![]() 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. Stay tuned to this space for the next instalment.One of the earliest applications of fractals came about well before the term was even used. For example, the Mandelbrot set is well known for providing different “scenes” based on the colour scheme used for its display.īy the Numbers is a weekly series that reflects on the lighter side of student life, research, and innovation in the Faculty of Mathematics at the University of Waterloo. The simplistic to the complex range of rules that govern fractal creation are downright alluring for artists. When they are combined with EM radiation concepts, multi-band antennas can also be operated. Curves like the Hilbert curve can be used to design high-performance and low-profile antennas. The typically self-similar nature of fractals is also helpful in creating and operating antennas. Since fractals allow us to convey seemingly random patterns with little data, working with image resolution and even 3D model creation becomes hugely data-efficient using fractal image coding (FIC) and other applications. This form of fractal analysis makes distinguishing between healthy cells and signs of concern much easier. Since healthy human blood vessel cells typically grow in an orderly fractal pattern, cancerous cells, which grow in an abnormal fashion, become easier to detect. Knowledge of fractals is especially useful in medical diagnoses, including for cancer. As a large fractal city absorbs its neighbouring towns and villages, the pattern developed resembles a self-similar structure that seems random at first but is a dynamic network that may prove to be more efficient than modern “pre-planned” cities. Some cities tend to grow in fractal patterns over time and are called fractal cities. Join us as we explore the top five applications of fractals. Check out researchers like William Gilbert, professor emeritus in the Department of Pure Mathematics, and the Waterloo Fractal Coding and Analysis Group. Here in the Faculty of Mathematics, students and researchers are actively working on fractal geometry. This method to capture roughness has uses in a wide variety of fields ranging from programming to medicine. Separate from Euclidean geometry, fractal geometry addresses the more non-uniform shapes found in nature, such as mountains, clouds and trees.įractals provide a systematic method to capture the “roughness” of some objects. Watch video on YouTube What is the length of Britain’s coastline? How does a frost crystal grow? How many questions are there in the problem set? Finding an answer to each of these requires some exploration of fractals.Ī fractal is a recursively created never-ending pattern that is usually self-similar in nature.
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