![]() Repeat steps a-c for each of the 48 lines of the resulting figure. Repeat steps a-c for each of the 12 smaller lines of the resulting figure. Replace the middle segment with an equilateral triangle that points outward and has that segment as its base. Divide the line into three equal segments.ī. Do the following for each of the three lines in the triangle:Ī. For example, consider the procedure described by Swedish mathematician Niels Fabian Helge von Koch in 1904 for creating a fractal known as the “Koch snowflake”: Iteration simply means repeating something over and over, and is the way all fractals are defined. More importantly, they can be used as a starting point to make new fractals by loading them and changing the options. They aren’t much to look at, but they describe the infinite fractal and can be reloaded and re-rendered. These are the options we’ve selected in our fractal program in a textual form that we can save in a file or copy/paste to a social media site to share with others. ![]() The second important term: Parameters params for short. For our purposes here, rendering is the magic that takes all of the options we’ve selected in our fractal program (the formulas, colors, location, etc.) and turns them into a picture. This simple definition belies the complexity of the process and the artistic decisions associated with rendering, but elaborating on those details will need to wait for another post. To render a fractal simply means to create an image of the fractal. So any image we create using a fractal program can only be an approximation of the underlying fractal. But first we make an observation and define two important terms. In this essay, we’ll explore what these mean and how they work to produce fractals. Three basic concepts are key to producing all types of fractals: iteration, formulas, and orbits. ![]()
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