Posted: Apr 23, 2012 11:08 pm
The particularly beautiful thing about the Mandelbrot set is that it arises from asking some very straightforward mathematical questions.
You start by talking about quadratic sequences. These are sequences where each value is generated as a fixed quadratic function of the predecessor. For example, if you take the quadratic function
[math]
And you start with 0. Then you get the sequence 0, -3,30, 1647, ...
Now you want to know how these sequences behave in the long term. The main question you ask is: for which formulas and for which initial values does the sequence grow without limit?
It turns out that you can reduce the problem to asking about the behaviour of sequences whose quadratic formula has the form
[math]
Thus, we only need to concern ourselves with values of c. So our question is: given a value of c, what starting values keep the sequence from growing without limit?
The answer is a class of fractals called the Julia sets. These sets are profoundly beautiful in their own right, but an interesting feature of them is that some are made of disjoint pieces and some are not. So the next question is: for what values of c are the Julia sets made of one solid block?
The answer is the Mandelbrot set.
Simple questions. Infinitely complex answers.
You start by talking about quadratic sequences. These are sequences where each value is generated as a fixed quadratic function of the predecessor. For example, if you take the quadratic function
[math]
And you start with 0. Then you get the sequence 0, -3,30, 1647, ...
Now you want to know how these sequences behave in the long term. The main question you ask is: for which formulas and for which initial values does the sequence grow without limit?
It turns out that you can reduce the problem to asking about the behaviour of sequences whose quadratic formula has the form
[math]
Thus, we only need to concern ourselves with values of c. So our question is: given a value of c, what starting values keep the sequence from growing without limit?
The answer is a class of fractals called the Julia sets. These sets are profoundly beautiful in their own right, but an interesting feature of them is that some are made of disjoint pieces and some are not. So the next question is: for what values of c are the Julia sets made of one solid block?
The answer is the Mandelbrot set.
Simple questions. Infinitely complex answers.