Download presentation
Presentation is loading. Please wait.
Published byAugustus Gordon Modified over 9 years ago
1
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
2
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
3
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.
4
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 (1571-1630)
5
6 February 2009Jarvis MacLeod Wachowiak5 HISTORICAL OVERVIEW ANCIENT EGYPT & GREECE
6
6 February 2009Jarvis MacLeod Wachowiak6 HISTORICAL OVERVIEW THE RENAISSANCE “DE DIVINA PROPORTIONE” (1509) WRITER: FRA LUCA PACIOLI ILLUSTRATOR: LEONARDO DA VINCI
7
6 February 2009Jarvis MacLeod Wachowiak7 HISTORICAL OVERVIEW THE MODERN ERA: ARTISTS OF THE 19 TH AND 20 TH CENTURIES SEURAT, DALI, MONDRIAN, COLVILLE
8
6 February 2009Jarvis MacLeod Wachowiak8 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?)
9
6 February 2009Jarvis MacLeod Wachowiak9 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)
10
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
11
6 February 2009Jarvis MacLeod Wachowiak11
12
6 February 2009Jarvis MacLeod Wachowiak12
13
6 February 2009Jarvis MacLeod Wachowiak13
14
6 February 2009Jarvis MacLeod Wachowiak14
15
6 February 2009Jarvis MacLeod Wachowiak15
16
6 February 2009Jarvis MacLeod Wachowiak16
17
6 February 2009Jarvis MacLeod Wachowiak17
18
6 February 2009Jarvis MacLeod Wachowiak18
19
6 February 2009Jarvis MacLeod Wachowiak19
20
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. 582-500 B.C.)
21
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
25
“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.”
29
FRACTALS KATARIN MACLEOD Math Talk 2009 FRACTALS
30
FRACTALS KATARIN MACLEOD Math Talk 2009 James Bond and an experience with fractals!
31
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
32
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
33
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/
34
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
35
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
36
FRACTALS KATARIN MACLEOD Math Talk 2009 Iterative function systems These have a fixed geometric replacement rule – Koch snowflake, Sierpinski triangle
37
FRACTALS KATARIN MACLEOD Math Talk 2009 Random Fractals Generated by stochastic rather than deterministic process Brownian motion, Levy flight, diffusion- limited aggregation
38
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”
39
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/?versio n=1.5.0_06&browser=MSIE&vendor=Sun_Microsystems_Inc.http://www.shodor.org/interactivate/activities/KochSnowflake/?versio n=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
40
Math Talk 2009 L-Systems
41
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
42
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.
43
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.
44
Math Talk 2009 A More Complicated Example Alphabet: X, F Constants: +, -, [, ] Axiom: X Production rules: X → F-[[X]+X]+F[+FX]-X F → FF
45
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
46
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
47
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
48
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.
49
Math Talk 2009 Example 1 Alphabet: F Constants: +, - ( 25 ) Axiom: X Production rules: X → F-[[X]+X]+F[+FX]-X F → FF
50
Math Talk 2009 Example 1 Alphabet: F Constants: +, - ( 25 ) Axiom: X Production rules: X → F-[[X]+X]+F[+FX]-X F → FF Iteration1546732
51
Math Talk 2009 Example 2 Alphabet: F Constants: +, - ( 90 ) Axiom: F-F-F-F Production rules: F → FF-F-F-F-F-F+F
52
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
53
Math Talk 2009 Example 3 Alphabet: F Constants: +, - ( 25 ) Axiom: F Production rules: F → FF+[+F-F-F]-[-F+F+F]
54
Math Talk 2009 Example 3 Alphabet: F Constants: +, - ( 25 ) Axiom: F Production rules: F → FF+[+F-F-F]-[-F+F+F] Iteration154632
55
Math Talk 2009 Example 4 Alphabet: F Constants: +, - ( 120 ) Axiom: F+F+F Production rules: F → F+F-F-F+F
56
Math Talk 2009 Example 4 Alphabet: F Constants: +, - ( 120 ) Axiom: F+F+F Production rules: F → F+F-F-F+F Iteration1546732
57
Math Talk 2009 http://www.royalhigh.edin.sch.uk/departments/departments/CDT/ahgc_0708_blender_vegetation.html 3D Trees Generated with an L-System
58
Math Talk 2009 Trees and Bushes http://www-ui.is.s.u-tokyo.ac.jp/~ijiri/SketchLSystem/
59
Math Talk 2009 “Hairy” Plants Fuhrer, M.; Jensen, H.W.; Prusinkiewicz, P. “Modeling Hairy Plants”, Graphical Models 68(4), 333-342, 2006..
60
Math Talk 2009 Fractal Mountains http://www.math.ucdavis.edu/~kapovich/fractal.gif http://myweb.cwpost.liu.edu/aburns/gallery/newgall.htm
61
Math Talk 2009 3D L-Systems http://mathforum.org/advanced/robertd/lsys3d.html
62
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
63
6 February 2009Jarvis MacLeod Wachowiak63
Similar presentations
© 2024 SlidePlayer.com. Inc.
All rights reserved.