3) Remove line segments that are no longer on the outer edge of the snowflake. The Koch Curve has the seemingly paradoxical property of having an infinitely long perimeter (edge) that bounds a finite (non-infinite) area. The area of an equilateral triangle with side s is given by the equation to the left. Only upper and lower bounds have been invented.[5]. 135 Click here to receive email alerts on new articles. The area of each of these new triangles is added to the total area: Below is the equation showing how the area of the snowflake increases with each depth. An example Koch Snowflake is shown on the right. Squares can be used to generate similar fractal curves. Let's see what happens to the area on each step. A turtle graphic is the curve that is generated if an automaton is programmed with a sequence. How do we go about calculating the area at every depth? The typical way to generate fractals is with recursion. This page was last edited on 27 October 2020, at 21:26. First let's consider what happens to the number of sides. (It's clear that the area can't grow ubounded forever; The shape is bounded by a hexagon-like shape. © 2009-2016 DataGenetics Privacy Policy, If we encounter a one, we step forward one space, then turn 60°, Combining these, the first 2 elements are, Combining these, the first 4 elements are, Combining these, the first 8 elements are. On the next iteration, there are 12 sides, each of length 1/3 unit (Each of the three straight sides of triangle is replaced with four new segments). Basically the Koch Snowflake are just three Koch curves combined to a regular triangle. Lets apply these rules to T4 = 0110100110010110, and see what we get …. If so, what is this value? Fractals are never-ending infinitely complex shapes. Another variation. Now all the exponents are the same order and we can combine them and pull the 4 inside each of the terms. Each iteration multiplies the number of sides in the Koch snowflake by four, so the number of sides after n iterations is given by: If the original equilateral triangle has sides of length s, the length of each side of the snowflake after n iterations is: an inverse power of three multiple of the original length. Its fractal dimension equals, Extension of the quadratic type 1 curve. [7]. The ThueâMorse sequence (or ProuhetâThueâMorse sequence), is an infinite binary sequence obtained by starting with 0 and successively appending the Boolean complement of the sequence obtained thus far. What about the area?

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