At its core, fractal geometry is based on scientific and mathematical principles, such as chaos theory, the study of deterministic chaos. Although it sounds paradoxical at first, it is not all that bizarre. While calculus and geometry are useful tools in natural sciences, nature itself rarely adheres to perfect geometric forms. As mathematician Benoit Mandelbrot once said, “Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.” In other words, nature is intrinsically unpredictable and characterized by chaos. Yet total randomness is impossible to study scientifically. Fractal geometry serves as the middle ground between these two sides of the spectrum, making it an incredibly effective tool for understanding the natural world.

In both nature and biology, fractals reveal striking similarities. The fractal branching of our lungs, from the trachea to our bronchial tubes, and further out to the bronchioles and alveoli, mirrors the branching of trees. This is a logical similarity, as both systems depend on large surface areas to carry out respiration and efficiently exchange oxygen and CO2. Additionally, spirals are a common fractal observed at all scales, from the motion of turbulent fluids to plants, crustaceans, monsoons, and even galaxies. 

Fractals aren’t just observed in nature; fractals can also be generated using simple equations on computers. For instance, the Mandelbrot Set, one of the most well-known fractals, is generated through the equation Zn+1 = Zn2 + C. Most initial values of ‘C’ tend to diverge into infinity, but as the computation process progresses, some values of ‘C’ become smaller or alternate between a fixed set of values. These are the points inside the Mandelbrot set. Outside the set, all the values of ‘C’ cause the equation to go to infinity, with the colors indicating the speed at which they expand. Variations of this same equation yield a myriad of other fractals. Though abstract, the study of “man-made” fractals offers visually impressive insights.

The applications of fractal geometry are wide-ranging, encompassing fields such as medicine, engineering, and material science. Researchers are more frequently employing fractal analysis to study the human body and diseases, like cancer, as well as to design cooling circuits for computer chips. Fractals are utilized to engineer high-strength wires and fractal vises in workshops. Fractal geometry enables us to simulate complicated environments from our computers, advancing research in numerous scientific domains. Whether analyzing the stock market or taking a scenic hike, fractals can truly be found everywhere.