Euler's Formula Generalized

Dear World,

This fractal I discovered can exist in any number of dimensions, so I was working out how to do that.  I've gotten to the point where I can generate a 3D fractal, but I can only draw a 2D cross-section of it.

In my research, though, I think I figured out what Euler's formula actually is: a special case.  I have come up with "Euler's Formula Generalized" (EFG), a one-page paper.

So it turns out that Euler's Identity is actually a blueprint that gives you all the information you need to sprout a second dimension out of the real numbers.  That's exactly what we've been doing with it, but we haven't been to certain as to how.

This generalization makes it clear how the constants e, i, and pi all work together.  This is because  EFG will give you a different set of constants for any given dimension.  When you plug '1' into the EFG function, you get the familiar e, i, and pi.  However, when you plug in '2', you get three different numbers.  These new constants are just like e, i, and pi, but they will operate orthogonally -- so we could name them f, j, and rho.

This is important because we need to keep the coordinates for each dimension separate while we do linear math on them (except when the math combines them on purpose, that is).

Using EFG, you can generate a certain kind of N-dimensional space.  If you use N=2 then you get the complex plane we are familiar with, but that's just the first step.

For example, N=3.  This space can be imagined easily: take a complex plane and rotate it in the third dimension around the x-axis.

Each new dimension is really a new plane, but all of the planes share the real axis, so they only each contribute a single dimension, each of which will be orthogonal to all the others and the real axis.

I don't know how useful it would be for anyone else, but it's perfect for generating my fractal in N-dimensions.  I've implemented this idea geometrically in 3D and got an image of the cross section of the fractal along the YZ plane.  It's very cool (and symmetrical, of course!).

This is all new in my brain so I'm not sure I'm explaining it well, but I'm pretty sure it's correct.  I hope someone will prove or disprove this theory -- or conjecture, or whatever it qualifies as.  To that end, I am posting my paper and my Mathematica notebook.

I'll get back to the fractal.  I still have to publish the generator code.  I'm sure tons of nobody are anxiously waiting, heh.

My paper can be found here
My Mathematica notebook can be found here

Now I'm going to head out and play in the rain for a while.

Cheers,
Dave

Update: I've been too busy to do anything with this. Hope to dig deeper soon. I did realize that the logarithmic intervals are an unnecessary complication. They made the math look nicer but they just obfuscate what's happening underneath. They still work tho, mathematically.

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