Magical Mathematics

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Magical Mathematics

• Frank Thuijsman

University College Maastricht Fall, 2004
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Course Contents

• Logic, Reasoning and Proofs
• Numbers, Sets and Counting
• Relations and Functions
• Examples and Exercises
• Today Examples

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All aboard for a little trip
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M.C. Escher Print Gallery (1956)
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Completed Print Gallery (2002)
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Completion Method
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Completed Straight
animation
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Completed and Twisted
animation
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The Droste effect
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Sierpinski Triangle
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Kochs Snowflake
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Fractals
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Back to Basics

? 1
0.9999999999999999999999
10 x 9.9999999999999999999999 x
0.9999999999999999999999 9 x
9.0000000000000000000000 x 1.00000000000000000
00000 1 0.9999999999999999999999 1/3
0.3333333333333333333333
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Numbers, Larger and Smaller
Some powerful examples
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Numbers, Natural and Peculiar
Natural numbers 1, 2, 3, 4, 5, 6, Integers ,
-3, -2, -1, 0, 1, 2, 3, Fractions, e.g. 3/4,
-2/5, -1, 0, 1, 2, Reals, e.g. v2, e, p, F,
-2/5, -1, 0, 1, 2, Complex numbers, e.g. 1 i
v2, -3/75 i Fibonacci numbers 1, 1, 2, 3, 5, 8,
13, 21,
(v2)2 2
e 1 1 1/2 1/6 1/24 1/120
e limn?8 (11/n)n
? 3.1416926 The area of a circle of radius 1
F (1v5)/2 1,618034 The Golden Section
i2 -1 (1 i v2)(1 - i v2) 1 i v2 - i v2 -
i2 (v2)2 1 2 3
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The Golden Section F
If we take AB x and AP 1, then we get x/1
1/(x-1) which gives us x2 x 1 0 Solving
this equation we find x (1v5)/2 1,618034
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Golden Section F
Since F 1 1/F, we have F 1 1/(1 1/F)
which leads to
Since F 2 1 F, we also have F v(1 F)
which yields
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F and Pentagram
F divides EC by golden section EF FC EC EF
Proof EF FC EB DC EC ED EC EF
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F and Pentagram
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F and Crop Circles
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F and Human Body
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F and Art
Leonardo da Vinci Vitruvius
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F and Art
Leonardo da Vinci Annunciation
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F and Architecture
Notre Dame Cathedral in Paris
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F and Architecture
Cheops Pyramid
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F and Architecture
Parthenon
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F and Fibonacci Numbers
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,
377,
Squares with such sides
Fractions 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13,
34/21, 55/34, 89/55, F
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F and Fibonacci Numbers
Why do we have 1/1, 2/1, 3/2, 5/3, 8/5, 13/8,
21/13, 34/21, 55/34, 89/55, F ?
Let x1 1, x2 1, x3 2, x4 3, x5 5, x6
8, Then xn2 xn1 xn and thus xn2 / xn
1 1 (xn / xn1) If y limn?8 xn1 / xn ,
then we get y 1 1/y Or y2 y 1 0, the
same equation as seen before!
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The Fibonacci Spiral
All are quarter circles
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Fibonacci Numbers and Rabbits
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Fibonacci Numbers and Nature
8 spirals
13 spirals
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,
377,
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Fibonacci Numbers and Nature
21 spirals
34 spirals
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,
377,
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Counting how many people?
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Counting how many people?
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Counting how many people?
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Counting how many people?
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Pythagoras
Proof of the Theorem of Pythagoras
Pythagoras Tree
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Lorenz Butterfly
An example of chaos
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The hull of points in the plane
How?
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Shortest tour along all points
How?
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Time for Coffee
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