In mathematics, the term chaos game originally referred to a method of creating a fractal, using a polygon and an initial point selected at random inside it. The fractal is created by iteratively creating a sequence of points, starting with the initial random point, in which each point in the sequence is a given fraction of the distance between the previous point and one of the vertices of the polygon; the vertex is chosen at random in each iteration. Repeating this iterative process a large number of times, selecting the vertex at random on each iteration, and throwing out the first few points in the sequence, will often (but not always) produce a fractal shape. Using a regular triangle and the factor 1/2 will result in the Sierpinski triangle, while creating the proper arrangement with four points and a factor 1/2 will create a display of a "Sierpinski Tetrahedron", the three-dimensional analogue of the Sierpinski triangle. As the number of points is increased to a number N, the arrangement forms a corresponding (N-1)-dimensional Sierpinski Simplex. The term has been generalized to refer to a method of generating the attractor, or the fixed point, of any iterated function system (IFS). Starting with any point x0, successive iterations are formed as xk+1 = fr(xk), where fr is a member of the given IFS randomly selected for each iteration. The iterations converge to the fixed point of the IFS. Whenever x0 belongs to the attractor of the IFS, all iterations xk stay inside the attractor and, with probability 1, form a dense set in the latter. The "chaos game" method plots points in random order all over the attractor. This is in contrast to other methods of drawing fractals, which test each pixel on the screen to see whether it belongs to the fractal. The general shape of a fractal can be plotted quickly with the "chaos game" method, but it may be difficult to plot some areas of the fractal in detail. With the aid of the "chaos game" a new fractal can be made and while making the new fractal some parameters can be obtained. These parameters are useful for applications of fractal theory such as classification and identification. The new fractal is self-similar to the original in some important features such as fractal dimension. (Wikipedia).
In this video, we explore the differences between starting with a random dot in a regular hexagon and iterating the procedure of choosing a hexagon vertex at random and moving either half the distance from the current dot to the chosen vertex OR two thirds the distance from the current dot
From playlist Fractals
In this short, we show what happens when iterating the procedure of choosing a hexagon vertex at random and moving wo thirds the distance from the current dot to the chosen vertex. If you like this video, check out my others and consider subscribing. Thanks! #chaos #chaosgame #hexagon #
From playlist Fractals
Unapologetic research stream 1
Broadcasted live on Twitch -- Watch live at https://www.twitch.tv/simuleios
From playlist research
Coding Challenge #123.1: Chaos Game Part 1
In this multi-part coding challenge, I visualize the "Chaos Game". Code: https://thecodingtrain.com/challenges/123-chaos-game p5.js Web Editor Sketches: 🕹️ The Chaos Game Part 1: https://editor.p5js.org/codingtrain/sketches/BJqKF9KRQ 🕹️ The Chaos Game Part 2: https://editor.p5js.org/codin
From playlist Coding Challenges
Coding Challenge #123.2: Chaos Game Part 2
In this multi-part coding challenge, I visualize the "Chaos Game". Code: https://thecodingtrain.com/challenges/123-chaos-game p5.js Web Editor Sketches: 🕹️ The Chaos Game Part 1: https://editor.p5js.org/codingtrain/sketches/BJqKF9KRQ 🕹️ The Chaos Game Part 2: https://editor.p5js.org/codin
From playlist Coding Challenges
Chaos games and fractals Day 9
Broadcasted live on Twitch -- Watch live at https://www.twitch.tv/simuleios
From playlist Fractal
Live Stream #161: Chaos Game & Quick, Draw! API
In this live stream, I code the Chaos Game, which is inspired by my visit at Thinkercon. Plus, I use the Quick, Draw! API to get drawings from the Quick, Draw! dataset. 🔗 https://thecodingtrain.com/CodingChallenges/123.1-chaos-game 🔗 https://thecodingtrain.com/CodingChallenges/123.2-chaos
From playlist Live Stream Archive
An Apple a Day Can Solve World Hunger (Chaos Theory: Butterfly Effect) #SoME2
Thanks to RJTheLammie and an anonymous maths teacher for helping with this project. They both contributed massively, so they deserve as much, if not more credit for this video. This was an absolute pain to create. 20 minutes of coded simulations, Manim animations, suffering, video editing
From playlist Summer of Math Exposition 2 videos
Fractals and chaos games day 10
Broadcasted live on Twitch -- Watch live at https://www.twitch.tv/simuleios
From playlist Fractal
RubyConf 2018 - It's Down! Simulating Incidents in Production by Kelsey Pederson
RubyConf 2018 - It's Down! Simulating Incidents in Production by Kelsey Pederson Who loves getting paged at 3am? No one. In responding to incidents -- either at 3am or the middle of the day -- we want to feel prepared and practiced in resolving production issues. In this talk, you'll lea
From playlist RubyConf 2018
François Delarue - Stochastic control for large population driven by correlated noises
François Delarue (Université de Nice) I will discuss recent advances in large population stochastic control, in the spirit of the pioneering by Lasry and Lions and by Caines and Malhamé in 2006. The basic point is to seek approximate equilibria over families of interacting players when t
From playlist Schlumberger workshop on Topics in Applied Probability
Chaos games and fractals Day 1
Working on Iterated Function Systems! -- Watch live at https://www.twitch.tv/simuleios
From playlist Fractal