CH 33: Nature = a+bi and Other Infinite Details
One of the most powerful areas of mathematics is surely geometry. This puts the visual and spatial into numbers, creates order behind shapes, and has allowed almost all of modern society to emerge. For many centuries, the only geometry known and believed is what is now called Euclidean Geometry. This was named after the Greek geometer Euclid. The fundamental quality of this geometry is the idealizing of shapes. This means that when we speak of a triangle there are certain expectations required, like the fact that the sum of the angles must equal 180 degrees. What new geometries have done is to consider the realistic conditions of nature and create geometry around this imperfect reality.
This has led to the development of fractal geometry, which is a better approximation of nature. For example, the branch structure of trees and the design of leaves all conform to fractal patterns. The idea of fractals is geometric shapes that repeat endlessly as you zoom in on any part of the fractal. So imagine some geometric form and imagine that as you zoom in on it the shape emerges again and no matter how much or where you zoom you encounter the same shape again and again. This is a powerful result and not only does it approximate reality well, it often yields results that are very aesthetically pleasing. The key to creating fractals like the ones you might see on a poster often use complex numbers, which are then plotted on the complex plane. A complex number is like a “regular” number except that in addition to being a Real number, it also has an imaginary component. Imaginary numbers are those that contain the component of i, where i is equal to the square root of -1. Now you might say, wait, I thought you couldn’t by definition, have a negative square root of a number! You are generally correct, but the imaginary numbers have allowed for the creation of a plane that did not previously exist.
Welcome to Fractal Island