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NUMERIC Mathematics Lecture Series Nipissing University, North Bay, Ontario

NUMERIC Mathematics Lecture Series Nipissing University, North Bay, Ontario. Dr. Daniel Jarvis Mathematics & Visual Arts Education Professor Katarin MacLeod Mathematics & Physics Education Dr. Mark Wachowiak Computer Science. Exploring the Math and Art Connection 6 February 2009.

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NUMERIC Mathematics Lecture Series Nipissing University, North Bay, Ontario

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  1. NUMERIC Mathematics Lecture SeriesNipissing University, North Bay, Ontario Dr. Daniel JarvisMathematics & Visual Arts Education Professor Katarin MacLeod Mathematics & Physics Education Dr. Mark Wachowiak Computer Science Exploring the Math and Art Connection 6 February 2009

  2. Workshop Overview • Introduction: Exploring the Math/Art Connection • Golden Section: Ratio/Proportion in Ancient Greece • Activity 1: Creating your own golden section bookmark • Tesselations: Transformations in 20th Century Europe • Activity 2: Creating your own tessellation pattern • Fractals: Iterations in 21st Century • Activity 3: Creating your own fractal designs • Technology: Simulations from Nature • Video Clips: : “Donald Duck In Mathmagicland” (1959) and “Life by the Numbers” with Danny Glover (2006) • Resources: Galleries, Artists, Books, Conferences, and Stuff • Questions and Comments Jarvis MacLeod Wachowiak

  3. AN INTRODUCTION TO RATIO In mathematics, a ratio is defined as a comparison of two numbers. A proportion is simply a comparison of two ratios. Perhaps the most famous mathematical ratio/proportion is what is known as the Golden Section or the “Divine Proportion.” This proportion is derived from dividing a line segment into two segments with the special property that the ratio of the small segment to the large segment is the same as the ratio of the long segment to the entire line segment. Jarvis MacLeod Wachowiak

  4. Geometry has two great treasures: One is the Theorem of Pythagoras; the other the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel. Kepler (1571-1630) Jarvis MacLeod Wachowiak

  5. HISTORICAL OVERVIEW ANCIENT EGYPT & GREECE Jarvis MacLeod Wachowiak

  6. HISTORICAL OVERVIEW THE RENAISSANCE “DE DIVINA PROPORTIONE” (1509) WRITER: FRA LUCA PACIOLI ILLUSTRATOR: LEONARDO DA VINCI Jarvis MacLeod Wachowiak

  7. HISTORICAL OVERVIEW THE MODERN ERA: ARTISTS OF THE 19TH AND 20TH CENTURIES SEURAT, DALI, MONDRIAN, COLVILLE Jarvis MacLeod Wachowiak

  8. HISTORICAL OVERVIEW SONY PLASMA GRAND WEGA $17 000 CANADIAN 16:9 ASPECT RATIO (APPROX. 1.78) APPLE IMAC $2500 CANADIAN 36.8/22.8 = 1.614 (GOLDEN APPLES?) Jarvis MacLeod Wachowiak

  9. TEACHING HOW TO FIND THE GOLDEN SECTION [I] ALGEBRAICALLY x 1 = 1.61803 (Phi) NOW, BEGINNING WITH ANY GIVEN LENGTH (L): NEXT LARGEST SECTION LENGTH X (1.61803) NEXT SMALLEST SECTION LENGTH/(1.61803) OR MORE SIMPLY, LENGTH X (0.61803) Jarvis MacLeod Wachowiak

  10. TEACHING HOW TO FIND THE GOLDEN SECTION • [II] GEOMETRICALLY • BEGIN WITH A SQUARE; EXTEND ONE SIDE • FROM MIDPOINT, CUT AN ARC FROM FAR CORNER TO EXTENDED LINE • COMPLETE RECTANGLE & INTERNAL SQUARES Jarvis MacLeod Wachowiak

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  20. THERE IS GEOMETRY IN THE HUMMING OF THE STRINGS. THERE IS MUSIC IN THE SPACING OF THE SPHERES. PYTHAGORAS (C.A. 582-500 B.C.) Jarvis MacLeod Wachowiak

  21. RELATED PHENOMENA • DYNAMIC SYMMETRY: ROOT RECTANGLES IN GREEK DESIGN (AS OPPOSED TO STATIC) • PLATONIC SOLIDS: REGULARITY, RECIPROCITY, & GOLDEN RECTANGLES • GOLDEN SHAPES: PENTAGRAM, GOLDEN TRIANGLE, ELLIPSE, & SPIRAL • FIBONACCI SEQUENCE & THE LIMIT • PATTERNS IN NATURE: FRACTALS & CHAOS Jarvis MacLeod Wachowiak

  22. “I used a square as the base shape. I did a tessellation by translation. The one side of the mobile is my tessellation, repeated on an angle. On the other side is a collage of tessellations and patterns. In the center there is a self-portrait of M.C.Esher. Most of the tessellations you see were done by him. I got the pictures from the Internet.”

