As in nature, the number of recursive iterations of each pattern was limited by the scale of the material they were working with. Simple, right? Don't worry, we'll go over all the pieces soon enough.Įarly African and Navajo artists noticed the beauty in these recursive patterns and sought to emulate them in many aspects of their everyday lives, including art and town planning. Now take all of that, and we can plainly see that a pure fractal is a geometric shape that is self-similar through infinite iterations in a recursive pattern and through infinite detail. Fractal geometry throws this concept a curve by creating irregular shapes in fractal dimension the fractal dimension of a shape is a way of measuring that shape's complexity. Most of us grew up being taught that length, width and height are the three dimensions, and that's that. Ever go into a store's dressing room and find yourself surrounded by mirrors? For better or worse, you're looking at an infinitely recursive image of yourself.įinally, a note about geometry.
For example, if you look at the interior of a nautilus shell, you'll see that each chamber of the shell is basically a carbon copy of the preceding chamber, just smaller as you trace them from the exterior to the interior.įractals are also recursive, regardless of scale. Fractals are created by repeating this equation through a feedback loop in a process called iteration, where the results of one iteration form the input value for the next. These self-similar patterns are the result of a simple equation, or mathematical statement. If fractals have really been around all this time, why have we only been hearing about them in the past 40 years or so? Mandelbrot saw this and used this example to explore the concept of fractal dimension, along the way proving that measuring a coastline is an exercise in approximation. Weird, but rather than converging on a particular number, the perimeter moves towards infinity. Each bump is, of course, longer than the original segment, yet still contains the finite space within. This fractal involves taking a triangle and turning the central third of each segment into a triangular bump in a way that makes the fractal symmetric.
Measure with a yardstick, you get one number, but measure with a more detailed foot-long ruler, which takes into account more of the coastline's irregularity, and you get a larger number, and so on.Ĭarry this to its logical conclusion and you end up with an infinitely long coastline containing a finite space, the same paradox put forward by Helge von Koch in the Koch Snowflake. He reasoned that the length of a coastline depends on the length of the measurement tool. Lewis Fry Richardson was an English mathematician in the early 20th century studying the length of the English coastline. One of the earliest applications of fractals came about well before the term was even used.
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