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

6 February 2009Jarvis MacLeod Wachowiak2 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 20 th Century Europe  Activity 2: Creating your own tessellation pattern  Fractals: Iterations in 21 st 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

6 February 2009Jarvis MacLeod Wachowiak3 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.

6 February 2009Jarvis MacLeod Wachowiak4 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 ( )

6 February 2009Jarvis MacLeod Wachowiak5 HISTORICAL OVERVIEW ANCIENT EGYPT & GREECE

6 February 2009Jarvis MacLeod Wachowiak6 HISTORICAL OVERVIEW THE RENAISSANCE “DE DIVINA PROPORTIONE” (1509) WRITER: FRA LUCA PACIOLI ILLUSTRATOR: LEONARDO DA VINCI

6 February 2009Jarvis MacLeod Wachowiak7 HISTORICAL OVERVIEW THE MODERN ERA: ARTISTS OF THE 19 TH AND 20 TH CENTURIES SEURAT, DALI, MONDRIAN, COLVILLE

6 February 2009Jarvis MacLeod Wachowiak8 HISTORICAL OVERVIEW SONY PLASMA GRAND WEGA $ CANADIAN 16:9 ASPECT RATIO (APPROX. 1.78) APPLE IMAC $2500 CANADIAN 36.8/22.8 = (GOLDEN APPLES?)

6 February 2009Jarvis MacLeod Wachowiak9 TEACHING HOW TO FIND THE GOLDEN SECTION [I] ALGEBRAICALLY x 1 = (Phi) NOW, BEGINNING WITH ANY GIVEN LENGTH (L): NEXT LARGEST SECTION LENGTH X ( ) NEXT SMALLEST SECTION LENGTH/( ) OR MORE SIMPLY, LENGTH X ( )

6 February 2009Jarvis MacLeod Wachowiak10 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

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6 February 2009Jarvis MacLeod Wachowiak20 THERE IS GEOMETRY IN THE HUMMING OF THE STRINGS. THERE IS MUSIC IN THE SPACING OF THE SPHERES. PYTHAGORAS (C.A B.C.)

6 February 2009Jarvis MacLeod Wachowiak21 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

“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.”

FRACTALS KATARIN MACLEOD Math Talk 2009 FRACTALS

FRACTALS KATARIN MACLEOD Math Talk 2009 James Bond and an experience with fractals!

FRACTALS KATARIN MACLEOD Math Talk 2009 Fractal Basics 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

FRACTALS KATARIN MACLEOD Math Talk 2009 Mendelbrot Fractal Born November 20, 1924 Z  Z 2 + C, where c = a + bi a)Fine structures at arbitrary small scales b)To irregular to be use Euclidean geometry c)Usually has a Hausdorff dimension (greater than its topological dimension) d)Has a simple and recursive definition

FRACTALS KATARIN MACLEOD Math Talk 2009 Koch Snowflake (1904) 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 ’.

FRACTALS KATARIN MACLEOD Math Talk 2009 Sierpinski Triangle (1915) Described by Polish mathematician Waclaw Sierpinski. Is only self-similar therefore it is not a ‘true fractal’

FRACTALS KATARIN MACLEOD Math Talk 2009 Escape-time fractals Known as ‘orbits’ Defined formula or recurrence relation Examples: Mandelbrot set, Julia set, Burning ship fractal, Nova Fractal

FRACTALS KATARIN MACLEOD Math Talk 2009 Iterative function systems These have a fixed geometric replacement rule – Koch snowflake, Sierpinski triangle

FRACTALS KATARIN MACLEOD Math Talk 2009 Random Fractals Generated by stochastic rather than deterministic process Brownian motion, Levy flight, diffusion- limited aggregation

FRACTALS KATARIN MACLEOD Math Talk 2009 Strange attractors Generated by iteration of a map or solution of a system of a system of initial valued differential equations that exhibit chaos. “Future Legends”

FRACTALS KATARIN MACLEOD Math Talk 2009 References & resources n=1.5.0_06&browser=MSIE&vendor=Sun_Microsystems_Inc. n=1.5.0_06&browser=MSIE&vendor=Sun_Microsystems_Inc

Math Talk 2009 L-Systems

L-System Aristid Lindenmayer (1925–1989). –Biologist and botanist. Studied the growth patterns of algae.

Math Talk 2009 L-System 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.

Math Talk 2009 Grammars 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.

Math Talk 2009 A More Complicated Example Alphabet: X, F Constants: +, -, [, ] Axiom: X Production rules: X → F-[[X]+X]+F[+FX]-X F → FF

Math Talk 2009 A More Complicated Example Production rules: X → F-[[X]+X]+F[+FX]-X F → FF Steps: 0X 1F-[[X]+X]+F[+FX]-X 2FF-[[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

Math Talk 2009 What Does it Mean? Suppose that we want to see “what the word looks like”. Now suppose we have one of these:

Math Talk 2009 Turtle Graphics F means “move forward”. + means “turn counterclockwise by a certain angle.” - means turn “clockwise by the same angle.”

Math Talk 2009 Turtle Graphics (2) [ means “remember location”. ] means “return to the point in memory”. X means “do nothing”. This is just a placeholder.

Math Talk 2009 Example 1 Alphabet: F Constants: +, - (  25  ) Axiom: X Production rules: X → F-[[X]+X]+F[+FX]-X F → FF

Math Talk 2009 Example 1 Alphabet: F Constants: +, - (  25  ) Axiom: X Production rules: X → F-[[X]+X]+F[+FX]-X F → FF Iteration

Math Talk 2009 Example 2 Alphabet: F Constants: +, - (  90  ) Axiom: F-F-F-F Production rules: F → FF-F-F-F-F-F+F

Math Talk 2009 Example 2 Alphabet: F Constants: +, - (  90  ) Axiom: F-F-F-F Production rules: F → FF-F-F-F-F-F+F Iteration1432

Math Talk 2009 Example 3 Alphabet: F Constants: +, - (  25  ) Axiom: F Production rules: F → FF+[+F-F-F]-[-F+F+F]

Math Talk 2009 Example 3 Alphabet: F Constants: +, - (  25  ) Axiom: F Production rules: F → FF+[+F-F-F]-[-F+F+F] Iteration154632

Math Talk 2009 Example 4 Alphabet: F Constants: +, - (  120  ) Axiom: F+F+F Production rules: F → F+F-F-F+F

Math Talk 2009 Example 4 Alphabet: F Constants: +, - (  120  ) Axiom: F+F+F Production rules: F → F+F-F-F+F Iteration

Math Talk D Trees Generated with an L-System

Math Talk 2009 Trees and Bushes

Math Talk 2009 “Hairy” Plants Fuhrer, M.; Jensen, H.W.; Prusinkiewicz, P. “Modeling Hairy Plants”, Graphical Models 68(4), ,

Math Talk 2009 Fractal Mountains

Math Talk D L-Systems

6 February 2009Jarvis MacLeod Wachowiak62 Video Clips Math/Art Videos: “Donald Duck In Mathmagicland” (1959) with host Donald Duck “Life by the Numbers” (2006) with host Danny Glover

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