Excel quad iteration

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Excel quad iteration M-set iterator Movie maker
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The Fractal Geometry of the Mandelbrot Set
How the computer has revolutionized mathematics
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The Fractal Geometry of the Mandelbrot Set
You need to know
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The Fractal Geometry of the Mandelbrot Set
You need to know
How to count
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The Fractal Geometry of the Mandelbrot Set
You need to know
How to count
How to add
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Many people know the pretty pictures...
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but few know the even prettier mathematics.
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Oh, that's nothing but the 3/4 bulb ....
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...hanging off the period 16 M-set.....
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...lying in the 1/7 antenna...
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...attached to the 1/3 bulb...
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...hanging off the 3/7 bulb...
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...on the northwest side of the main cardioid.
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Oh, that's nothing but the 3/4 bulb, hanging
off the period 16 M-set, lying in the 1/7
antenna of the 1/3 bulb attached to the 3/7
bulb on the northwest side of the main cardioid.
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Start with a function
2
x constant
29
Start with a function
2
x constant
and a seed
x
0
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Then iterate
2
x x constant
1
0
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Then iterate
2
x x constant
1
0
2
x x constant
2
1
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Then iterate
2
x x constant
1
0
2
x x constant
2
1
2
x x constant
3
2
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Then iterate
2
x x constant
1
0
2
x x constant
2
1
2
x x constant
3
2
2
x x constant
4
3
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Then iterate
2
x x constant
1
0
2
x x constant
2
1
Orbit of x
2
0
x x constant
3
2
2
x x constant
4
3
etc.
Goal understand the fate of orbits.
35
2
Example x 1 Seed 0
x 0
0
x
1
x
2
x
3
x
4
x
5
x
6
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2
Example x 1 Seed 0
x 0
0
x 1
1
x
2
x
3
x
4
x
5
x
6
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2
Example x 1 Seed 0
x 0
0
x 1
1
x 2
2
x
3
x
4
x
5
x
6
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2
Example x 1 Seed 0
x 0
0
x 1
1
x 2
2
x 5
3
x
4
x
5
x
6
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2
Example x 1 Seed 0
x 0
0
x 1
1
x 2
2
x 5
3
x 26
4
x
5
x
6
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2
Example x 1 Seed 0
x 0
0
x 1
1
x 2
2
x 5
3
x 26
4
x big
5
x
6
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2
Example x 1 Seed 0
x 0
0
x 1
1
x 2
2
x 5
3
x 26
4
x big
5
x BIGGER
6
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2
Example x 1 Seed 0
x 0
0
x 1
1
x 2
2
Orbit tends to infinity
x 5
3
x 26
4
x big
5
x BIGGER
6
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2
Example x 0 Seed 0
x 0
0
x
1
x
2
x
3
x
4
x
5
x
6
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2
Example x 0 Seed 0
x 0
0
x 0
1
x
2
x
3
x
4
x
5
x
6
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2
Example x 0 Seed 0
x 0
0
x 0
1
x 0
2
x
3
x
4
x
5
x
6
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2
Example x 0 Seed 0
x 0
0
x 0
1
x 0
2
x 0
3
x
4
x
5
x
6
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2
Example x 0 Seed 0
x 0
0
x 0
1
x 0
2
A fixed point
x 0
3
x 0
4
x 0
5
x 0
6
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2
Example x - 1 Seed 0
x 0
0
x
1
x
2
x
3
x
4
x
5
x
6
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2
Example x - 1 Seed 0
x 0
0
x -1
1
x
2
x
3
x
4
x
5
x
6
