Planet Friendship First 16 days on Planet Friendship First 16 days on Planet Friendship First 16 days on Planet Friendship First 16 days on Planet Friendship Birth, survival, and death The rules on Planet Friendship are very precise. Birth, survival, and death The rules on Planet Friendship are very precise. Birth, survival, and death The rules on Planet Friendship are very precise. Birth, survival, and death The rules on Planet Friendship are very precise. Example Example Example Example Exercise: According to the rules of friendification, which of the following arrangements of Budlings can occur? Exercise: Starting with a single Budling, use these new rules to find their state after ten days. (Don't just calculate | find the pattern!) What about different generation rules? Exercise: Starting with a single Budling, use these new rules to find their state after ten days. (Don't just calculate | find the pattern!) What about different generation rules? What about different generation rules? Exercise: Starting with a single Budling, use these new rules to find their state after ten days. (Don't just calculate | find the pattern!) It's easier to see what's going on if we use colors. I Live alien ! Black cell I Dead alien ! White cell These patterns of cells are called cellular automata. Less aliens, more colors These patterns of cells are called cellular automata. Less aliens, more colors It's easier to see what's going on if we use colors. I Live alien ! Black cell I Dead alien ! White cell These patterns of cells are called cellular automata. Less aliens, more colors It's easier to see what's going on if we use colors. I Live alien ! Black cell I Dead alien ! White cell Less aliens, more colors It's easier to see what's going on if we use colors. I Live alien ! Black cell I Dead alien ! White cell These patterns of cells are called cellular automata. A cellular automaton starts at an initial state of live and dead cells, and uses transition rules to create a new state. Even simple transition rules can generate chaotic patterns. Cellular Automata Even simple transition rules can generate chaotic patterns. Cellular Automata A cellular automaton starts at an initial state of live and dead cells, and uses transition rules to create a new state. Even simple transition rules can generate chaotic patterns. Cellular Automata A cellular automaton starts at an initial state of live and dead cells, and uses transition rules to create a new state. Cellular Automata A cellular automaton starts at an initial state of live and dead cells, and uses transition rules to create a new state. Even simple transition rules can generate chaotic patterns. Stephen Wolfram Creator of Mathematica and Elementary Cellular Automata [VIDEO] A New Kind Of Science Free online: https://www.wolframscience.com/nks/ Automatic patterns in real life Automatic patterns in real life Automatic patterns in real life This is an example of an elementary cellular automaton: two states (black/white), and the state of each cell only depends on the cells to its left and right. Exercise: Iterate Rule 30 ten times, starting with just one black cell. Rule 30: A closer look This is an example of an elementary cellular automaton: two states (black/white), and the state of each cell only depends on the cells to its left and right. Exercise: Iterate Rule 30 ten times, starting with just one black cell. Rule 30: A closer look Exercise: Iterate Rule 30 ten times, starting with just one black cell. Rule 30: A closer look This is an example of an elementary cellular automaton: two states (black/white), and the state of each cell only depends on the cells to its left and right. Rule 30: A closer look This is an example of an elementary cellular automaton: two states (black/white), and the state of each cell only depends on the cells to its left and right. Exercise: Iterate Rule 30 ten times, starting with just one black cell. It's probably easier to get a computer to do this. Go to WolframAlpha and use Wolfram Language to compute! Enter this: Rule Initial state Iterations Try different rules and initial states! Fun With Programming Go to WolframAlpha and use Wolfram Language to compute! Enter this: Rule Initial state Iterations Try different rules and initial states! Fun With Programming It's probably easier to get a computer to do this. Enter this: Rule Initial state Iterations Try different rules and initial states! Fun With Programming It's probably easier to get a computer to do this. Go to WolframAlpha and use Wolfram Language to compute! Rule Initial state Iterations Fun With Programming It's probably easier to get a computer to do this. Go to WolframAlpha and use Wolfram Language to compute! Enter this: Try different rules and initial states! Initial state Iterations Fun With Programming It's probably easier to get a computer to do this. Go to WolframAlpha and use Wolfram Language to compute! Enter this: Rule Try different rules and initial states! Iterations Fun With Programming It's probably easier to get a computer to do this. Go to WolframAlpha and use Wolfram Language to compute! Enter this: Rule Initial state Try different rules and initial states! Fun With Programming It's probably easier to get a computer to do this. Go to WolframAlpha and use Wolfram Language to compute! Enter this: Rule Initial state Iterations Try different rules and initial states! Wolfram calls this Rule 182 because 10110110 is the binary representation of the number 182. (1)27 +(0)26 +(1)25 +(1)24 +(0)23 +(1)22 +(1)21 +(0)20 = 182: Another example: the binary expansion of 122 is 1111010, so Rule 122 is Why is it called \Rule 30"? (1)27 +(0)26 +(1)25 +(1)24 +(0)23 +(1)22 +(1)21 +(0)20 = 182: Another example: the binary expansion of 122 is 1111010, so Rule 122 is Why is it called \Rule 30"? Wolfram calls this Rule 182 because 10110110 is the binary representation of the number 182. Another example: the binary expansion of 122 is 1111010, so Rule 122 is Why is it called \Rule 30"? Wolfram calls this Rule 182 because 10110110 is the binary representation of the number 182. (1)27 +(0)26 +(1)25 +(1)24 +(0)23 +(1)22 +(1)21 +(0)20 = 182: Why is it called \Rule 30"? Wolfram calls this Rule 182 because 10110110 is the binary representation of the number 182. (1)27 +(0)26 +(1)25 +(1)24 +(0)23 +(1)22 +(1)21 +(0)20 = 182: Another example: the binary expansion of 122 is 1111010, so Rule 122 is Exercises I Find the binary expansion of 90, and use it to write down the transition rules for Rule 90. I The following transition rules represent which rule number? I How many elementary cellular automata are there in total? Solution: 90 = 01011010 I Which cellular automaton has the following transition rules? Solution: 10111110 = 190, so this is Rule 190. I How many elementary cellular automata are there in total? Solution: There are 28 = 256! Solutions I Find the binary expansion of 90, and use it to write down the transition rules for Rule 90. I Which cellular automaton has the following transition rules? Solution: 10111110 = 190, so this is Rule 190. I How many elementary cellular automata are there in total? Solution: There are 28 = 256! Solutions I Find the binary expansion of 90, and use it to write down the transition rules for Rule 90. Solution: 90 = 01011010 Solution: 10111110 = 190, so this is Rule 190. I How many elementary cellular automata are there in total? Solution: There are 28 = 256! Solutions I Find the binary expansion of 90, and use it to write down the transition rules for Rule 90. Solution: 90 = 01011010 I Which cellular automaton has the following transition rules? I How many elementary cellular automata are there in total? Solution: There are 28 = 256! Solutions I Find the binary expansion of 90, and use it to write down the transition rules for Rule 90.
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