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