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The Koch snowflake (also known as the Koch curve, Koch star, or Koch island [1] [2]) is a fractal curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Curve Without Tangents, Constructible from Elementary Geometry" [3] by the Swedish mathematician Helge von Koch.
Three anti-snowflakes arranged in a way that a koch-snowflake forms in between the anti-snowflakes. 1.2619: Koch curve: 3 Koch curves form the Koch snowflake or the anti-snowflake. 1.2619: boundary of Terdragon curve: L-system: same as dragon curve with angle = 30°.
The first four iterations of the Koch snowflake, which has a Hausdorff dimension of approximately 1.2619. The same rule applies to fractal geometry but less intuitively. To elaborate, a fractal line measured at first to have one length, when remeasured using a new stick scaled by 1/3 of the old may be 4 times as many scaled sticks long rather ...
Iterated function systems (IFS) – use fixed geometric replacement rules; may be stochastic or deterministic; [44] e.g., Koch snowflake, Cantor set, Haferman carpet, [45] Sierpinski carpet, Sierpinski gasket, Peano curve, Harter-Heighway dragon curve, T-square, Menger sponge
This equation is easily solved for D, yielding the ratio of logarithms (or natural logarithms) appearing in the figures, and giving—in the Koch and other fractal cases—non-integer dimensions for these objects. The Hausdorff dimension is a successor to the simpler, but usually equivalent, box-counting or Minkowski–Bouligand dimension.
The four most common special cases are formed with triangles, squares, pentagons, and hexagons, but it can be extended to any polygon. [1]: 2 Its boundary is the von Koch curve of varying types – depending on the n-gon – and infinitely many Koch curves are contained within. The fractals occupy zero area yet have an infinite perimeter.
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English: The first six iterations of a fractal square resembling a koch snowflake. A second square is placed on each side of the original square at one half the size - As the number of iterations approaches infinity, the area of the fractal approaches exactly twice the area of the original square.