Fractal Geometry James Harris Figure 1: Fractal Art - Pixabay All content on this page, as well as other resources, can be accessed online via the Fractal Geometry website: www.mrjamesharris.com/educ5507-fractal-geometry What are fractals? “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” (Benoit B. Mandelbrot, 1982, The Fractal Geometry of Nature). Fractals are complex shapes and patterns that look similar on many different scales. They seem to appear in nature and are increasingly being used by engineers, architects, musicians, scientists, artists and mathematicians to create innovative and unique designs and models. Generally speaking, a fractal can be defined as “a rough or fragmented geometric shape that can be subdivided into parts, each of which is (at least approximately) a reduced-size copy of itself.” Fractals are formed by iteration, that is, by carrying out a procedure or algorithm on a shape repeatedly. This iterative procedure can be seen in the two examples of fractals below. The Von Koch Snowflake, is formed by taking an equilateral triangle and repeatedly adding on a smaller equilateral triangle that is 1/3 of the size of the original. Figure 2: The Von Koch Snowflake – Your Maths The Sierpinski Triangle is formed by repeatedly joining the midpoints of each side of an equilateral triangle and removing the contained area. Figure 3: The Sierpinski Triangle – link Fractals generally exhibit three important characteristics: 1. Self-similarity; 2. Non-integer dimensionality; and 3. Nowhere differentiable. Self-similarity Self-similarity is not a unique property of fractals. Many geometric shapes of Euclidean origin also exhibit self-similarity. However, fractals exhibit identical or approximately identical features at different scales. Zooming in or out from a fractal object should look the same or similar to the existing shape. This can be seen in mathematical constructs such as the Sierpinski Triangle and Von Koch Snowflake but can also be seen in nature, for example, when looking at a fern leaf or a nautilus shell. Figure 4: Fern Leaf - Pinterest Figure 5: Nautilus Shell - Howstuffworks Non-integer Dimensionality Unlike Euclidean geometry which deals with objects of integer dimensions, fractal geometry describes objects that can have non-integer dimensionality. The fractal dimension is related to the way a shape scales when the lengths are changed. This means that while Euclidean shapes like lines have a dimension of 1 and squares have a dimension of 2, the Sierpinski Triangle has a fractal dimension of approximately 1.5849. Additionally, the coastline of Britain (see Figure 6) was calculated to have a fractal dimension of 1.25, while the cross-section of a broccoli has a fractal dimension of approximately 2.7. A good introduction to fractal dimensions can be found here. Figure 6: The Coastline of Britain - Wikimedia Commons Figure 7: Weiestrass Function - Wikipedia Nowhere Differentiable Up until the mid-19th century, mathematicians were generally only interested in “well- behaved” functions. These were functions that were considered to be continuous and differentiable. However, with the discovery of the first Weierstrass function (see Figure 7) in 1872, “monstrous” functions that were continuous everywhere but not differentiable anywhere were discovered. These functions incorporated an inherent roughness that would become synonymous with fractal geometry. The Development of Fractal Geometry Fractal geometry is a relatively modern field of geometry that gained prominence in the 1970s when Polish mathematician Benoit Mandelbrot coined the term “fractal”. However, the origins of fractal geometry were laid out by a series of European mathematicians at least 100 years earlier, beginning in the late 19th century. Karl Weierstrass Weierstrass was a German mathematician who in 1872 defined the first fractal function (Debnath, 2006). While investigating analytic functions, Weierstrass published a proof that defined a continuous function that was non-differentiable everywhere. ! � � = �! cos(�! ��) !!! These types of functions would become known as Weierstrass Functions. The graph of this function (shown below) was the first example of a fractal. Figure 8: Karl Weierstrass (1815-1897) - Alchetron Figure 9: Weierstrass function - Harris Georg Cantor Cantor was a Russian/German mathematician who is known as the father of Set Theory. Cantor's work included the invention of Set Theory, as well as important discoveries relating to infinite numbers (Trochet, 2009). In 1883, Cantor developed a set, later to be known as the “Cantor Set”, which exhibited self-similarity and was non- differentiable almost everywhere. It is important to note that self-similarity, non-integer dimensionality and fractals were not defined at this time so neither Cantor nor Weierstrass would have thought of their discoveries as fractals. Figure 11: Georg Cantor (1845-1918) - Britannica Figure 10: Cantor Set - Wikimedia Commons Felix Hausdorff Hausdorff was a German mathematician, philosopher