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2
Example x - 1 Seed 0
x 0
0
x -1
1
x 0
2
x
3
x
4
x
5
x
6
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2
Example x - 1 Seed 0
x 0
0
x -1
1
x 0
2
x -1
3
x
4
x
5
x
6
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2
Example x - 1 Seed 0
x 0
0
x -1
1
x 0
2
x -1
3
x 0
4
x
5
x
6
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2
Example x - 1 Seed 0
x 0
0
x -1
1
x 0
2
x -1
A two- cycle
3
x 0
4
x -1
5
x 0
6
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2
Example x - 1.1 Seed 0
x 0
0
x
1
x
2
x
3
x
4
x
5
x
6
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2
Example x - 1.1 Seed 0
x 0
0
x -1.1
1
x
2
x
3
x
4
x
5
x
6
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2
Example x - 1.1 Seed 0
x 0
0
x -1.1
1
x 0.11
2
x
3
x
4
x
5
x
6
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2
Example x - 1.1 Seed 0
x 0
0
x -1.1
1
x 0.11
2
x
3
time for the computer!
x
4
x
5
x
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Excel OrbDgm
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Observation
For some real values of c, the orbit of 0 goes
to infinity, but for other values, the orbit of
0 does not escape.
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Complex Iteration
2
Iterate z c
complex numbers
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2
Example z i Seed 0
z 0
0
z
1
z
2
z
3
z
4
z
5
z
6
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2
Example z i Seed 0
z 0
0
z i
1
z
2
z
3
z
4
z
5
z
6
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2
Example z i Seed 0
z 0
0
z i
1
z -1 i
2
z
3
z
4
z
5
z
6
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2
Example z i Seed 0
z 0
0
z i
1
z -1 i
2
z -i
3
z
4
z
5
z
6
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2
Example z i Seed 0
z 0
0
z i
1
z -1 i
2
z -i
3
z -1 i
4
z
5
z
6
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2
Example z i Seed 0
z 0
0
z i
1
z -1 i
2
z -i
3
z -1 i
4
z -i
5
z
6
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2
Example z i Seed 0
z 0
0
z i
1
z -1 i
2
z -i
3
z -1 i
4
z -i
5
2-cycle
z -1 i
6
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2
Example z i Seed 0
i
1
-1
-i
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2
Example z i Seed 0
i
1
-1
-i
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2
Example z i Seed 0
i
1
-1
-i
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2
Example z i Seed 0
i
1
-1
-i
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2
Example z i Seed 0
i
1
-1
-i
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2
Example z i Seed 0
i
1
-1
-i
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2
Example z i Seed 0
i
1
-1
-i
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2
Example z i Seed 0
i
1
-1
-i
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2
Example z 2i Seed 0
z 0
0
z
1
z
2
z
3
z
4
z
5
z
6
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2
Example z 2i Seed 0
z 0
0
z 2i
1
z -4 2i
2
Off to infinity
z 12 - 14i
3
z -52 336i
4
z big
5
z BIGGER
6
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Same observation
Sometimes orbit of 0 goes to infinity, other
times it does not.
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The Mandelbrot Set
All c-values for which the orbit of 0 does NOT
go to infinity.
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Algorithm for computing M
Start with a grid of complex numbers
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Algorithm for computing M
Each grid point is a complex c-value.
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Algorithm for computing M
Compute the orbit of 0 for each c. If the orbit
of 0 escapes, color that grid point.
red fastest escape
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Algorithm for computing M
Compute the orbit of 0 for each c. If the orbit
of 0 escapes, color that grid point.
orange slower
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Algorithm for computing M
Compute the orbit of 0 for each c. If the orbit
of 0 escapes, color that grid point.

