Fractal Explorer

Fractal Explorer (FE) is just one of the many free fractal generating programs out there. Its power lies in the fact that it balances math and art in fractals. With this free program, you can create many beautiful images, and it allows the user to appreciate fractals from a mathematical perspective (by changing seeds, constants, setting limits on input/output, and even defining you own equations).

Fractal Explorer will entertain you for a good 2 days.

What is a Fractal?

Lets start off with Vocabulary: (in the real system)
seed a start value
orbit--repeatedly applying a function to a seed/sequence

Line 1) X₀

Line 2) X₁ = f(X₀)

Line 3) X₂ = f(X₁) = f(f(X₀))

Line 4) X₃ = f(X₂) = f(f(f(X₀)))

So all the lines are orbits, line 1 is the seed, lines 2 to infinity are iteration of f

A seed is said to be fixed if we plug it into the function, and plug the output back into the function, and then put THAT output into the function, and so on for infinity, and the output is always about constant ( later, we will color that starting value black).

Now, if we continually insert the "range" back in the "domain," and we find that the range goes to infinity, then that point doesn't converge.

Now the cool part: a hallmark of Chaos is when there is a sensitive dependence on the initial condition. For example, a seed of 2.7 might have a fixed value, but a seed of 2.71 will go to infinity.

A Fractal is just an iterated function with a complex seed (real and imaginary).