Preliminary Discussion
only if time admits
We start in z = 1.02 + 0.1i and, in each step, use z 7→ z 2 to construct
the next point.
We start in z = 1.02 + 0.1i and, in each step, use z 7→ z 2 to construct
the next point. That is, square the modulus and double the argument!
We start in z = 1.02 + 0.1i and, in each step, use z 7→ z 2 to construct
the next point. That is, square the modulus and double the argument!
Im
Re
We start in z = 1.02 + 0.1i and, in each step, use z 7→ z 2 to construct
the next point. That is, square the modulus and double the argument!
Im
Re
We start in z = 1.02 + 0.1i and, in each step, use z 7→ z 2 to construct
the next point. That is, square the modulus and double the argument!
Im
Re
We start in z = 1.02 + 0.1i and, in each step, use z 7→ z 2 to construct
the next point. That is, square the modulus and double the argument!
Im
Re
We start in z = 1.02 + 0.1i and, in each step, use z 7→ z 2 to construct
the next point. That is, square the modulus and double the argument!
Im
Re
We start in z = 1.02 + 0.1i and, in each step, use z 7→ z 2 to construct
the next point. That is, square the modulus and double the argument!
Im
Re
We start in z = 1.02 + 0.1i and, in each step, use z 7→ z 2 to construct
the next point. That is, square the modulus and double the argument!
Im
Re
We start in z = 1.02 + 0.1i and, in each step, use z 7→ z 2 to construct
the next point. That is, square the modulus and double the argument!
Im
Re
Main Part
c by Rama, cc-by-sa-2.0-fr
Photography Benoit Mandelbrot
1924 – 2010
c by Rama, cc-by-sa-2.0-fr
Photography We start in z = 0 and, in each step, use
z 7→ z 2 + c to construct the next point.
We start in z = 0 and, in each step, use
z 7→ z 2 + c to construct the next point.
c=1:
We start in z = 0 and, in each step, use
z 7→ z 2 + c to construct the next point.
c=1:
0, 1, 2, 5, 26, 677, . . .
We start in z = 0 and, in each step, use
z 7→ z 2 + c to construct the next point.
c=1:
0, 1, 2, 5, 26, 677, . . .
goes to infinity
We start in z = 0 and, in each step, use
z 7→ z 2 + c to construct the next point.
c=1:
c=0:
0, 1, 2, 5, 26, 677, . . .
goes to infinity
We start in z = 0 and, in each step, use
z 7→ z 2 + c to construct the next point.
c=1:
0, 1, 2, 5, 26, 677, . . .
c=0:
0, 0, 0, 0, 0, 0, . . .
goes to infinity
We start in z = 0 and, in each step, use
z 7→ z 2 + c to construct the next point.
c=1:
0, 1, 2, 5, 26, 677, . . .
goes to infinity
c=0:
0, 0, 0, 0, 0, 0, . . .
remains bounded
We start in z = 0 and, in each step, use
z 7→ z 2 + c to construct the next point.
c=1:
0, 1, 2, 5, 26, 677, . . .
goes to infinity
c=0:
0, 0, 0, 0, 0, 0, . . .
remains bounded
c = –1 :
We start in z = 0 and, in each step, use
z 7→ z 2 + c to construct the next point.
c=1:
0, 1, 2, 5, 26, 677, . . .
goes to infinity
c=0:
0, 0, 0, 0, 0, 0, . . .
remains bounded
c = –1 :
0, −1, 0, −1, 0, −1, . . .
We start in z = 0 and, in each step, use
z 7→ z 2 + c to construct the next point.
c=1:
0, 1, 2, 5, 26, 677, . . .
goes to infinity
c=0:
0, 0, 0, 0, 0, 0, . . .
remains bounded
c = –1 :
0, −1, 0, −1, 0, −1, . . .
remains bounded
We start in z = 0 and, in each step, use
z 7→ z 2 + c to construct the next point.
c=1:
0, 1, 2, 5, 26, 677, . . .
goes to infinity
c=0:
0, 0, 0, 0, 0, 0, . . .
remains bounded
c = –1 :
0, −1, 0, −1, 0, −1, . . .
remains bounded
c=i:
..................
goes to infinity
We start in z = 0 and, in each step, use
z 7→ z 2 + c to construct the next point.
c=1:
0, 1, 2, 5, 26, 677, . . .
goes to infinity
c=0:
0, 0, 0, 0, 0, 0, . . .
remains bounded
c = –1 :
0, −1, 0, −1, 0, −1, . . .
remains bounded
c=i:
..................
goes to infinity
c = -i :
..................
goes to infinity
We start in z = 0 and, in each step, use
z 7→ z 2 + c to construct the next point.