yellow green blue violet
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Algorithm for computing M
Compute the orbit of 0 for each c. If the orbit
of 0 does not escape, leave that grid point
black.
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Algorithm for computing M
Compute the orbit of 0 for each c. If the orbit
of 0 does not escape, leave that grid point
black.
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The eventual orbit of 0
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The eventual orbit of 0
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The eventual orbit of 0
3-cycle
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The eventual orbit of 0
3-cycle
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The eventual orbit of 0
3-cycle
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The eventual orbit of 0
3-cycle
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The eventual orbit of 0
3-cycle
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The eventual orbit of 0
3-cycle
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The eventual orbit of 0
3-cycle
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The eventual orbit of 0
3-cycle
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The eventual orbit of 0
3-cycle
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The eventual orbit of 0
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The eventual orbit of 0
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The eventual orbit of 0
4-cycle
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The eventual orbit of 0
4-cycle
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The eventual orbit of 0
4-cycle
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The eventual orbit of 0
4-cycle
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The eventual orbit of 0
4-cycle
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The eventual orbit of 0
4-cycle
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The eventual orbit of 0
4-cycle
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The eventual orbit of 0
4-cycle
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The eventual orbit of 0
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The eventual orbit of 0
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The eventual orbit of 0
5-cycle
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The eventual orbit of 0
5-cycle
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The eventual orbit of 0
5-cycle
112
The eventual orbit of 0
5-cycle
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The eventual orbit of 0
5-cycle
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The eventual orbit of 0
5-cycle
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The eventual orbit of 0
5-cycle
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The eventual orbit of 0
5-cycle
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The eventual orbit of 0
5-cycle
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The eventual orbit of 0
5-cycle
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The eventual orbit of 0
5-cycle
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The eventual orbit of 0
2-cycle
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The eventual orbit of 0
2-cycle
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The eventual orbit of 0
2-cycle
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The eventual orbit of 0
2-cycle
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The eventual orbit of 0
2-cycle
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The eventual orbit of 0
fixed point
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The eventual orbit of 0
fixed point
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The eventual orbit of 0
fixed point
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The eventual orbit of 0
fixed point
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The eventual orbit of 0
fixed point
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The eventual orbit of 0
fixed point
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The eventual orbit of 0
fixed point
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The eventual orbit of 0
fixed point
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The eventual orbit of 0
goes to infinity
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The eventual orbit of 0
goes to infinity
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The eventual orbit of 0
goes to infinity
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The eventual orbit of 0
goes to infinity
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The eventual orbit of 0
goes to infinity
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The eventual orbit of 0
goes to infinity
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The eventual orbit of 0
goes to infinity
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The eventual orbit of 0
goes to infinity
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The eventual orbit of 0
goes to infinity
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The eventual orbit of 0
goes to infinity
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The eventual orbit of 0
goes to infinity
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The eventual orbit of 0
gone to infinity
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How understand the periods of the
bulbs?
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How understand the periods of the
bulbs?
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junction point
three spokes attached
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junction point
three spokes attached
Period 3 bulb
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Period 4 bulb
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Period 5 bulb
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Period 7 bulb
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Period 13 bulb
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Filled Julia Set
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Filled Julia Set
Fix a c-value. The filled Julia set is all of
the complex seeds whose orbits do NOT go to
infinity.
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2
Example z
Seed
In Julia set?
0
