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.
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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. Compare your curve with your neighbour. Are they different or the same? 2. Place your two curves together and tape the ends together. Draw the larger curve that has been formed. What orientation should you place your two curves at before taping them? Does it matter? Is there a way we can find out? What iteration does this step represent? As a group of four 1. Compare your curves with another group. Are they different or the same? 2. Place your two curves together and tape the ends together. Draw the larger curve that has been formed. 3. Take some time to walk around the classroom and compare each group's dragon curve. What orientation should you place your two curves at before taping them? Does it matter? Is there a way we can find out? What iteration does this step represent? Do all of the dragon curves look identical? Should they all look the same? Could we find a way to join up all of the dragon curves to form one giant dragon curve?
Figure 24: The Heighway Dragon Curve - Wikipedia Teachers Notes • It may make life easier for students if they tape their curve onto a piece of paper to hold the right angles in place. Just make sure that they tape the end near the edge of the paper so that they can join it up to another students curve! • The students’ dragon curves should all look the same. Obviously there may be differences if students have folded their paper incorrectly or sticky-taped their curves to another student backwards. Teachers need to stress that all folding and taping orientation should be done to ensure always consistently being right-over-left. • Towards the end of the lesson, show students a picture of the dragon curve after multiple iterations (such as the one below). This picture is a great example of fractals demonstrating self-similarity on many scales. Each section of the dragon curve is a scaled down version of the larger version. Ask the students if they can spot the repeated shape and the translations that are taking place on each repeat. • Another interesting concept to mention to students is the Golden Dragon which is similar to the dragon curve but introduces the concept of the Golden Ratio into the folding as an enrichment idea for high ability students. • There is a great Numberphile video detailing the dragon curve that is worth showing your students following the completion of the activity. There is a link to it here.
Figure 25: 14th iteration of the Dragon curve - Hollymath
Teaching Resource 2: Exploring the Mandelbrot Set Suitable for Years 10 to 12 Note: the Mandelbrot set deals with complex numbers. For a recap or introduction to complex numbers head here.
Figure 26: The Mandelbrot Set - Wikipedia Aim Investigate complex numbers and the iteration of quadratic functions by exploring convergent and divergent terms in the Mandelbrot set. Background The Mandelbrot set is arguably the most famous fractal. Named after Benoit Mandelbrot, the set is often used by mathematicians and graphic artists to create stunning artworks. But what actually is the Mandelbrot set? The Mandelbrot set is a set of complex numbers for which the values of a specific iterative function do not diverge. The function has the simple quadratic form of: 2 zn+1 = zn + c
For the Mandelbrot set, c is any number in the complex plane and z0 is zero. By selecting a number c in the complex plane and completing repeated iterations of the function, the values of this number will either diverge to infinity or remain bounded. If the values do not diverge to infinity, then the number c is considered to be in the Mandelbrot set. Investigation When creating Mandelbrot artwork, any number in the Mandelbrot set (i.e., any number that remains bounded) is generally coloured in black. Numbers that diverge are then shaded according to how rapidly they diverge. Hence, in order to create a piece of artwork, we must first determine: 1. Which numbers are in the Mandelbrot set, that is, which numbers remain bounded under iteration of the quadratic function; 2. Which numbers are not in the Mandelbrot set, that is, which numbers diverge under iteration of the quadratic function; and 3. If a number is divergent, how quickly the number diverges.
Let us investigate what happens with c = 0. For the Mandelbrot set, z0 is always zero. This gives the first three terms of our iteration to be: 2 2 z1 = z0 + c = 0 + 0 = 0 2 2 z2 = z1 + c = 0 + 0 = 0 2 2 z3 = z2 + c = 0 + 0 = 0
This is obviously convergent as the value z n will always by 0, which means that the number c = 0, or more technically, the complex number c = 0 + 0i, is in the Mandelbrot set (see Figure 27). What about when c = i? Again z0 is always zero. This time the first four terms of our iteration will be: 2 2 z1 = z0 + c = 0 + i = i 2 2 z2 = z1 + c = i + i = i - 1 2 2 z3 = z2 + c = (i - 1) + i = -i 2 2 z4 = z3 + c = -i + i = i - 1 This also convergent as the values will continually oscillate between i - 1 and -i. Therefore, the complex number c = 0 + i is in the Mandelbrot set (see Figure 28).
