Thursday, July 15, 2004
Fractals, Fractals everywhere
I wanted to be doubly sure I understood the definition of a fractal before I posted this, so I've been doing some research into fractals. More specifically, I've been looking at fractal sequences, and I think I'm sure enough about the definition of fractal sequences to state the following:
Let Z = all positive integers greater than 0.
BigOmega(Z) is a fractal sequence.
BigOmega(Z2), Z2 = all odd positive integers greater than 0, is also a fractal sequence.
I haven't, for some reason, been able to find that stated on the 'net. So I'm stating it here without argument, but these sequences are very much self-similar. If you start with a prime p, and take the set of all all the numbers in the sequence divisible by that prime, BigOmega of that set will be the same as BigOmega(Z) + BigOmega(p) . This is true of the set of all positive integers, all even integers, and all odd integers.
Let Z = all positive integers greater than 0.
BigOmega(Z) is a fractal sequence.
BigOmega(Z2), Z2 = all odd positive integers greater than 0, is also a fractal sequence.
I haven't, for some reason, been able to find that stated on the 'net. So I'm stating it here without argument, but these sequences are very much self-similar. If you start with a prime p, and take the set of all all the numbers in the sequence divisible by that prime, BigOmega of that set will be the same as BigOmega(Z) + BigOmega(p) . This is true of the set of all positive integers, all even integers, and all odd integers.