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2
Example z
Seed
In Julia set?
0
Yes
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2
Example z
Seed
In Julia set?
0
Yes
1
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2
Example z
Seed
In Julia set?
0
Yes
1
Yes
168
2
Example z
Seed
In Julia set?
0
Yes
1
Yes
-1
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2
Example z
Seed
In Julia set?
0
Yes
1
Yes
-1
Yes
170
2
Example z
Seed
In Julia set?
0
Yes
1
Yes
-1
Yes
i
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2
Example z
Seed
In Julia set?
0
Yes
1
Yes
-1
Yes
i
Yes
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2
Example z
Seed
In Julia set?
0
Yes
1
Yes
-1
Yes
i
Yes
2i
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2
Example z
Seed
In Julia set?
0
Yes
1
Yes
-1
Yes
i
Yes
No
2i
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2
Example z
Seed
In Julia set?
0
Yes
1
Yes
-1
Yes
i
Yes
No
2i
5
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2
Example z
Seed
In Julia set?
0
Yes
1
Yes
-1
Yes
i
Yes
No
2i
5
No way
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Filled Julia Set for z
2
i
1
-1
All seeds on and inside the unit circle.
177
Other filled Julia sets
Choose c from some component of the Mandelbrot
set, then use the same algorithm as
before colored points escape to 8 and so are not
in the filled Julia set black points form the
filled Julia set.
M-set computer
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If c is in the Mandelbrot set, then the filled
Julia set is always a connected set.
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Other filled Julia sets
But if c is not in the Mandelbrot set, then the
filled Julia set is totally disconnected.
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Amazingly, the orbit of 0 knows it all
Theorem For z2 c If the orbit of 0
goes to infinity, the Julia set is a Cantor set
(totally disconnected, fractal dust), and c
is not in the Mandelbrot set. But if the orbit
of 0 does not go to infinity, the Julia set is
connected (just one piece), and c is in the
Mandelbrot set.
M-set movie maker --- frame 200
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Animations
In and out of M
Saddle node
Period doubling
Period 4 bifurcation
arrangement of the bulbs
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How do we understand the arrangement of the
bulbs?
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How do we understand the arrangement of the
bulbs?
Assign a fraction p/q to each bulb hanging off
the main cardioid q period of the bulb.
184
?/3 bulb
shortest spoke
principal spoke
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1/3 bulb
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1/3 bulb
1/3
187
1/3 bulb
1/3
188
1/3 bulb
1/3
189
1/3 bulb
1/3
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1/3 bulb
1/3
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1/3 bulb
1/3
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1/3 bulb
1/3
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1/3 bulb
1/3
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1/3 bulb
1/3
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1/3 bulb
1/3
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??? bulb
1/3
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1/4 bulb
1/3
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1/4 bulb
1/3
1/4
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1/4 bulb
1/3
1/4
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1/4 bulb
1/3
1/4
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1/4 bulb
1/3
1/4
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1/4 bulb
1/3
1/4
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1/4 bulb
1/3
1/4
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1/4 bulb
1/3
1/4
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1/4 bulb
1/3
1/4
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1/4 bulb
1/3
1/4
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??? bulb
1/3
1/4
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2/5 bulb
1/3
1/4
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2/5 bulb
1/3
2/5
1/4
210
2/5 bulb
1/3
2/5
1/4
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2/5 bulb
1/3
2/5
1/4
212
2/5 bulb
1/3
2/5
1/4
213
2/5 bulb
1/3
2/5
1/4
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??? bulb
1/3
2/5
1/4
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3/7 bulb
1/3
2/5
1/4
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3/7 bulb
1/3
2/5
1/4
3/7
217
3/7 bulb
1/3
2/5
1/4
3/7
218
3/7 bulb
1/3
2/5
1/4
3/7
219
3/7 bulb
1/3
2/5
1/4
3/7
220
3/7 bulb
1/3
2/5
1/4
3/7
221
3/7 bulb
1/3
2/5
1/4
3/7
222
3/7 bulb
1/3
2/5
1/4
3/7
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??? bulb
1/3
2/5
1/4
3/7
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1/2 bulb
1/3
2/5
1/4
3/7
1/2
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1/2 bulb
1/3
2/5
1/4
3/7
1/2
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1/2 bulb
1/3
2/5
1/4
3/7
1/2
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1/2 bulb
1/3
2/5
1/4
3/7
1/2
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??? bulb
1/3
2/5
1/4
3/7
1/2
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2/3 bulb
1/3
2/5
1/4
3/7
1/2
2/3
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2/3 bulb
1/3
2/5
1/4
3/7
1/2
2/3
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2/3 bulb
1/3
2/5
1/4
3/7
1/2
2/3
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2/3 bulb
1/3
2/5
1/4
3/7
1/2
2/3
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2/3 bulb
1/3
2/5
1/4
3/7
1/2
2/3
234
2/3 bulb
1/3
2/5
1/4
3/7
1/2
2/3
235
How to count
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How to count
1/4
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How to count
1/3
1/4
238
How to count
1/3
2/5
1/4
239
How to count
1/3
2/5
1/4
3/7
240
How to count
1/3
2/5
1/4
3/7
1/2
241
How to count
1/3
2/5
1/4
3/7
1/2
2/3
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How to count
1/3
2/5
1/4
3/7
1/2
2/3
The bulbs are arranged in the exact order of the
rational numbers.
243
How to count
1/3
32,123/96,787
2/5
1/4
3/7
1/101
1/2
2/3
The bulbs are arranged in the exact order of the
rational numbers.
244
Animations
Mandelbulbs
Spiralling fingers
245
How to add
246
How to add
1/2
247
How to add
1/3
1/2
248
How to add
1/3
2/5
1/2
249
How to add
1/3
2/5
3/7
1/2
250
1/2 1/3 2/5


