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Fractal Math and Graphics Sylvia Carlisle Nancy Van Cleave Talk Fractal Math and Graphics Talk Overview I Fractals — Origins and Definition I Natural Phenomenon Sylvia Carlisle Nancy Van Cleave I Motivation for Studying Fractals Mathematics Department Mathematics and Computer Rose–Hulman Science Department I Koch’s Snowflake Eastern Illinois University I Sierpinski’s Curves October 18, 2013 I Mathematics of Fractals I Resources and Conclusion Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Discovery — Lewis Fry Richardson Coastline Paradox I English mathematician, physicist, meteorologist, psychologist, and pacifist, 1881 – 1953 I Decided to research the relation between the probability of two countries going to war and the length of their common border I He observed the coastline paradox: the length of a coastline depends on the level of detail at which you measure it. I Empirically, the smaller the unit of measure (e.g., ruler vs yardstick), the longer the measured length becomes. 200 KM 100 KM 50 KM Measure Measure Measure Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Definition Coastline Fractal from Google: Fractal: a curve or geometric figure, each part of which has the same statistical character as the whole. Another way of putting it: a fractal is a figure consisting of an identical motif which repeats on an ever–reducing scale. Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Chou Romanesco Same Pattern At Macro and Micro Levels Fractal forms are complex shapes which look more or less the same at a wide variety of scale factors, and are everywhere in nature. Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Another Example — Ferns Frost Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Peoria Via Google Earth Peoria A Bit Closer. Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 And Closer Yet. Mississippi River Via Google Earth Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 A Bit Closer. And Closer Yet. Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Some Reasons for Studying Fractals More Reasons for Studying Fractals I Brings visual beauty and art to math and science; allows I An interesting way to teach students to look for and creativity understand patterns I Student motivation — there’s a lot about fractals they can I Provide practice with fractions and exponents understand I Improve problem solving skills in a colorful environment I Provides interesting examples of iterative functions and recursive algorithms Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 And Yet More Reasons Koch Snowflake Rules I Fractals are useful in modeling structures (such as eroded coastlines or snowflakes) in which similar patterns recur at progressively smaller scales, and in describing partly random or chaotic phenomena such as: 1. crystal growth, 2. fluid turbulence, and 3. galaxy formation. I They are used to solve real–world problems, for instance in engineering with Fractal Control of Fluid Dynamics Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Koch Iterations Over a Triangle Koch Curve Math Iteration 1 2 3 4 n Number of Segments per Side 1 4 16 1 Length of Segment 1 3 Total Length of Curve 3 4 What is the length of the nth iteration? Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Further Iterations and Combinations of Koch Snowflakes Snowflake by Hexagon Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Class Exercise Example Sierpinski’s Triangle — Iterations 1 & 2 1. The software I use to on average once out of create graphics took 4 every 10 mouse clicks, mouse clicks to change the about how many problems color of each of the small occurred? hexagons in the figure at right. Given that I copied the red group of hexagons, how many clicks did it take to change the colors on the copies? 2. The drawing program is a little flakey and tends to have some problems when I “mis-click.” If I mis-click Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Sierpinski’s Triangle — Iterations 3 & 4 Sierpinski’s Triangle — Iteration 5 Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Exploring the Math of Sierpinski’s Triangle Possible Student Activities 1. Find a pattern for the number of shaded Triangles added at Iteration 1 2 3 4 n each iteration. Determine a formula for the number of shaded triangles at the nth iteration. Number of Shaded Triangles Added 1 3 Total Number of Shaded Triangles 1 2. Find a pattern for the area of one of the shaded Triangles 1 added at each iteration. Determine a formula for the area of Area of Smallest Shaded Triangle 4 one of the added shaded Triangles at the nth iteration. 1 Total Shaded Area 4 3. Find a pattern in the values for the total shaded area. th Assume the original triangle has area 1. Determine a formula for the total shaded area at the n iteration. Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Combining Koch’s Snowflake & Sierpinski’s Triangle Consider Pascal’s Triangle 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 Binomial Coefficients Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Color Empty and Even Triangles Seemingly Off Topic. The Chaos Game 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56Surprise!70 56 28 8 1 Pascal’s Triangle contains Sierpinski’s Triangle How many more rows to get to another complete Sierpinski Triangle? Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 The Result — Sierpinski’s Triangle! That Triangle Is Everywhere! Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Sierpinski’s Carpet Sierpinski’s Quilt Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Math & Sierpinski’s Carpet Pentagon Fractal Iteration 1 2 3 n Number of Shaded Squares Added 1 8 Iteration 1 2 3 n Total Number of Shaded Squares 1 Total Number of Shaded Pentagons 1 5 Area of Smallest Shaded Square 1 9 Number of Unshaded Pentagons 5 25 Total Shaded Area 1 9 Total Number of Pentagons 6 30 Assume the original square has area 1. Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Hexagon Fractal Octagon Fractal Iteration 1 2 3 n Total Number of Shaded Hexagons 1 6 Number of Unshaded Hexagons 5 36 Quad–Koch? Total Number of Hexagons 6 42 Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Another Octagon Fractal Proof Without Words What is the sum of 1 1 1 1 1 1 + of + of of + ... 4 4 4 4 4 4 or 1 1 1 1 1 1 1 1 + + + + = + + + + ... 4 16 64 256 ··· 41 42 43 44 Let us start with one unit, divided into fourths: Octo–Carpet? Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 1 And Keep Shading 4 of Successive Regions One–fourth of One–fourth Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Ah! I See Where This Is Headed! Creeping Forward by Halves 1 1 1 1 1 1 1 1 + + + + = + + + + ... 2 4 8 16 ··· 2 22 23 24 The pattern continues to repeat at reduced scale. Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 By Thirds. Making Your Own Fractals 1 1 1 1 1 1 1 1 + + + + = + + + + ... Spiraling Into A Fractal 3 9 27 81 ··· 3 32 33 34 Make a doodle, make a fractal! Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Carlisle & Van Cleave— 63rd ICTM, October 17—19, 2013 Individualized Fractal Name Generating Fractals with Computer Programs NANCY The Julia Set NANCY NANCY NANCY NANCYNANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCYNANCY NANCY NANCYNANCY NANCY NANCYNANCY NANCY NANCYNANCY NANCY NANCY NANCY NANCY NANCY NANCYNANCY NANCY NANCYNANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCYNANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCYNANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCY NANCYNANCY NANCY
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