Hello!
As I have not posted in a long while, I'll post some eye candy. Context:
For those that don't know, the Mandelbrot set

is a set of complex numbers defined in the following manner. Consider the complex function
. Now for a particular
, consider the sequence

If

does not diverge, then
.
Here is an image of
. Every white pixel represents a point in
, while the colored pixels represent the number of iterations before it was certain the sequence would diverge.
How do we know at what point a sequence will diverge?Assume

and
. Then

as

for
.
Now assume

with no particular restriction on
. We show that
.

as

for
.
Now if
, and still

as

for
.
So we have shown that the magnitude will be strictly increasing if at any point

and
. If
, then
, and from that point on the magnitude will be strictly increasing. In all, the sequence will diverge when
.
Most importantly, here are some deep zoom images I generated. (I might share the application itself when it runs faster)
I implemented a "real-time" Mandelbrot set viewer in C++ with AMP for GPU multithreading. A major optimization(among the countless micro optimizations) I have yet to implement uses the following fact: The Mandelbrot set is simply connected. In other words, If the boundary of a square(or any other shape) is a subset

then the interior of that square(or any other shape) is in
.
While the Mandelbrot set is generated by those points which do not diverge under the iteration of the complex function
, what set is generated by those points who converge? Surely it must be a subset of
. The answer
is...simpler than it seems! Respond in the comments if you figured it out.
Thanks for reading.

This post has been edited 3 times. Last edited by always_correct, Jul 10, 2017, 9:42 PM