251
1/2 2/5 3/7


252
Undergrads who add fractions this way will be
subject to a minimum of five years in jail where
they must do at least 500 integrals per
day. Only PhDs in mathematics are allowed to
add fractions this way.
253
Heres an interesting sequence
2
2
1/2
0/1
254
Watch the denominators
1/3
2
2
1/2
0/1
255
Watch the denominators
1/3
2/5
2
2
1/2
0/1
256
Watch the denominators
1/3
3/8
2/5
2
2
1/2
0/1
257
Watch the denominators
1/3
3/8
5/13
2/5
2
2
1/2
0/1
258
Whats next?
1/3
3/8
5/13
2/5
2
2
1/2
0/1
259
Whats next?
8/21
1/3
3/8
5/13
2/5
2
2
1/2
0/1
260
The Fibonacci sequence
13/34
8/21
1/3
3/8
5/13
2/5
2
2
1/2
0/1
261
The Farey Tree
262
The Farey Tree
How get the fraction in between with the smallest
denominator?
263
The Farey Tree
How get the fraction in between with the smallest
denominator?
Farey addition
264
The Farey Tree
265
The Farey Tree
266
The Farey Tree
....
essentially the golden number
267
Another sequence
(denominators only)
2
1
268
Another sequence
(denominators only)
3
2
1
269
Another sequence
(denominators only)
3
4
2
1
270
Another sequence
(denominators only)
3
4
5
2
1
271
Another sequence
(denominators only)
3
4
5
2
6
1
272
Another sequence
(denominators only)
3
4
5
2
6
7
1
273
sequence
Devaney
3
4
5
2
6
7
1
274
The Dynamical Systems and Technology
Project at Boston University
website math.bu.edu/DYSYS
Mandelbrot set explorer Applets for
investigating M-set Applets for other complex
functions Chaos games, orbit diagrams, etc.
Have fun!
275
Other topics
Farey.qt
Farey tree
D-sequence
Far from rationals
Continued fraction expansion
Website
276
Continued fraction expansion
Lets rewrite the sequence 1/2, 1/3,
2/5, 3/8, 5/13, 8/21, 13/34, .....
as a continued fraction
277
Continued fraction expansion
1 2
1 2

the sequence 1/2, 1/3, 2/5, 3/8, 5/13, 8/21,
13/34,.....
278
Continued fraction expansion
1 3
1 2


1 1
the sequence 1/2, 1/3, 2/5, 3/8, 5/13, 8/21,
13/34,.....
279
Continued fraction expansion
2 5
1 2


1 1

1 1
the sequence 1/2, 1/3, 2/5, 3/8, 5/13, 8/21,
13/34,.....
280
Continued fraction expansion
3 8
1 2


1 1

1 1

1 1
the sequence 1/2, 1/3, 2/5, 3/8, 5/13, 8/21,
13/34,.....
281
Continued fraction expansion
5

1 2


1 1
13

1 1

1 1

1 1
the sequence 1/2, 1/3, 2/5, 3/8, 5/13, 8/21,
13/34,.....
282
Continued fraction expansion
8

