Update

Dear World,

It turns out I only had half the picture -- literally:

(Zoomed-in a bit)

If we replace the constants in the generation functions with their imaginary counterparts, we get two additional symmetry functions.

The trick is to consider endpoints as having positions relative to their 'parent' point of symmetry.  For the original functions, this was the same as using absolute positions, but for these new functions it creates a much more organic pattern.

When we map all four functions together, something pretty amazing happens.

I will update the paper with an addendum and upload images for the new functions on their own.  I will also upload the source code to my generator once I clean it up (a little).

Here are images for generations 6-10
Here is what the original functions looks like without the new ones
Here is what the new functions look like without the original ones


Edit: Updated pics / Added pics of individual function pairs as promised

I made this

Dear World,

I made this and I thought someone might be interested.

Publishing as a non-academic isn't easy, so I opted to publish myself.  Or you can consider this a public peer-review stage.  I just want to see if this is anything new and/or useful.

If this idea is novel and anyone is interested, I would love some help.  I'm closer to idiot savant than mathematician, so I have more questions than answers.

If this is something anyone is already aware of, I would appreciate it if you made me aware of it as well.

Here is a link to the paper (PDF)

Here is a link to the Mathematica Notebook

Here is a link to the fractal image (PNG)

Cheers,
Dave