15.5: Complex Recursive Sequences
( \newcommand{\kernel}{\mathrm{null}\,}\)
We will now explore recursively defined sequences of complex numbers.
A recursive relationship is a formula which relates the next value,
The sequence of values produced is the recursive sequence.
Given the recursive relationship
recursive sequence.
Solution
We are given the starting value,
if
Notice this defines
Now letting
(z_{2}=z_{1}+2=6+2=8\)
Continuing,
The previous example generated a basic linear sequence of real numbers. The same process can be used with complex numbers.
Given the recursive relationship
recursive sequence.
Solution
We are given
Notice this sequence is exhibiting an interesting pattern – it began to repeat itself.
Mandelbrot Set
The Mandelbrot Set is a set of numbers defined based on recursive sequences
For any complex number
If this sequence always stays close to the origin (within 2 units), then the number
Determine if
Solution
We start with
We can already see that these values are getting quite large. It does not appear that
Determine if
Solution
We start with
While not definitive with this few iterations, it does appear that this value is remaining small, suggesting that
Determine if
- Answer
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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[1].
The boundary of this shape exhibits quasi-self-similarity, in that portions look very similar to the whole.
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
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[2], 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.
[1] en.Wikipedia.org/wiki/File:Mandelset_hires.png
[2] This series was generated using Scott’s Mandelbrot Set Explorer