1 2


1 1
21

1 1

1 1

1 1

1 1
the sequence 1/2, 1/3, 2/5, 3/8, 5/13, 8/21,
13/34,.....
283
Continued fraction expansion

13
1 2


1 1
34

1 1

1 1

1 1

1 1

1 1
the sequence 1/2, 1/3, 2/5, 3/8, 5/13, 8/21,
13/34,.....
284
Continued fraction expansion

13
1 2


1 1
34

1 1

1 1

1 1

1 1

1 1
essentially the 1/golden number
the sequence 1/2, 1/3, 2/5, 3/8, 5/13, 8/21,
13/34,.....
285
We understand what happens for
1 a


1 b

1 c

1 d

1 e

1 f

1 g
etc.
where all entries in the sequence a, b, c,
d,.... are bounded above. But if that
sequence grows too quickly, were in trouble!!!
286
The real way to prove all this
Need to measure the size of bulbs the
length of spokes the size of the ears.
287
There is an external Riemann map
C - D C - M taking the exterior of the
unit disk to the exterior of the Mandelbrot set.
288
takes straight rays in C - D to the
external rays in C - M
external ray of angle 1/3
1/3
0
1/2
2/3
289
Suppose p/q is periodic of period k
under doubling mod 1
period 2
period 3
period 4
290
Suppose p/q is periodic of period k
under doubling mod 1
period 2
period 3
period 4
Then the external ray of angle p/q lands at the
root point of a period k bulb in the
Mandelbrot set.
291
0 is fixed under angle doubling, so lands at the
cusp of the main cardioid.
1/3
0
2/3
292
1/3 and 2/3 have period 2 under doubling, so
and land at the root of the period 2
bulb.
1/3
2
0
2/3
293
And if lies between 1/3 and 2/3, then
lies between and .
1/3
2
0
2/3
294
So the size of the period 2 bulb is, by
definition, the length of the set of rays
between the root point rays, i.e., 2/3-1/31/3.
1/3
2
0
2/3
295
1/15 and 2/15 have period 4, and are smaller than
1/7....
1/3
2/7
1/7
3
3/7
2/15
1/15
2
0
3
4/7
6/7
2/3
5/7
296
1/15 and 2/15 have period 4, and are smaller than
1/7....
1/3
2/7
1/7
3
3/7
2/15
1/15
2
0
3
4/7
6/7
2/3
5/7
297
3/15 and 4/15 have period 4, and are between 1/7
and 2/7....
1/3
2/7
1/7
3
3/7
2/15
1/15
2
0
3
4/7
6/7
2/3
5/7
298
3/15 and 4/15 have period 4, and are between 1/7
and 2/7....
1/3
2/7
1/7
3
3/7
2/15
1/15
2
0
3
4/7
6/7
2/3
5/7
299
3/15 and 4/15 have period 4, and are between 1/7
and 2/7....
1/7
2/7
300
3/15 and 4/15 have period 4, and are between 1/7
and 2/7....
3/15
4/15
1/7
2/7
301
So what do we know about M?
All rational external rays land at a single
point in M.
302
So what do we know about M?
All rational external rays land at a single
point in M.
Rays that are periodic under doubling land at
root points of a bulb.
Non-periodic rational rays land at Misiurewicz
points (how we measure length of antennas).
303
So what do we know about M?
Highly irrational rays also land at unique
points, and we understand what goes on
here. Highly irrational" far from
rationals, i.e.,
304
So what do we NOT know about M?
But we don't know if irrationals that are
close to rationals land. So we won't
understand quadratic functions until we figure
this out.
305
MLC Conjecture
The boundary of the M-set is locally connected
--- if so, all rays land and we are in heaven!.
But if not......
306
The Dynamical Systems and Technology
Project at Boston University
website math.bu.edu/DYSYS
Have fun!
307
A number is far from the rationals if
308
A number is far from the rationals if
309
A number is far from the rationals if
This happens if the continued fraction
expansion of has only bounded terms.