On A Generally Fractal Family

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Recently, my fellow students and I have been caught up in the craze that is sweeping through the users of Professor B------'s clockwork calculating engine[1]; namely the charting of sets of two dimensional points that have fractal planar boundaries, being those that in some sense have a fractional dimension[2]. Of particular interest have been the results of repeated applications of quadratic functions to complex numbers; specifically in measuring how quickly, if at all, they escape a region surrounding the starting point, by which charts may be constructed that many of the collegiate consider so delightful as to constitute art painted by mathematics itself!

The Mandelbrot Set

The most favoured is the Mandelbrot set which is founded upon the recursion
\[ \begin{align*} z_0 &= 0\\ z_k &= z_{k-1}^2 + c \end{align*} \]
for a complex value \(c\), which is a member of the set if the magnitude of
\[ z_k = x_k + i \times y_k \]
defined by
\[ \left|z_k\right| = \sqrt{x_k^2 + y_k^2} \]
is finite for all \(k\).

Whilst we can't very well wait for the magnitude of \(z_k\) to grow all the way to infinity, we can exploit the fact that if it ever exceeds two then it must eventually do so.

To figure why this should be inevitable we must first note that the magnitude of the sum of two complex numbers is bounded by
\[ \bigg||a|-|b|\bigg| \leqslant |a+b| \leqslant |a|+|b| \]
with the lower being exact if the one lies in the opposite direction to the other and the upper if they lie in the same direction. Secondly, the magnitude of the square of a complex number is equal to the square of its magnitude, as we can see with
\[ \begin{align*} (x + i \times y)^2 &= \left(x^2-y^2\right) + i \times 2 \times x \times y\\ \left|(x + i \times y)^2\right|^2 &= \left(x^2-y^2\right)^2 + \left(2 \times x \times y\right)^2 = x^4 - 2 \times x^2 \times y^2 + y^4 + 4 \times x^2 \times y^2\\ &= x^4 + 2 \times x^2 \times y^2 + y^4 = \left(x^2+y^2\right)^2 = \left|x + i \times y\right|^4\\ \left|(x + i \times y)^2\right| &= \left|x + i \times y\right|^2 \end{align*} \]
Now, if the magnitude of \(z_k\) is greater than two then we may define it as
\[ \left|z_k\right| = 2 + \epsilon \]
for some strictly positive \(\epsilon\) and consequently, if the magnitude of \(c\) is no greater than two, we have
\[ \begin{align*} \left|z_{k+1}\right| = \left|z_k^2 + c\right| &\geqslant \bigg|\left|z_k^2\right| - \left|c\right|\bigg| = \bigg|\left|z_k\right|^2 - \left|c\right|\bigg| = \bigg|(2+\epsilon)^2 - \left|c\right|\bigg| \geqslant \bigg|4 + 4 \times \epsilon + \epsilon^2 - 2\bigg| = \bigg|2 + 4 \times \epsilon + \epsilon^2\bigg|\\ &> 2 + 4 \times \epsilon \end{align*} \]
The magnitudes of \(z_k\) must therefore grow exponentially as \(k\) increases and we may thusly conclude that, if in any event they exceed two, \(c\) cannot be a member of the Mandelbrot set.

