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At each iteration of the chaos game, the point x k+1 can be placed anywhere along the line connecting the point x k and the vertex of choice, v. Defining r as the ratio between the two distances d(x k ,x k+1 ) and d(x k ,v), it is possible to find the optimal value of r, i.e., r opt , for every N-sided regular polygon, that produces a fractal ...
Construction of an IFS by the chaos game (animated) IFS being made with two functions. Hutchinson showed that, for the metric space R n {\displaystyle \mathbb {R} ^{n}} , or more generally, for a complete metric space X {\displaystyle X} , such a system of functions has a unique nonempty compact (closed and bounded) fixed set S . [ 3 ]
The term chaos game refers to a method of generating the fixed point of any iterated function system (IFS). Starting with any point x 0, successive iterations are formed as x k+1 = f r (x k), where f r is a member of the given IFS randomly selected for each iteration. Hence the chaos game is a randomized fixed-point iteration.
The first iterate (blue) is the initial condition, which essentially forms a circle. Animation shows the first to the sixth iteration of the circular initial conditions. It can be seen that mixing occurs as we progress in iterations. The sixth iteration shows that the points are almost completely scattered in the phase space.
DeepSeek is legitimately impressive, but the level of hysteria is an indictment of so many. The $5M number is bogus. It is pushed by a Chinese hedge fund to slow investment in American AI startups ...
Animated slapstick comedy and a whole lot of chaos. Despite their rambunctious rivalry, they occasionally show a softer side for one another and sometimes team up to beat other enemies. Image ...
In mathematics, a chaotic map is a map (an evolution function) that exhibits some sort of chaotic behavior.Maps may be parameterized by a discrete-time or a continuous-time parameter.
The Game of Life, also known as Conway's Game of Life or simply Life, is a cellular automaton devised by the British mathematician John Horton Conway in 1970. [1] It is a zero-player game, [2] [3] meaning that its evolution is determined by its initial state, requiring no further input. One interacts with the Game of Life by creating an initial ...