The Many Lessons in Fractals
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1 The Many Lessons in Fractals Lisa Oberbroeckling MAA Mathfest Washington, D.C. L. Oberbroeckling (Loyola U MD) Fractals Mathfest, 05 Aug / 15
2 Course Programming in Mathematics (MA302): 3 credit course L. Oberbroeckling (Loyola U MD) Fractals Mathfest, 05 Aug / 15
3 Course Programming in Mathematics (MA302): 3 credit course Required of all mathematics majors; elective for statistics majors L. Oberbroeckling (Loyola U MD) Fractals Mathfest, 05 Aug / 15
4 Course Programming in Mathematics (MA302): 3 credit course Required of all mathematics majors; elective for statistics majors Pre-requisite: intro. to computer science L. Oberbroeckling (Loyola U MD) Fractals Mathfest, 05 Aug / 15
5 Course Programming in Mathematics (MA302): 3 credit course Required of all mathematics majors; elective for statistics majors Pre-requisite: intro. to computer science Co-requisite: intro. to linear algebra L. Oberbroeckling (Loyola U MD) Fractals Mathfest, 05 Aug / 15
6 Course Programming in Mathematics (MA302): 3 credit course Required of all mathematics majors; elective for statistics majors Pre-requisite: intro. to computer science Co-requisite: intro. to linear algebra Assignment and project based L. Oberbroeckling (Loyola U MD) Fractals Mathfest, 05 Aug / 15
7 Course Programming in Mathematics (MA302): 3 credit course Required of all mathematics majors; elective for statistics majors Pre-requisite: intro. to computer science Co-requisite: intro. to linear algebra Assignment and project based Currently use MATLAB L. Oberbroeckling (Loyola U MD) Fractals Mathfest, 05 Aug / 15
8 Linear Transformations Given vertices of a polygon, draw original and transformed shapes L. Oberbroeckling (Loyola U MD) Fractals Mathfest, 05 Aug / 15
9 Linear Transformations Given vertices of a polygon, draw original and transformed shapes Rotation Reflection Scaling (contraction and dilation) Composition A given 2 2 matrix L. Oberbroeckling (Loyola U MD) Fractals Mathfest, 05 Aug / 15
10 Linear Transformations Original Original 2 Transormed 2 Transormed (a) Rotation (b) Reflection L. Oberbroeckling (Loyola U MD) Fractals Mathfest, 05 Aug / 15
11 Affine Transoformations and Fern Fractals Upload matrices for affine transformations, then apply to polygon. L. Oberbroeckling (Loyola U MD) Fractals Mathfest, 05 Aug / 15
12 Affine Transoformations and Fern Fractals Upload matrices for affine transformations, then apply to polygon. Four Fern Fractal Affine Transformations L. Oberbroeckling (Loyola U MD) Fractals Mathfest, 05 Aug / 15
13 Fern Fractals and Code Improvement Create three fern fractal functions L. Oberbroeckling (Loyola U MD) Fractals Mathfest, 05 Aug / 15
14 Fern Fractals and Code Improvement Create three fern fractal functions (Good for control statements, loops, random numbers) L. Oberbroeckling (Loyola U MD) Fractals Mathfest, 05 Aug / 15
15 Fern Fractals and Code Improvement Create three fern fractal functions (Good for control statements, loops, random numbers) 1 Traditional way: plot each point as it is calculated L. Oberbroeckling (Loyola U MD) Fractals Mathfest, 05 Aug / 15
16 Fern Fractals and Code Improvement Create three fern fractal functions (Good for control statements, loops, random numbers) 1 Traditional way: plot each point as it is calculated 2 Store each point in matrix, then plot the matrix L. Oberbroeckling (Loyola U MD) Fractals Mathfest, 05 Aug / 15
17 Fern Fractals and Code Improvement Create three fern fractal functions (Good for control statements, loops, random numbers) 1 Traditional way: plot each point as it is calculated 2 Store each point in matrix, then plot the matrix 3 Preallocate the size of the matrix L. Oberbroeckling (Loyola U MD) Fractals Mathfest, 05 Aug / 15
18 Fern Fractals and Code Improvement Fern 1 with n=10000 iterations took seconds L. Oberbroeckling (Loyola U MD) Fern 2 with n=10000 iterations took seconds Fractals Fern 3 with n=10000 iterations took seconds Mathfest, 05 Aug / 15
19 Complex Numbers Introduce complex numbers L. Oberbroeckling (Loyola U MD) Fractals Mathfest, 05 Aug / 15
20 Complex Numbers Introduce complex numbers Addition and subtraction L. Oberbroeckling (Loyola U MD) Fractals Mathfest, 05 Aug / 15