We start in z = 0 and, in each step, use
z 7→ z 2 + c to construct the next point.
Im
Depending on c, we
= go to infinity.
= remain bounded.
Re
We start in z = 0 and, in each step, use
z 7→ z 2 + c to construct the next point.
Im
Depending on c, we
= go to infinity.
= remain bounded.
Re
We start in z = 0 and, in each step, use
z 7→ z 2 + c to construct the next point.
Im
Depending on c, we
= go to infinity.
= remain bounded.
Re
We start in z = 0 and, in each step, use
z 7→ z 2 + c to construct the next point.
Im
Depending on c, we
= go to infinity.
= remain bounded.
Re
We start in z = 0 and, in each step, use
z 7→ z 2 + c to construct the next point.
Im
Depending on c, we
= go to infinity.
= remain bounded.
Re
We start in z = 0 and, in each step, use
z 7→ z 2 + c to construct the next point.
Im
Depending on c, we
= go to infinity.
= remain bounded.
Re
We start in z = 0 and, in each step, use
z 7→ z 2 + c to construct the next point.
Im
Depending on c, we
= go to infinity.
= remain bounded.
Re
We start in z = 0 and, in each step, use
z 7→ z 2 + c to construct the next point.
Im
Depending on c, we
= go to infinity.
= remain bounded.
Re
We start in z = 0 and, in each step, use
z 7→ z 2 + c to construct the next point.
Im
Depending on c, we
= go to infinity.
= remain bounded.
Re
We start in z = 0 and, in each step, use
z 7→ z 2 + c to construct the next point.
Im
Depending on c, we
= go to infinity.
= remain bounded.
Re
We start in z = 0 and, in each step, use
z 7→ z 2 + c to construct the next point.
Im
Depending on c, we
= go to infinity.
= remain bounded.
Re
We start in z = 0 and, in each step, use
z 7→ z 2 + c to construct the next point.
Im
Depending on c, we
= go to infinity.
= remain bounded.
Re
We start in z = 0 and, in each step, use
z 7→ z 2 + c to construct the next point.
Im
Depending on c, we
= go to infinity.
= remain bounded.
Re
We start in z = 0 and, in each step, use
z 7→ z 2 + c to construct the next point.
Im
Depending on c, we
= go to infinity.
= remain bounded.
Re
We start in z = 0 and, in each step, use
z 7→ z 2 + c to construct the next point.
Im
Depending on c, we
= go to infinity.
= remain bounded.
Re
We start in z = 0 and, in each step, use
z 7→ z 2 + c to construct the next point.
Im
Depending on c, we
= go to infinity.
= remain bounded.
Re
We start in z = 0 and, in each step, use
z 7→ z 2 + c to construct the next point.
Im
Depending on c, we
= go to infinity.
= remain bounded.
Re
We start in z = 0 and, in each step, use
z 7→ z 2 + c to construct the next point.
Im
Depending on c, we
= go to infinity.
= remain bounded.
Re
We start in z = 0 and, in each step, use
z 7→ z 2 + c to construct the next point.
Im
Depending on c, we
= go to infinity.
= remain bounded.
Re
We start in z = 0 and, in each step, use
z 7→ z 2 + c to construct the next point.
Im
Depending on c, we
= go to infinity.
= remain bounded.
Re
We start in z = 0 and, in each step, use
z 7→ z 2 + c to construct the next point.
Im
Depending on c, we
= go to infinity.
= remain bounded.
Re
We start in z = 0 and, in each step, use
z 7→ z 2 + c to construct the next point.
Im
Depending on c, we
= go to infinity.
= remain bounded.
Re
We start in z = 0 and, in each step, use
z 7→ z 2 + c to construct the next point.
Im
Depending on c, we
= go to infinity.
= remain bounded.
Re
We start in z = 0 and, in each step, use
z 7→ z 2 + c to construct the next point.
Im
Depending on c, we
= go to infinity.
= remain bounded.
Re
We start in z = 0 and, in each step, use
z 7→ z 2 + c to construct the next point.
Im
Depending on c, we
= go to infinity.
= remain bounded.
Re
We start in z = 0 and, in each step, use
z 7→ z 2 + c to construct the next point.
Im
Depending on c, we
= go to infinity.
= remain bounded.
Re
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