If, upon the other hand, the magnitude of \(c\) is greater than two then we may define it as
\[ \left|c\right| = 2 + \epsilon \]
so that
\[ \left|z_1\right| = 2 + \epsilon \]
and therefore
\[ \begin{align*} \left|z_2 \right| = \left|z_1^2 + c\right| &\geqslant \bigg|\left|z_1^2\right| - \left|c\right|\bigg| = \bigg|(2+\epsilon)^2 - (2+\epsilon)\bigg| = \bigg|\left(4 + 4 \times \epsilon + \epsilon^2\right) - (2+\epsilon)\bigg| = \bigg|2 + 3 \times \epsilon + \epsilon^2\bigg|\\ &> 2 + 3 \times \epsilon \end{align*} \]
Assuming that
\[ \left|z_k\right| = 2 + n \times \epsilon \]
for \(k\) greater than zero and some \(n\) greater than one yields
\[ \begin{align*} \left|z_{k+1} \right| = \left|z_k^2 + c\right| &\geqslant \bigg|\left|z_k^2\right| - \left|c\right|\bigg| = \bigg|(2 + n \times \epsilon)^2 - (2+\epsilon)\bigg|\\ &= \bigg|\left(4 + 4 \times n \times \epsilon + n^2 \times \epsilon^2\right) - (2+\epsilon)\bigg| = \bigg|2 + (4 \times n - 1) \times \epsilon + n^2 \times \epsilon^2\bigg|\\ &> 2 + (4 \times n - 1) \times \epsilon > 2 + 3 \times n \times \epsilon \end{align*} \]
which again yields an exponential flight and eliminates \(c\) from the set.

Unfortunately we have no general means of reckoning whether or not a given value of \(c\) will ultimately yield a value of \(z_k\) with a magnitude greater than two and so this is a somewhat hollow victory. We might assume that if the magnitude hasn't exceeded two after a sufficiently large number of steps then it will never do so, but regrettably this is by no means a certainty.

Having little choice in the matter, this is precisely the manner by which my fellow students and I have charted the Mandelbrot set, as demonstrated by deck 1 which plots the elements that we haven't ruled out of the set in black and the number of steps that were taken to rule out those that we had from deep red at the quickest to bright yellow at the slowest.

Deck 1: The Mandelbrot Set

Julia Sets

If instead we fix the value of \(c\), the values of \(x_0\) for which the magnitude of each and every \(x_k\) remain finite are known as a Julia set, an example of which is provided by deck 2.

Deck 2: A Julia Set

Here the white dots represent values that are close to those within the set, taking the greatest number of steps to diverge. The elements of Julia sets are not necessarily sparsely situated, as demonstrated by deck 3 which plots its elements in black.

Deck 3: A Connected Julia Set

The simplest of such Julia sets is that defined by \(c\) equal to zero. It is trivially the case that the only complex numbers with magnitudes that do not tend to infinity upon repeated squaring are those with a magnitude no greater than one, as demonstrated by deck 4.

Deck 4: The Trivial Julia Set

Figure 1 shows the Julia sets for values of \(c\) with real and complex parts ranging from minus one to plus one in steps of one half, with the trivial set at its centre.

Figure 1: A Catalogue Of Julia Sets

This clearly reveals symmetries both within each set and between them. Firstly, that the square of minus \(z\) is trivially equal to the square of \(z\) results in the half turn rotational symmetry possessed by each set. Secondly, the fact that
\[ \begin{align*} (x + y \times i)^2 + a + b \times i &= \left(x^2-y^2+a\right) + (2 \times x \times y + b) \times i\\ (x - y \times i)^2 + a - b \times i &= \left(x^2-y^2+a\right) - (2 \times x \times y + b) \times i \end{align*} \]
accounts for the reflective symmetry through the real axis between sets.

Note that the sparse Julia sets are precisely those defined by a value of \(c\) that lies outside of the Mandelbrot set whereas the dense sets are defined by those that lie within it. Those whose defining value lies close to the boundary typically have the more complex structure, as illustrated by deck 5 for a value outside of it

Deck 5: Neighbouring The Outer Boundary

and by deck 6 for a value within.

Deck 6: Neighbouring The Inner Boundary

Note that since their defining constants are close the region of slowest escape in the first has a discernibly similar shape to the elements of the second.

My fellow students and I are particularly interested in generalisations of these sets, but for now our studies must take priority.
\(\Box\)

References

[1] On An Age Of Wonders, www.thusspakeak.com, 2014.

[2] Finally On Natural Analogarithms, www.thusspakeak.com, 2018.

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