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
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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. - PowerPoint PPT Presentation
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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
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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
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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.
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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)
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HISTORICAL OVERVIEW
ANCIENTEGYPT&GREECE
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HISTORICAL OVERVIEW
THE RENAISSANCE“DE DIVINA PROPORTIONE” (1509)
WRITER: FRA LUCA PACIOLIILLUSTRATOR: LEONARDO DA VINCI
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HISTORICAL OVERVIEW
THE MODERN ERA:
ARTISTS OF THE 19TH AND 20TH CENTURIES
SEURAT, DALI, MONDRIAN,COLVILLE
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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?)
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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)
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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
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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.)
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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’.