Wednesday, November 12, 2008

Request a Fractal!

In a previous post, I explained how to make fractals using Newton's method. These fractals can be generated by a java program I wrote for a high school project. The main input is a mathematical function.

So you give me a mathematical function, and I will make a fractal out of it!

You do not need to understand how Newton's Method works to make a request.

The rules:
  • Be creative! A simple function (like f(x) = x2) might not produce anything interesting. More complicated functions may not produce anything interesting either, but I can adjust them to make them interesting.
  • You may use any of the following in your function:
    • Any arithmetic: addition, subtraction, multiplication, division, in absolutely any combination.
    • Imaginary numbers (represented by "i")
    • Trigonometric functions: sin, cos, tan, etc.
    • Inverse trigonometric functions: arcsin, arccos, arctan, etc.
    • Exponentials: e^x, x^(-1/2), x^x, x^i, and so on.
    • Logarithms
    • Any, absolutely any combination of the above. Even something like x*(cos(tan(x))^(i/x))-log(i+arctan(x)) is possible. But please don't suggest anything that crazy.
More complicated functions tend to generate more complicated fractals; simpler functions tend to generate simpler (more elegant?) fractals. For examples refer to the bottom of my previous post.

Oh, and here's one more example to throw out there. This Mandelbrot look-alike was generated with the function (x+1)*sqrt(x).

I will post a bunch of these at a later date.

4 comments:

intrinsicallyknotted said...

I'd like to see hyperbolic sine, cosine, etc.

miller said...

Yep, you can see those too, although I should mention that they're not much different from the regular trig functions (because cosh(x) = cos(ix), sinh(x) = isin(ix) and so forth).

Anonymous said...

how about e^-(ixcosx)+e^(xsinx)

miller said...

A bold request! But I like the results. Wait 'til you see.