Fractals are typically not self-similar

This paragraph introduces the concept of fractals, highlighting their infinite, repeating patterns and the fascination they hold for programmers due to their simplicity and complexity. It clarifies that fractals are often misunderstood as being perfectly self-similar, using examples like the Von Koch snowflake and the Sierpinski triangle. The paragraph also introduces Benoit Mandelbrot's broader definition of fractals, which is not solely based on self-similarity but also on modeling the roughness found in nature. The concept of fractal dimension is introduced, challenging the traditional understanding of dimension as only applicable to whole numbers, and suggesting that fractal dimension can be any positive real number.

This paragraph delves deeper into the concept of fractal dimension by examining how the mass of self-similar shapes changes as they are scaled down. It uses the examples of a line, a square, a cube, and a Sierpinski triangle to illustrate how the scaling factor relates to the dimension of the shape. The dimension is defined as the power to which the scaling factor must be raised to achieve the mass reduction. The Sierpinski triangle and the von Koch curve are used to demonstrate how their dimensions can be calculated using logarithms. The paragraph also introduces the idea that fractal dimension is a measure of roughness and how it can be applied to non-self-similar shapes.

The third paragraph discusses the need for a more mathematically rigorous approach to defining fractal dimension, especially for non-self-similar shapes. It introduces the box-counting method, where the number of grid squares touching a shape is counted and compared as the shape is scaled. The paragraph uses the example of a disk to show how its dimension can be empirically verified through this method. The concept is then applied to fractals like the Sierpinski triangle and the coastline of Britain, demonstrating how the number of boxes touched increases with the scaling factor raised to the power of the fractal dimension. The paragraph also explains how to empirically compute the dimension of a shape using logarithms and linear regression on a log-log plot.

The final paragraph emphasizes the practical application of fractal dimensions, particularly in describing the roughness of natural shapes like the coastline of Britain. It explains that fractals are shapes with non-integer dimensions that maintain their roughness at various scales. The paragraph also discusses the nuances of measuring fractal dimensions, noting that the dimension can vary depending on the scale of observation. It highlights the importance of fractal dimensions in distinguishing natural objects from man-made ones and provides examples of how different natural phenomena can be quantified using fractal dimensions, such as the jaggedness of coastlines and the roughness of ocean surfaces.