Generating Fractals With Complex Numbers
Learning Outcomes
- Identify the difference between an imaginary number and a complex number
- Identify the real and imaginary parts of a complex number
- Plot a complex number on the complex plane
- Perform arithmetic operations on complex numbers
- Graph physical representations of arithmetic operations on complex numbers as scaling or rotation
- Generate several terms of a recursive relation
- Determine whether a complex number is part of the set of numbers that make up the Mandelbrot set
Complex Recursive Sequences
Some fractals are generated with complex numbers. The Mandlebrot set, which we introduced briefly at the beginning of this module, is generated using complex numbers with a recursive sequence. Before we can see how to generate the Mandelbrot set, we need to understand what a recursive sequence is.
Recursive Sequence
A
recursive relationship is a formula which relates the next value,
zn+1, in a sequence to the previous value,
zn. In addition to the formula, we need an initial value,
z0.
The sequence of values produced is the recursive sequence.
Example
Given the recursive relationship
zn+1=zn+2,z0=4, generate several terms of the recursive sequence.
We are given the starting value,
z0=4. The recursive formula holds for any value of
n, so if
n=0, then
zn+1=zn+2 would tell us
z0+1=z0+2, or more simply,
z1=z0+2.
Notice this defines
z1 in terms of the known
z0, so we can compute the value:
z1=z0+2=4+2=6.
Now letting
n=1, the formula tells us
z1+1=z1+2, or
z2=z1+2. Again, the formula gives the next value in the sequence in terms of the previous value.
z2=z1+2=6+2=8
Continuing,
z3=z2+2=8+2=10z4=z3+2=10+2=12
The previous example generated a basic linear sequence of real numbers. The same process can be used with complex numbers.
Example
Given the recursive relationship
zn+1=zn⋅i+(1−i),z0=4, generate several terms of the recursive sequence.
We are given
z0=4. Using the recursive formula:
z1=z0⋅i+(1−i)=4⋅i+(1−i)=1+3iz2=z1⋅i+(1−i)=(1+3i)⋅i+(1−i)=i+3i2+(1−i)=i−3+(1−i)=−2z3=z2⋅i+(1−i)=(−2)⋅i+(1−i)=−2i+(1−i)=1−3iz4=z3⋅i+(1−i)=(1−3i)⋅i+(1−i)=i−3i2+(1−i)=i+3+(1−i)=4z5=z4⋅i+(1−i)=4⋅i+(1−i)=1+3i
Notice this sequence is exhibiting an interesting pattern—it began to repeat itself.
In the following video we show more worked examples of how to generate the terms of a recursive, complex sequence.
https://youtu.be/lOyusyTsLTs
Mandelbrot Set
The Mandelbrot Set is a set of numbers defined based on recursive sequences.
Mandelbrot Set
For any complex number
c, define the sequence
zn+1=zn2+c,z0=0
If this sequence always stays close to the origin (within 2 units), then the number
c is part of the
Mandelbrot Set. If the sequence gets far from the origin, then the number
c is not part of the set.
Example
Determine if
c=1+i is part of the Mandelbrot set.
We start with
z0=0. We continue, omitting some detail of the calculations
z1=z02+1+i=0+1+i=1+iz2=z12+1+i=(1+i)2+1+i=1+3iz3=z22+1+i=(1+3i)2+1+i=−7+7iz4=z32+1+i=(−7+7i)2+1+i=1−97i
We can already see that these values are getting quite large. It does not appear that
c=1+i is part of the Mandelbrot set.
Example
Determine if
c=0.5i is part of the Mandelbrot set.
We start with
z0=0. We continue, omitting some detail of the calculations
z1=z02+0.5i=0+0.5i=0.5iz2=z12+0.5i=(0.5i)2+0.5i=−0.25+0.5iz3=z22+0.5i=(−0.25+0.5i)2+0.5i=−0.1875+0.25iz4=z32+0.5i=(−0.1875+0.25i)2+0.5i=−0.02734+0.40625i
While not definitive with this few iterations, it does appear that this value is remaining small, suggesting that 0.5
i is part of the Mandelbrot set.
Try It
Determine if
c=0.4+0.3i is part of the Mandelbrot set.
Answer:
z1=z02+0.4+0.3i=0+0.4+0.3i=0.4+0.3iz2=z12+0.4+0.3i=(0.4+0.3i)2+0.4+0.3iz3=z22+0.5i=(−0.25+0.5i)2+0.5i=−0.1875+0.25iz4=z32+0.5i=(−0.1875+0.25i)2+0.5i=−0.02734+0.40625i
If all complex numbers are tested, and we plot each number that is in the Mandelbrot set on the complex plane, we obtain the shape to the right.[footnote]http://en.wikipedia.org/wiki/File:Mandelset_hires.png[/footnote]
The boundary of this shape exhibits quasi-self-similarity, in that portions look very similar to the whole.
Watch the following video for more examples of how to determine whether a complex number is a member of the Mandelbrot set.
https://youtu.be/ORqk5jAFpWg
In addition to coloring the Mandelbrot set itself black, it is common to the color the points in the complex plane surrounding the set. To create a meaningful coloring, often people count the number of iterations of the recursive sequence that are required for a point to get further than 2 units away from the origin. For example, using c=1+i above, the sequence was distance 2 from the origin after only two recursions.
For some other numbers, it may take tens or hundreds of iterations for the sequence to get far from the origin. Numbers that get big fast are colored one shade, while colors that are slow to grow are colored another shade. For example, in the image below[footnote]This series was generated using Scott’s Mandelbrot Set Explorer[/footnote], light blue is used for numbers that get large quickly, while darker shades are used for numbers that grow more slowly. Greens, reds, and purples can be seen when we zoom in—those are used for numbers that grow very slowly.
The Mandelbrot set, for having such a simple definition, exhibits immense complexity. Zooming in on other portions of the set yields fascinating swirling shapes.
Additional Resources
A much more extensive coverage of fractals can be found on the Fractal Geometry site. This site includes links to several Java software programs for exploring fractals.
If you are impressed with the Mandelbrot set, check out this TED talk from 2010 given by Benoit Mandelbrot on fractals and the art of roughness.Licenses & Attributions
CC licensed content, Original
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