  23. FRACTALS

  24. James Bond and an experience with fractals!

  25. A rough or fragmented geometric shape Exhibits self-similarity First introduced in 1975 by Benoit Mandelbrot Term is derived from Latin (fractus) meaning broken or fractured. Based on a mathematics equation that undergoes iterations whereby the equation is recursive Fractal Basics

  26. Born November 20, 1924 Z Z2 + C, where c = a + bi Fine structures at arbitrary small scales To irregular to be use Euclidean geometry Usually has a Hausdorff dimension (greater than its topological dimension) Has a simple and recursive definition Mendelbrot Fractal

  27. Begin with an equilateral triangle and then replace the middle of each third of every line segment with a pair of line segments that form an equilateral ‘bump’. Koch Snowflake (1904) http://www.shodor.org/interactivate/activities/KochSnowflake/

  28. Described by Polish mathematician Waclaw Sierpinski. Is only self-similar therefore it is not a ‘true fractal’ Sierpinski Triangle (1915) http://www.arcytech.org/java/fractals/sierpinski.shtml

  29. Known as ‘orbits’ Defined formula or recurrence relation Examples: Mandelbrot set, Julia set, Burning ship fractal, Nova Fractal Escape-time fractals

  30. These have a fixed geometric replacement rule – Koch snowflake, Sierpinski triangle Iterative function systems

  31. Generated by stochastic rather than deterministic process Brownian motion, Levy flight, diffusion-limited aggregation Random Fractals

  32. Generated by iteration of a map or solution of a system of a system of initial valued differential equations that exhibit chaos. Strange attractors http://www.fractal-vibes.com/fvc/Frame01.php3 “Future Legends”

  33. http://www.pbs.org/wgbh/nova/fractals/program.html http://en.wikipedia.org/wiki/Fractal http://serendip.brynmawr.edu/playground/sierpinski.html http://www.shodor.org/interactivate/activities/KochSnowflake/?version=1.5.0_06&browser=MSIE&vendor=Sun_Microsystems_Inc. http://www.geocities.com/CapeCanaveral/2854/ http://local.wasp.uwa.edu.au/~pbourke/fractals/burnship/ http://library.thinkquest.org/26242/full/types/ch14.html http://www.ocf.berkeley.edu/~trose/rossler.html http://groups.csail.mit.edu/mac/users/rauch/islands/ http://wapedia.mobi/en/L%C3%A9vy_flight http://apricot.polyu.edu.hk/~lam/dla/dla.html http://www.fractal-vibes.com/fvc/Frame01.php3 References & resources

  34. L-Systems

  35. Aristid Lindenmayer (1925–1989). Biologist and botanist. Studied the growth patterns of algae. L-System http://cage.rug.ac.be/~bh/L-systemen/Lindenmayer.htm

  36. L-systems were devised to provide a mathematical description of the development of simple multi-cellular organisms, and to demonstrate relationships between plant cells. These systems are also used to describe higher plants and complex branching. L-System

  37. An alphabet is needed. A set of fixed symbols known as constants. A initial word that starts everything. This is called an axiom. A set of production rules that describes how the word is to be built. Words are built iteratively, applying the production rules at each iteration to form longer, more complex words. Grammars

  38. Alphabet: X, F Constants: +, -, [, ] Axiom: X Production rules: X → F-[[X]+X]+F[+FX]-X F → FF A More Complicated Example

  39. Production rules: X → F-[[X]+X]+F[+FX]-X F → FF Steps: 0 X 1 F-[[X]+X]+F[+FX]-X 2 FF-[[F-[[X]+X]+F[+FX]-X]+F-[[X]+X]+F[+FX]- X]+FF[+FF F-[[X]+X]+F[+FX]-X]- F-[[X]+X]+F[+FX]-X A More Complicated Example

  40. Suppose that we want to see “what the word looks like”. Now suppose we have one of these: What Does it Mean? http://www.waynet.org/waynet/spotlight/2004/images/07/turtle640.jpg

  41. F means “move forward”. + means “turn counterclockwise by a certain angle.” - means turn “clockwise by the same angle.” Turtle Graphics http://www.terrapinswim.vicid.com/images/images/328/0/online_button.png

  42. [ means “remember location”. ] means “return to the point in memory”. X means “do nothing”. This is just a placeholder. Turtle Graphics (2)

  43. Alphabet: F Constants: +, - (25) Axiom: X Production rules: X → F-[[X]+X]+F[+FX]-X F → FF Example 1

  44. Alphabet: F Constants: +, - (25) Axiom: X Production rules: X → F-[[X]+X]+F[+FX]-X F → FF Example 1 Iteration 1 5 4 6 7 3 2

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