Hyperchaotic functions

[Charleswehner]

I described the dawn of Fractals elsewhere in these forums. Z to a linear power is too tame. Z to n to a constant (like n-squared or n-cubed) begins to show the properties of Fractals. Z to a constant to the n (like Z to the 2 to the n) delivers the classic Julia/Mandelbrot sets. Z to the factorial of n is much too fierce. (n is the palette number).

Here was my original factorial-exponentiation image:

I moved in closer:

Then closer still:

Then I picked up a vector for a Julia set. However, all I got was:

or

or the same in black, for example.

When I zoomed out, however, I got the FOOTPRINTS OF THE PINK PANTHER:

More clues for Clouseau. Nothing useful for Fractals, though.

Charles

[gandreas]

More clues for Clouseau. Nothing useful for Fractals, though.

Not necessarily.

Have you tried adding a simple (complex) constant to each iteration of the expression to see how it behaves?

After all, if the constant in Z² + C is zero, the julia set will be just a circle - tweaking C creates all sorts of magic.

Also, there are a wide variety of techniques such as traps (especially "off axis" traps, or rotating/moving traps) that can produce interesting images even if the basic image isn't all that interesting.

Finally, "hypercomplex" already has a specific meaning (which would encompass split-complex, dual numbers, quaternions, octernions, etc... - basically multidimensional numbers closed under addition and multiplication other than "plain" complex numbers), so technically, these aren't hypercomplex functions...

[Charleswehner]

Finally, "hypercomplex" already has a specific meaning .......

so technically, these aren't hypercomplex functions...

I was aware of this - and planned to correct the mistype. I have changed it to hyperchaotic functions. I have been examining the behaviour of functions with various degrees of "Chaoticness" (or "Chaoticity" or whatever). This function, with factorial growth is excessively chaotic.

I changed the Julia vector, and got a change from pink to yellow to black. I am fully aware that subtle changes to the vector may reveal something interesting. However, the factorial function is very reluctant to deliver any interesting Julia sets.

Charles

[gandreas]

However, the factorial function is very reluctant to deliver any interesting Julia sets.

That's unfortunate - it really seems like there should be something cool in there...