NUMERIC Mathematics Lecture Series Nipissing University, North Bay, Ontario

NUMERIC Mathematics Lecture SeriesNipissing University, North Bay, Ontario

Exploring the Math and Art Connection6 February 2009

Dr. Daniel JarvisMathematics & Visual Arts

Education

Professor Katarin MacLeodMathematics & Physics

Education

Dr. Mark WachowiakComputer Science

6 February 2009 Jarvis MacLeod Wachowiak 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

6 February 2009 Jarvis MacLeod Wachowiak 3

AN INTRODUCTION TO RATIOIn 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 2009 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)

6 February 2009 Jarvis MacLeod Wachowiak 5

HISTORICAL OVERVIEW

ANCIENTEGYPT&GREECE

6 February 2009 Jarvis MacLeod Wachowiak 6

HISTORICAL OVERVIEW

THE RENAISSANCE“DE DIVINA PROPORTIONE” (1509)

WRITER: FRA LUCA PACIOLIILLUSTRATOR: LEONARDO DA VINCI

6 February 2009 Jarvis MacLeod Wachowiak 7

HISTORICAL OVERVIEW

THE MODERN ERA:

ARTISTS OF THE 19TH AND 20TH CENTURIES

SEURAT, DALI, MONDRIAN,COLVILLE

6 February 2009 Jarvis MacLeod Wachowiak 8

HISTORICAL OVERVIEW

SONY PLASMAGRAND WEGA$17 000 CANADIAN16:9 ASPECT RATIO(APPROX. 1.78)

APPLE IMAC$2500 CANADIAN36.8/22.8 = 1.614(GOLDEN APPLES?)

6 February 2009 Jarvis MacLeod Wachowiak 9

TEACHING HOW TO FIND THE GOLDEN SECTION

[I] ALGEBRAICALLY

1

11

1 0

1 1 4 1 1

2 1

1 5

2161803398

2

2

2

x

x

xx x

x x

x

x

x

( ) ( )( )

( )

. ...

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)

6 February 2009 Jarvis MacLeod Wachowiak 10

TEACHING HOW TO FIND THE GOLDEN SECTION

[II] GEOMETRICALLYBEGIN WITH A SQUARE; EXTEND ONE SIDE

FROM MIDPOINT, CUT AN ARC FROM FAR CORNER TO EXTENDED LINE

COMPLETE RECTANGLE & INTERNAL SQUARES

6 February 2009 Jarvis MacLeod Wachowiak 11

6 February 2009 Jarvis MacLeod Wachowiak 12

6 February 2009 Jarvis MacLeod Wachowiak 13

6 February 2009 Jarvis MacLeod Wachowiak 14

6 February 2009 Jarvis MacLeod Wachowiak 15

6 February 2009 Jarvis MacLeod Wachowiak 16

6 February 2009 Jarvis MacLeod Wachowiak 17

6 February 2009 Jarvis MacLeod Wachowiak 18

6 February 2009 Jarvis MacLeod Wachowiak 19

6 February 2009 Jarvis MacLeod Wachowiak 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.)

6 February 2009 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 LIMITPATTERNS 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

FRACTALSFRACTALS

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 Z2 + C, where c = a + bia) Fine structures at arbitrary small scalesb) To irregular to be use Euclidean geometryc) 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’.

http://www.shodor.org/interactivate/activities/KochSnowflake/

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’

http://www.arcytech.org/java/fractals/sierpinski.shtml

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.

http://www.fractal-vibes.com/fvc/Frame01.php3“Future Legends”

FRACTALS KATARIN MACLEOD

Math Talk 2009

References & resources

• 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

Math Talk 2009

L-SystemsL-Systems

Math Talk 2009

L-System

• Aristid Lindenmayer (1925–1989). – Biologist and botanist.

• Studied the growth patterns of algae.

http://cage.rug.ac.be/~bh/L-systemen/Lindenmayer.htm

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:

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

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:

http://www.waynet.org/waynet/spotlight/2004/images/07/turtle640.jpg

Math Talk 2009

Turtle Graphics• F means “move forward”.

• + means “turn counterclockwise by a certain angle.”

• - means turn “clockwise by the same angle.”

http://www.terrapinswim.vicid.com/images/images/328/0/online_button.png

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 1546732

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

Iteration 1432

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]

Iteration 154632

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 1546732

Math Talk 2009

http://www.royalhigh.edin.sch.uk/departments/departments/CDT/ahgc_0708_blender_vegetation.html

3D Trees Generated with an L-System

Math Talk 2009

Trees and Bushes

http://www-ui.is.s.u-tokyo.ac.jp/~ijiri/SketchLSystem/

Math Talk 2009

“Hairy” Plants

Fuhrer, M.; Jensen, H.W.; Prusinkiewicz, P. “Modeling Hairy Plants”, Graphical Models 68(4), 333-342, 2006..

Math Talk 2009

Fractal Mountains

http://www.math.ucdavis.edu/~kapovich/fractal.gifhttp://myweb.cwpost.liu.edu/aburns/gallery/newgall.htm

Math Talk 2009

3D L-Systems

http://mathforum.org/advanced/robertd/lsys3d.html

6 February 2009 Jarvis MacLeod Wachowiak 62

Video Clips

Math/Art Videos: “Donald Duck In Mathmagicland” (1959) with host Donald Duck

“Life by the Numbers” (2006) with host Danny Glover

6 February 2009 Jarvis MacLeod Wachowiak 63