Fractals!
Bullock Math Academy
March 22, 2014
Brian Shelburne
Dept. of Math & CS
Wittenberg Univ.
Website
http://www4.wittenberg.edu/academics/mathcomp/shelburne/Fractals/Fractalindex.html
Bullock Math Academy – Saturday March 22, 2014
Kahn Academy Video - Doodling in Math: Dragons
https://www.khanacademy.org/math/recreational-math/vi-hart/doodling-in-math/v/dragons
1.
2.
3.
4.
5.
Iteration
Generating Fractals
Measuring Areas and Lengths of Some Common Fractals
Self-Similarity and Fractal Dimension
The Mandelbrot Set Fractal
Prerequisites
Pythagorean Theorem for Right Triangles
FOIL
Exponents: 2×2×2 = 23 and (1/3)×(1/3) = (1/3)2.
1
2
Square Roots: 3 = 3 (and fractional exponents)
Formulas for Area and Perimeter
Ability to see and describe patterns
Geometric Iteration
A Seed and a Rule
Orbits and Eventual Orbits
Seed: A square with side 1
Rule: Shrink each side by half
More Geometric Iteration
Seed
Rule: Rotate 90° clockwise
Seed
Rule?
Orbit:
Numeric Iteration
Seed: 12.3 → x
Rule: (1/3)∙x → x
Seed: anything → x
Rule: 𝑥 → x
Seed: 0.67 → x
Rule: 3.5∙x∙(1-x) → x
0.67 [sto→] x
3.5∙x∙(1-x) [sto→] x
Seed: 0.45 → x
Rule: 3.2∙x∙(1-x) → x
0.45 [sto→] x
3.2∙x∙(1-x) [sto→] x
You Try It
Seed:
Rule: Shrink each linear dimension by ½ and rotation 180°
What is the Orbit?
What is the Eventual Orbit?
The Fractal Plus
Seed:
Rule: Tic-Tac-Toe and Remove the 4 Corners
The Sierpinski Carpet Fractal
Seed:
Rule: Tic Tac Toe & Remove Middle Square
Orbit:
Fractals by Removals
Seed: An equilateral triangle
Rule: Connect the midpoints of the 3 sides and remove the
interior of the middle triangle
Assume the area of the Seed is 1. What is the area of the 2nd and
3rd iterations?
Self-Similar Copies - Another Method for Generating Fractals
Seed:
Generator:
Rule:
Dragon Fractal
The Dragon Fractal
Seed:
Rule:
Orbit:
i.e. “jag” right then left
The Koch Fractal
Seed: Line of length 1:
Rule: Divide by 3 – replace middle 3rd with a tent
The Thunderbolt Fractal
Seed: A Line of length 1:
Rule: Divide by 4 and replace two middle pieces with opposite
tents
Area and Perimeter
Sierpinski Carpet – Area and Perimeter
Iteration
0
1
2
3
…
n
Number of Squares
1
8
64
512
8n
Length of Side
1
1/3
1/9
1/27
(1/3)n
Area
1
8/9
Perimeter
4
4+4/3
Sierpinski Gasket – Area and Perimeter
Iteration
0
1
2
3
…
n
Number of Triangles 1
3
9
27
3n
Length of Side
1
1/2
1/4
1/8
(1/2)n
Area
1
3/4
Perimeter
3
9/2
Koch Curve – Perimeter
Iteration
0
1
2
3
Number
of Segments
1
4
16
64
4n
Length of
Segment
1
1/3
1/9
1/27
(1/3)n
Length
1
4/3
Area under curve
…
n
Self-Similarity & Fractal Dimension
Dimension
Line
Number of Copies
Scaling Factor
1
1
2
2
3
3
4
4
…
…
9
3
16
4
…
...
27
3
64
4
…
…
Square
Number of Copies
Scaling Factor
1
1
4
2
Cube
Number of Copies
Scaling Factor
1
1
8
2
Fractal Dimension
Sierpinski Gasket
Number of Copies
Scaling Factor
1
1
3
2
9
4
…
…
Sierpinski Carpet
Number of Copies
Scaling Factor
1
1
8
3
64
9
…
...
Koch Curve
Number of Copies
Scaling Factor
1
1
4
3
16
9
…
…
Logarithms: Solving 3 = 2n. What is n?
Examples:
Solve 9 = 3n.
Solve 16 = 4n.
Solve 2 = 4n.
Solve 32 = 4n.
Solve 10 = 2m. Since 23 = 8 and 24 = 16 it follows that 3 < m < 4.
If n is the unique number such that 3 = 2n , the we say n = log2 3.
Using your calculator log2 3 = log3/log2
The Mandelbrot Set
Complex Numbers (a brief introduction)
Let 𝑖 = −1. So 𝑖 2 = −1. The i sometimes stands for imaginary
although it’s not a very good name.
A Complex Number has a real part and an imaginary part
For example: (2 + 3i)
All the usual rules for arithmetic work for complex numbers (Law
of Least Astonishment)
To add or subtract complex numbers add or subtract real parts
and the imaginary parts separately
For example (2 + 3i) + (4 – i) = (6 + 2i)
To multiply complex numbers use First Outside Inside Last
For Example: (2 + 3i) × (4 – i) = 8 - 2i + 12i - 6i2 except i2 = -1 so
8 - 2i + 12i - 6i2 = 8 – 2i +12i + 6 = 14 -10i
You Try It:
Finally – In the same way you can plot real numbers on the
number line, you can plot complex numbers in the complex
plane (using the vertical y-axis to mark off the imaginary
distance) .
Complex Iteration
Seed: 1 → z
Rule: Multiply by each term by
1
2
Orbit: 1, +
3
𝑖
2
,
−1
2
+
3
𝑖
2
, -1,
1
2
−1
2
-0.5 + 0.866i
Geometric Picture
+
+
3
𝑖
2
+
:
− 3
𝑖
2
1
2
+
1
2
, +
+
+
-1
-0.5 - 0.866i
3
𝑖
2
×𝑧 →𝑧
− 3
𝑖,
2
1…
0.5 + 0.866i
+
1
+
+
0.5 - 0.866i
The Mandelbrot Iteration
Seed: Let z = 0 and let c be any complex number
Rule: Square z and add c ( z = z2 + c)
Try c = i
Try c = -1.76 + 0.01i
The Mandelbrot Set is the set of all complex numbers c such that
the iteration formula 0 →z; z2 + c → z does not escape to infinity!
Coloring the Mandelbrot Set
Black: Elements that do not escape – these points are in the
Mandelbrot Set
Colors: speed of escape – from red (fast) to magenta (slow)