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by Emily Fung. The Koch Snowflake was created by the Swedish mathematician Niels Fabian Helge von Koch. In his 1904 paper entitled "Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire" he used the Koch Snowflake to show that it is possible to have figures that are continuous everywhere but differentiable nowhere. Von Koch Snowflake Goal: To use images of a snowflake to determine a sequence of numbers that models various patterns (ie: perimeter of figure, number of triangles in figure, total area of figure, etc.).

(see [13]). This fractal is interesting because it is known that in the limit it has an infinite perimeter but its area  Construction of the von Koch curve (left), often simply called the "Koch curve" here. Fractal billiards, Koch snowﬂake billiard, rational polygonal bil-. liards ticularly, polygonal billiards or even, rational billiards— Jul 2, 2014 The von Koch snowflake is a fractal curve initially described by Helge The curve has infinite length inside a finite area,; As a result of, it has  Recursion and making a Koch Snowflake with Maple.

Each iteration, each side is divided into thirds and the central third is turned into a triangular bump, therefore the perimeter increases. However, the same area is contained in the shape. That’s crazy right?!

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Summing an infinite geometric series to finally find the finite area of a Koch SnowflakeWatch the next lesson: https://www.khanacademy.org/math/geometry/basi 2008-01-03 · Area of a Koch Snowflake Question: A Koch Snowflake is a fractal which can be built by starting with an equilateral triangle, removing the inner third of each side, building another equilateral triangle at the location where the side was removed, and then repeating the process indefinitely. The snowflake is actually a continuous curve without a tangent at any point.

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Then the n-th iteration adds $$3 \cdot 4^{n-1}$$ triangles. The Koch Snowflake The Koch Snowflake is a fractal identified by Helge Von Koch, that looks similar to a snowflake. Here are the diagrams of the first four stages of the fractal - 1. At any stage (n) the values are denoted by the following – Nn - number of sides Ln - length of each side Pn - length of perimeter An - Area of snowflake In 1904 the Swedish mathematician Helge von Koch(1870-1924) introduced one of the earliest known fractals, namely, the Koch Snowflake. It is a closed continuous curve with discontinuities in its derivative at discrete points. The simplest way to construct the curve 2012-06-25 · The Koch Snowflake is an iterated process.It is created by repeating the process of the Koch Curve on the three sides of an equilateral triangle an infinite amount of times in a process referred to as iteration (however, as seen with the animation, a complex snowflake can be created with only seven iterations - this is due to the butterfly effect of iterative processes).

Other properties. The Koch snowflake is self-replicating (insert image here!) with six copies around a central point and one larger copy at the center. Hence it is an an irreptile which is The Koch Snowflake, devised by Swedish mathematician Helge von Koch in 1904, is one of the earliest and perhaps most familiar fractal curves. On this page I shall explore the intriguing and somewhat surprising geometrical properties of this ostensibly simple curve, and have also included an AutoLISP program to enable you to construct the Koch Snowflake fractal curve on your own computer.
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The Von Koch Snowflake. If we fit three Koch curves together we get a Koch snowflake which has another interesting property. In the diagram below, I have added a circle around the snowflake. It can be seen by inspection that the snowflake has a smaller area than the circle as it fits completely inside it. It therefore has a finite area.

It is constructed by starting (at level 0) with the snowflake's "initiator", an equilateral triangle: At each successive level, each straight line is replaced with the snowflake's "generator": Here are two quite different algorithms for constructing a… \$ iudfwdo lv d pdwkhpdwlfdo vhw wkdw h[klelwv d uhshdwlqj sdwwhuq glvsod\hg dw hyhu\ vfdoh ,w lv dovr nqrzq dv h[sdqglqj v\pphwu\ ru hyroylqj v\pphwu\ ,i wkh uhsolfdwlrq lv h[dfwo\ wkh vdph dw hyhu\ History of Von Koch’s Snowflake Curve The Koch snowflake is a mathematical curve, which is believed to be one of the earliest fractal curves with description. In 1904, a Swedish mathematician, Helge von Koch introduced the construction of the Koch curve on his paper called, “On a continuous curve without tangents, constructible from elementary geometry”. Other articles where Von Koch’s snowflake curve is discussed: number game: Pathological curves: Von Koch’s snowflake curve, for example, is the figure obtained by trisecting each side of an equilateral triangle and replacing the centre segment by two sides of a smaller equilateral triangle projecting outward, then treating the resulting figure the same way, and so on.
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1. Start with an equilateral triangle. 2. Divide each side into three equal parts.

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Below is a graph showing how the area of the snowflake changes with increasing fractal depth, and how the length of the curve increases. The snowflake is actually a continuous curve without a tangent at any point. Von Koch curves and snowflakes are also unusual in that they have infinite perimeters, but finite areas. After writing another book on the prime number theorem in 1910, von Koch succeeded Mittag-Leffler as mathematics professor at the University of Stockholm in 1911. PERIMETER (p) Since all the sides in every iteration of the Koch Snowflake is the same the perimeter is simply the number of sides multiplied by the length of a side. p = n*length. p = (3*4 a )* (x*3 -a) for the a th iteration.