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While an optimally packed fractal appears only for a defined value of r, i.e., r opt, it is possible to play the chaos game using other values as well.If r>1 (the point x k+1 jumps at a greater distance than the distance between the point x k and the vertex v), the generated figure extends outside the initial polygon. [5]
This method is also called the chaos game, and is an example of an iterated function system. You can start from any point outside or inside the triangle, and it would eventually form the Sierpiński Gasket with a few leftover points (if the starting point lies on the outline of the triangle, there are no leftover points).
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.
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 .
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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 ]
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
Mathematical puzzles require mathematics to solve them. Logic puzzles are a common type of mathematical puzzle. Conway's Game of Life and fractals, as two examples, may also be considered mathematical puzzles even though the solver interacts with them only at the beginning by providing a set of initial conditions. After these conditions are set ...