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Animated creation of a Sierpinski triangle using a chaos game method The way the "chaos game" works is illustrated well when every path is accounted for. 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.
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.
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 ]
Chaos theory (or chaology [1]) is an interdisciplinary area of scientific study and branch of mathematics. It focuses on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions .
Fractal fern in four states of construction. Highlighted triangles show how the half of one leaflet is transformed to half of one whole leaf or frond.. Though Barnsley's fern could in theory be plotted by hand with a pen and graph paper, the number of iterations necessary runs into the tens of thousands, which makes use of a computer practically mandatory.
The farcical math is elegantly worked out by FastHorse, who maps out the ensuing chaos with elan. The production, directed by Michael John Garcés, quickly reaches cruising speed on a vivid set by ...
MIDTOWN EAST, Manhattan (PIX11) - A public elementary school in Manhattan is taking a dramatically different approach to homework, which encourages students to play more, and spend more time with ...