21 Complex Numbers Introduce complex numbers Addition and subtraction Multiplication and de Moivre s formula L. Oberbroeckling (Loyola U MD) Fractals Mathfest, 05 Aug / 15
22 Chaos Game Generate vertices of the polygon as sin θ + i cos θ L. Oberbroeckling (Loyola U MD) Fractals Mathfest, 05 Aug / 15
23 Chaos Game Generate vertices of the polygon as sin θ + i cos θ Draw polygon. L. Oberbroeckling (Loyola U MD) Fractals Mathfest, 05 Aug / 15
24 Chaos Game Generate vertices of the polygon as sin θ + i cos θ Draw polygon. Draw vertices using different colors and markers. L. Oberbroeckling (Loyola U MD) Fractals Mathfest, 05 Aug / 15
25 Chaos Game Generate vertices of the polygon as sin θ + i cos θ Draw polygon. Draw vertices using different colors and markers. Preallocate the vector. L. Oberbroeckling (Loyola U MD) Fractals Mathfest, 05 Aug / 15
26 Chaos Game Generate vertices of the polygon as sin θ + i cos θ Draw polygon. Draw vertices using different colors and markers. Preallocate the vector. Generate seed as a random complex number. L. Oberbroeckling (Loyola U MD) Fractals Mathfest, 05 Aug / 15
27 Chaos Game Generate vertices of the polygon as sin θ + i cos θ Draw polygon. Draw vertices using different colors and markers. Preallocate the vector. Generate seed as a random complex number. Play the game for n turns: L. Oberbroeckling (Loyola U MD) Fractals Mathfest, 05 Aug / 15
28 Chaos Game Generate vertices of the polygon as sin θ + i cos θ Draw polygon. Draw vertices using different colors and markers. Preallocate the vector. Generate seed as a random complex number. Play the game for n turns: Simulate a roll of the die. L. Oberbroeckling (Loyola U MD) Fractals Mathfest, 05 Aug / 15
29 Chaos Game Generate vertices of the polygon as sin θ + i cos θ Draw polygon. Draw vertices using different colors and markers. Preallocate the vector. Generate seed as a random complex number. Play the game for n turns: Simulate a roll of the die. Next element is X of the midpoint between current element and vertex chosen. L. Oberbroeckling (Loyola U MD) Fractals Mathfest, 05 Aug / 15
30 Chaos Game Generate vertices of the polygon as sin θ + i cos θ Draw polygon. Draw vertices using different colors and markers. Preallocate the vector. Generate seed as a random complex number. Play the game for n turns: Simulate a roll of the die. Next element is X of the midpoint between current element and vertex chosen. Plot points. L. Oberbroeckling (Loyola U MD) Fractals Mathfest, 05 Aug / 15
31 Chaos Game n = (a) X = 1/2 L. Oberbroeckling (Loyola U MD) (b) X = 3/8 Fractals (c) X = 1/3 Mathfest, 05 Aug / 15
32 Line Replacement Fractals n = 0, s n = 1, s n = 2, s n = 3, s n = 4, s n = 5, s L. Oberbroeckling (Loyola U MD) Fractals Mathfest, 05 Aug / 15
33 Line Replacement Fractals Given n-th iteration L. Oberbroeckling (Loyola U MD) Fractals Mathfest, 05 Aug / 15
34 Line Replacement Fractals Given n-th iteration Given pseudocode L. Oberbroeckling (Loyola U MD) Fractals Mathfest, 05 Aug / 15
35 Line Replacement Fractals Given n-th iteration Given pseudocode Figure the geometry for the points L. Oberbroeckling (Loyola U MD) Fractals Mathfest, 05 Aug / 15
36 Line Replacement Fractals Given n-th iteration Given pseudocode Figure the geometry for the points Plot the fractal L. Oberbroeckling (Loyola U MD) Fractals Mathfest, 05 Aug / 15
37 Line Replacement Fractals Given n-th iteration Given pseudocode Figure the geometry for the points Plot the fractal Calculate the enclosed area and perimeter for small n L. Oberbroeckling (Loyola U MD) Fractals Mathfest, 05 Aug / 15
38 Line Replacement Fractals Given n-th iteration Given pseudocode Figure the geometry for the points Plot the fractal Calculate the enclosed area and perimeter for small n Use geometric series for area and perimeter as n L. Oberbroeckling (Loyola U MD) Fractals Mathfest, 05 Aug / 15
39 Examples L. Oberbroeckling (Loyola U MD) Fractals Mathfest, 05 Aug / 15
40 Future Assignments could be expanded Done in lower level courses Explore other fractals L. Oberbroeckling (Loyola U MD) Fractals Mathfest, 05 Aug / 15
41 Thank you! Lisa Oberbroeckling math.loyola.edu/ loberbro L. Oberbroeckling (Loyola U MD) Fractals Mathfest, 05 Aug / 15
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