Mandelbrot and Julia Sets

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1 Mandelbrot and Julia Sets Peter Tiňo pxt School of Computer Science University of Birmingham, UK Mandelbrot and Julia Sets p./25

2 Complex numbers Imaginary number: i = Note: i 2 = Complex numbers: z = a + i b Each complex number z = a + i b can be represented as a 2-dimensional point (a, b) R 2. z = a + i b, z 2 = a 2 + i b 2 Addition: z + z 2 = (a + a 2 ) + i (b + b 2 ) Multiplication: z z 2 = (a a 2 b b 2 ) + i (a b 2 + a 2 b ) Modulus ( size ): z = z z = a 2 + b 2 Mandelbrot and Julia Sets p.2/25

3 Complex numbers Im b z z2 z z 3 0 z a Re z 4 z = a + i b z j, j =,2,3,4, are solutions of z 4 =. Mandelbrot and Julia Sets p.3/25

4 Mandelbrot set Pick a complex number c. Start at the origin: z 0 = 0 + i 0 and iterate: z n+ = z 2 n + c. Key question: Will the trajectory {z n } n=0 bounded set? diverge, or will it stay forever inside a Different answer for different c! The Mandelbrot set contains those complex numbers c for which the trajectory {z n } n=0 remains bounded. Actually, it can be shown that if at some stage z n > 2, then the trajectory will diverge. Mandelbrot and Julia Sets p.4/25

5 Mandelbrot set - algorithm For each complex number c in a subset of complex plane. Set z 0 = 0 + i 0 2. For n = 0 : N max Compute z n+ = z 2 n + c If z n+ > 2, then break out of loop 3. If n < N max, then color point c white (trajectory diverges) 4. If n = N max, then color point c black (trajectory stays bounded) Mandelbrot and Julia Sets p.5/25

6 Mandelbrot set - picture Mandelbrot and Julia Sets p.6/25

7 Explorations of Mandelbrot Set When zooming into the Mandelbrot set we can discover peculiar shapes and amazing forms of self-similarity! It is often instructive to color code each c not in the Mandelbrot set (e.g. those that give rise to a divergent trajectory) with the "escape rate" - how many iterations it takes to escape from the centered disk of radius 2, starting at the origin. Explore your own color schemes for the escape rates and zoom into different regions of the Mandelbrot set. Mandelbrot and Julia Sets p.7/25

8 Mandelbrot set - linear color scheme Mandelbrot set Mandelbrot and Julia Sets p.8/25

Mandelbrot set - square color scheme Mandelbrot set square escape time x 0 5 0 0.8 9 0.

9 Mandelbrot set - square color scheme Mandelbrot set square escape time x Mandelbrot and Julia Sets p.9/25

Mandelbrot set - log color scheme Mandelbrot set log escape time 0.8 6 0.

10 Mandelbrot set - log color scheme Mandelbrot set log escape time Mandelbrot and Julia Sets p.0/25

Mandelbrot set - linear mirror color scheme Mandelbrot set (colormap) 000 0.8 900 0.6 800 0.

11 Mandelbrot set - linear mirror color scheme Mandelbrot set (colormap) Mandelbrot and Julia Sets p./25

12 Mandelbrot set - square mirror color scheme Mandelbrot set square escape time (colormap) x Mandelbrot and Julia Sets p.2/25

13 Mandelbrot set - log mirror color scheme Mandelbrot set log escape time (colormap) Mandelbrot and Julia Sets p.3/25

14 Example of zooming in - log mirror color scheme Mandelbrot set level Mandelbrot and Julia Sets p.4/25

Zooming in - level 0.2 Mandelbrot set level 0.2 0.9 0.8 0.7 0.

15 Zooming in - level 0.2 Mandelbrot set level Mandelbrot and Julia Sets p.5/25

Zooming in - level 2 0.75 Mandelbrot set level 2 0.7 0.705 0.7 0.695.

16 Zooming in - level Mandelbrot set level Mandelbrot and Julia Sets p.6/25

Zooming in - level 3 0.706 Mandelbrot set level 3 0.706 0.706 0.706 0.706 0.705 0.705 0.705 0.705 0.705 0.705.249.

17 Zooming in - level Mandelbrot set level Mandelbrot and Julia Sets p.7/25

Zooming in - another example Mandelbrot set level 0 0.8 0.6 0.

18 Zooming in - another example Mandelbrot set level Mandelbrot and Julia Sets p.8/25

Nice! 0.236 Mandelbrot set level 0.236 0.236 0.236 0.236 0.236 0.236 0.7488 0.

19 Nice! Mandelbrot set level Mandelbrot and Julia Sets p.9/25

20 Julia sets There is only one Mandelbrot set. We always start iterations in z 0 = 0 + i 0, and ask for each c, whether z n+ = z 2 n + c diverges, or not. Now, we will have many different Julia sets, one for each c. For a given complex number c, investigate for which starting positions z 0 will z n+ = z 2 n + c diverge and for which z 0 it will stay confined to a bounded region of the complex plane. Julia set associated with a complex number c contains those initial positions z 0 for which the trajectory of z n+ = z 2 n + c remains bounded. Mandelbrot and Julia Sets p.20/25

21 Julia Set associated with c - algorithm For each complex number z 0 in a subset of complex plane. For n = 0 : N max Compute z n+ = z 2 n + c If z n+ > 2, then break out of loop 2. If n < N max, then color point z 0 white (trajectory diverges) 3. If n = N max, then color point z 0 black (trajectory stays bounded) As with Mandelbrot set, one can use different color schemes for initial conditions not in the Julia set. Mandelbrot and Julia Sets p.2/25

22 Julia sets - pictures Mandelbrot and Julia Sets p.22/25

A Julia set - log mirror color scheme c = 0.277334 i 0.

23 A Julia set - log mirror color scheme c = i Julia set level Mandelbrot and Julia Sets p.23/25

Zooming in - level Julia set level 0.7 0.6 0.5 0.4 0.3 0.2 0.

24 Zooming in - level Julia set level Mandelbrot and Julia Sets p.24/25

Zooming in - level 2 0.63 Julia set level 2 0.62 0.6 0.6 0.59 0.58 0.57 0.56 0.55 0.54 0.

25 Zooming in - level Julia set level Mandelbrot and Julia Sets p.25/25

Manipulating Images with Fractal Julia Sets

Manipulating Images with Fractal Julia Sets Stanley Spencer The Sycamores Queens Road Hodthorpe Worksop Nottinghamshire, England, S80 4UT pythagoras@bcs.org.uk Abstract The pictures associated with the