1.0 1.0
0.5 0.5 c=0+0i c=0+i
0.0 0.0
-0.5 -0.5
-1.0 -1.0
-2.0 -1.5 -1.0 -0.5 0.0 0.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 Figure 27: c=0+0i is inside the Mandelbrot set Figure 28: c=0+i is inside the Mandelbrot set
For the following numbers, calculate: 1. whether they are inside or outside the Mandelbrot set. 2. If they are outside the Mandelbrot set, calculate how many iterations it takes for the modulus of the complex number to be bigger than 2. 3. If they are inside the Mandelbrot set, calculate how many iterations it takes before the values start repeating themselves and plot them on the Argand plane (note: do not do more than 10 iterations). 1, -1, -i, 0.75i, -1.75, 0.25, 0.15 + 0.4i, 0.15 - 0.4i, -0.75i, 0.285 + 0.05i A spreadsheet can be downloaded from here to assist students in undertaking iterations of real functions if required. Note: Excel cannot handle complex number computations easily. A program could be run in Mathematica to perform iterations of complex functions if required. Around this time, it is worth students referring to an online Mandelbrot set applet (there are many, including XaoS or Gnofract4D) to visualise what is happening as they carry out the iterative procedure. Alternatively, a detailed poster, like this one from Mandelmap, may also be useful to aid the visualisation process.
Figure 29: Mandmap Poster - Mandelmap Questions Looking at a graphical visualise of the Mandelbrot set and the values that you have calculated, can you notice any symmetry existing within the set? Do all numbers that are in the Mandelbrot set begin repeating themselves? Are there numbers that appear to keep changing under continuous iteration?
Teachers Notes • This activity is only suitable for students that have been introduced to imaginary numbers. Ideally, high ability students will be introduced to i late in Year 10 and this would be an interesting way to consolidate understanding of complex numbers and quadratic functions. • For some of the more complicated complex numbers, students can use their graphics calculators to calculate the iterations. Hopefully, this will allow students to see that some numbers do not have a periodic orbit and instead are continually tending to a specific value. This is a neat way to introduce attractors. • Holly Krieger presents a great introduction to the Mandelbrot set, in a Numberphile video here. It is definitely worth watching with students to consolidate before the end of the lesson. Additionally, there is a video that continues on that introduces Julia sets which would be useful for students who have grasped the concept of the Mandelbrot set.
Rationale Most of the mathematics that is taught at school has very old roots that range from hundreds to thousands of years old. This can be seen in the 2000-year-old geometry of Euclid and 300-year-old probability of Fermat and Pascal taught in lower secondary maths class, to the 400-year-old Calculus of Newton and Leibniz taught in upper secondary classes (Merzbach & Boyer, 2011). Fractal geometry is a relatively modern field of mathematics. Although it has roots in the study of analytic functions about 100 years ago in the late 19th century, fractal geometry was truly developed from the 1970s when Benoit Mandelbrot coined the term ‘fractal’ to describe the geometric objects that arose in 1920s studies by Fatou and Julia into complex dynamic systems (Trochet, 2009). Introducing fractal geometry at a high school level, provides a unique opportunity to present students with a modern abstract mathematical concept that is highly visual, lends itself to the use of manipulatives and easily incorporates numerous facets of ICT (Lornell & Westerberg, 1999; Shiver, Willard & McDaniel, 2017). This teaching resource combines a brief introduction to the ideas of fractal geometry, an overview of the history of its development over the last 140 years, a couple of classroom activities suitable for a range of student abilities and several other resources that could be helpful in the classroom. This is by no means an exhaustive list and the number of other potential activities that could