 In this video, we're going to discuss cellular automata. We will firstly talk about what they are before looking at a classic example. We will then discuss individually the different classes of patterns that cellular automata can generate before wrapping up with a talk about their significance as a new approach to mathematical modeling. Cellular automata are algorithmic models that use computation to iterate on very simple rules. In so doing, these very simple rules can create complex emergent phenomena through the interaction between agents as they evolve over time. To illustrate the functioning of a cellular automaton, we will take an example from probably the most famous algorithm called the Game of Life, devised by the mathematician John Conway. The Game of Life is played on a grid of square cells. A cell can be live or dead. A live cell is shown by putting a mark on its square. A dead cell is shown by leaving the square empty. Each cell in the grid has a neighborhood consisting of all adjacent cells to it, and there are just three rules governing the behavior of an agent. One, any live cell with fewer than two live neighbors dies, as if caused by underpopulation. Two, any live cell with two or three live neighbors lives on to the next generation. Three, any live cell with more than three live neighbors dies as if by overcrowding. Four, any dead cell with exactly three live neighbors becomes a live cell as if by reproduction. So, let's input a starting condition and run the program to see what we get. This pattern is called still life for obvious reasons. Its product is probably the most simple class of pattern, called class 1, where nearly all of these patterns evolve quickly into a stable, homogeneous state and any randomness in the initial pattern disappears. The second class of pattern we make it is where the system evolves into an oscillating structure, the simplest of these being a blinker that has a period 2 oscillation. We can also have oscillating structures that cycle over prolonged periods of time. For example, a pulsar has a period 3 oscillation, but oscillators of many more periods are known to exist. Class 3 patterns are random, where nearly all initial patterns evolve in a semi-random or chaotic manner. Any stable structures that appear are quickly destroyed by the surrounding noise. Local changes to the initial pattern tend to spread indefinitely. Here we can get what are called gliders, where a group of cells appear to glide across the screen, and this is a good example of emergence, as we no longer see the simple rules that are producing them, but instead this emergent structure of an object gliding. Lastly, automata can also produce patterns that become complex and endure over a prolonged period of time with stable local structures. With these more complex patterns, cellular automata can simulate a variety of real-world systems, including biological and chemical ones. Since the advent of the game of life, new, similar cellular automata have been developed that can do all sorts of things, such as create fractal patterns, that is, self-similar structures that repeat themselves over various scale of magnitude. Other games create patterns that can reproduce themselves. We might ask, is there anything that these automata can't do? From the perspective of computation, the game of life can do anything that your computer can do. It can count to 100, calculate the volume of a cylinder, or if you wanted to figure out the cube root of 1,230, you could encode this into a set of cells on the automata and have it compute the value. The game of life, as simple as it is, has proven by computer science to be capable of universal computation. Lastly, we will talk about the significance of cellular automaton as a new approach to mathematical modeling. Von Neumann and Ulam originally introduced the concept in the mid-20th century, and then a few decades later, the popular game of life brought interest to the subject beyond academia. In the 80s, Stephen Wolfram engaged in a systematic study of cellular automata after which he published a book called A New Kind of Science, claiming that cellular automata could enable a new approach based upon the exploration of these algorithms. But what is behind this big statement about these simple programs creating a new kind of science? One assumption within modern science is that simple rules can only create simple phenomena, and thus, inversely, complex phenomena must be the product of complex rules. The advent of chaos theory during the past few decades revealed this to be an invalid assumption as simple systems like a double pendulum proved to be capable of generating complex and chaotic behavior. It is now increasingly accepted that complexity may not be the product of complex rules, but in fact emerge out of an interaction of simple rules as they evolve over time. Cellular automata are the tools that capture and embody this paradigm within science. Secondly, ever since the rise of modern science some 400 years ago, equations have been the dominant form of mathematical models through which we have encoded so-called scientific laws of nature. There are many valid applications for equation-based modeling, but they also have their limitations. They present a somewhat static picture of the world in a state of permanent equilibrium. This is most clearly exhibited within economic models that describe markets as always moving towards an equilibrium state between supply and demand. But in reality, many complex systems like ecosystems and societies are only at or near equilibrium when they are dead. Things are constantly changing, new technologies are invented, startups are disrupting the status quo, and so on. Non-equilibrium phenomena of this kind are not well modeled by equations and are best described through the evolutionary dynamics that shape them, with cellular automata again being well designed to capture this. We will wrap up then by saying that cellular automata are an alternative computation modeling method based upon algorithms that iterate on simple rules to try and simulate complex phenomena. The primary classification of cellular automata are numbered 1 to 4 in order of the complexity that they can sustain, as they go from stable to periodic oscillation to chaotic and complex patterns of behavior. We have also talked about how they are better suited to modeling phenomena that are the product of evolutionary dynamics and have emerged out of the interaction between their parts, as is typically the case for complex systems.