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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 ]
Animated creation of a Sierpiński triangle using the chaos game. If one takes a point and applies each of the transformations d A, d B, and d C to it randomly, the resulting points will be dense in the Sierpiński triangle, so the following algorithm will again generate arbitrarily close approximations to it: [4]
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.
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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.
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 ...
The '90s erotic thriller Wild Things featured a famous sex scene between Neve Campbell, Matt Dylan, and Denise Richards—the titillating and more common FFM iteration of an on-screen threesome.