and writer who made significant contributions to set theory and topology in the early 20th century. In a paper in 1918, Hausdorff introduced a new definition of dimensionality that allowed sets to have non-integer dimensional values (Trochet, 2009). This concept would be known as the Hausdorff Dimension and would be fundamental to the definition of fractals. Unfortunately, as a Jewish mathematician in Germany during World War II, Hausdorff committed suicide in 1942 rather than be forced into a concentration camp. Figure 12: Felix Hausdorff (1868-1942) - HIM Figure 13: Julia set - wordpress Pierre Fatou Fatou was a French mathematician and astronomer who is credited with creating the study of Complex Dynamics (Debnath, 2006). This involved studying systems defined by the iteration of complex functions. As part of a competition being run by the French Academy of Sciences, Fatou was the first person to introduce and study the complex dynamical system that would become known as the Julia Set. The inverse of the Julia Set is known as the Fatou Set. Figure 14: Pierre Fatou (1878-1929) - wordpress Waclaw Sierpinski Sierpinski was a Polish mathematician who contributed to set theory, number theory and function theory. In the 1920s, Sierpinski worked on recursively defined fractal functions that would later become known as Sierpinski curves. A number of other fractals are named after him, including the Sierpinski Triangle, the Sierpinski Carpet and the Sierpinski Sponge. Figure 15: Waclaw Sierpinski Figure 16: Menger sponge - Inverse of the Sierpinski sponge - Wikipedia (1882-1969) - Culture.pl Paul Levy Levy was a French mathematician who made contributions to probability theory (Trochet, 2009). In 1938, Levy published a comprehensive study on self- similarity, analysing and generalising self-similar curves such as the Von Koch Snowflake and the Levy-C Curve. During his time as a university professor at the famous École Polytechnique, he would instruct Benoit Mandelbrot. Figure 18: Paul Levy (1886-1971) - Wikipedia Figure 17: Levy C curve - Wikipedia Gaston Julia Julia was a French mathematician best know for his involvement with the development and study of complex dynamics, along with Pierre Fatou (Debnath, 2006). Julia was involved in World War I and in January 1915, was shot in the face resulting in the loss of his nose. Julia won the prize from the French Academy of Sciences for his 1918 paper describing the iteration of rational functions. This work was largely forgotten until, with the advent of modern computing, it was revisited by Benoit Mandelbrot in the 1970s. Figure 19: Gaston Julia (1893-1978) - Free Republic Benoit B. Mandelbrot Polish mathematician Benoit Mandelbrot is known as the father of Fractal Geometry (Trochet, 2009). By building on the work of those before him, Mandelbrot coined the term fractal and combined the concepts of self-similarity, non-integer dimensionality and roughness to develop the mathematics of fractal geometry. Mandelbrot studied in France under Paul Levy and was introduced to the works of Pierre Fatou and Gaston Julia. While working at IBM in the US, Mandelbrot was able to utilise modern computers to visualise the complex dynamical systems that Fatou and Julia had investigated, producing beautiful fractal imagery that captured the mathematical worlds imagination. Figure 20: Benoit B. Mandelbrot (1924-2010) - IBM Figure 21: The Mandelbrot Set - Wikipedia Teaching Resource 1: Creating the Dragon Curve Suitable for Years 7 to 9 Figure 22: Chinese Dragon - Gold Wallpaper Figure 23: Multiple Dragon Curves - Wikimedia Commons Aim Investigate the construction of fractal shapes by creating dragon curves from strips of paper. Materials Different coloured strips of card (approximately 30cm by 3cm), sticky tape, graph paper Background The dragon curve is a space-filling curve that was first studied by three physicists at NASA in the 1960's (Binimelis Bassa, 2017). Procedure Individually 1. Take your long strip of paper and fold it in half from right to left. This procedure is called iteration 1. 2. Unfold your paper so that each bend is at a right angle. Draw a picture of what the paper looks like on your graph paper and describe how the curve has changed following the first iteration. 3. Refold your paper and fold your strip in half again from right to left. This is called iteration 2. 4. Again, ensuring each bend is at right angles, draw a picture and describe your paper curve. 5. Carry out these steps for another two iterations, continuing to unfold and inspect your paper after each iteration. 6. Once finished, place your curve on your table ensuring all bends are folded at right angles. You should have a shape that looks like the 4th iteration shown below. How many line segments are present at each step of the folding? Can you see a pattern emerging? How many line segments do you think would be present if we carried out another iteration? Can you see any patterns in the orientation of the folds along your curve? As a pair 1.