be developed for the classroom relating to fractals is limited only to your imagination. Australian Curriculum Links The study of fractal geometry presents numerous opportunities for links to the Western Australian curriculum (SCSA, 2017). Mathematics • Describe translations, reflections in an axis and rotations of multiples of 90° on the Cartesian plane using coordinates. Identify line and rotational symmetries (ACMMG181); • Solve problems using ratio and scale factors in similar figures (ACMMG221); • understand the concepts and techniques in combinatorics, geometry and vectors (Year 11 Specialist); • apply reasoning skills and solve problems in combinatorics, geometry and vectors (Year 11 Specialist); • understand the concepts and techniques in trigonometry, real and complex numbers, and matrices (Year 11 Specialist). Others • Analyse and visualise data to create information and address complex problems (ACTDIP037); • Construct and use a range of representations, including graphs, keys and models to represent and analyse patterns or relationships in data using digital as appropriate (ACSIS129); • The experiences of men, women and children during the Industrial Revolution, and their changing way of life (ACDSEH081); • The short-term and long-term impacts of the Industrial Revolution, including global changes in landscapes, transport and communication (ACDSEH082); General Capabilities • Numeracy – both the activities described provide opportunities for students to recognise and apply their understanding of patterns and relationships in order to solve problems. Spatial reasoning plays a large role in students understanding fractal geometry. • Information and Communication Technology (ICT) Capability – students are given opportunities to utilise software, such as XaoS, GNOfract4D or Excel to investigate iteration of quadratic function and visualisation of fractals. • Critical and Creative Thinking – both activities provide investigations which require students to utilise and develop their critical thinking skills as they inquire and generate ideas related to either the Dragon Curve or the Mandelbrot set. Further Resources A comprehensive list of further teaching and learning resources is available from the Fractal Geometry website at www.mrjamesharris.com/educ5507-fractal-geometry
References Bardos, L. C. (n.d.). Cut Out Fold Up: Dragon Curve. Retrieved from: http://www.cutoutfoldup.com/216-dragon-curve.php Binimelis Bassa, M. I. (2017). Our Mathematical World: A New Way of Seeing the World - Fractal Geometry. National Geographic. Debnath, L. (2006). A Brief Historical Introduction to Fractals and Fractal Geometry. International Journal of Mathematical Education in Science and Technology, 37 (1), 29-50. Eglash, R. (n.d.). African Fractals. Retrieved from: http://csdt.rpi.edu/african/African_Fractals/index.html Fraboni, M. & Moller, T. (2008). Fractals in the Classroom. The Mathematics Teacher, 102(3), 197-199. Fractal Foundation (n.d.). Explore Fractals. Retrieved from: http://fractalfoundation.org/resources/ Haran, B. (n.d.). Numberphile: YouTube. Retrieved from: https://www.youtube.com/user/numberphile Lauwerier, H. (1991). Fractals: Endlessly Repeated Geometrical Figures. Princeton, New Jersey: Princeton University Press. Lornell, R. & Westerberg, J. (1999). Fractals in High School: Exploring a New Geometry. The Mathematics Teacher, 92(3), 260-269 Mandelbrot, B. B. (1982). The Fractal Geometry of Nature. San Francisco, California: W.H. Freeman and Company. Merzbach, U. C. & Boyer, C. B. (2011). A History of Mathematics (3rd ed.). Hoboken, New Jersey: John Wiley & Sons. Patrazalek, E. (n.d.). Fractals: Useful Beauty. Retrieved from: http://www.fractal.org/Bewustzijns-Besturings-Model/Fractals-Useful-Beauty.htm Shiver, J., Willard, T. & McDaniel, M. (2017). Do-It-Yourself Fractal Functions. The Mathematics Teacher, 110(9), 694-701. Trochet, H. (2009). A History of Fractal Geometry. Retrieved from: http://www- history.mcs.st-andrews.ac.uk/HistTopics/fractals.html Vacc, N. N. (1999). Exploring Fractal Geometry with Children. School Science and Mathematics, 99(2), 77-83.