[goldenkey]

This fractal is non-holomorphic which makes it kind of a dumpster fire..or a burning ship I guess... The easiest layman's definition of holomorphic is any function or operations on complex numbers that doesn't peek at the real and imaginary parts of the complex number. In algebra, there is no combination of algebraic operations [1] that can do so. But on computers, we have Re(z) and Im(z), which are easy to calculate since the complex number is simply stored as two parts, and it just pulls one of them out.

When you violate this sacred notion, of treating a complex number as a singular entity, indivisible, the operations or functions you get, no longer have nice properties, like being analytic [2]. You are no longer doing algebra, you have gone into the realm of chaotics over R^2 with an equivalence relation through rotations, which is a far different beast.

That's why other fractals that treat complex numbers as first class citizens, singular entities, and don't use hacks like Re or Im, are much prettier and have less branch cuts.

[1] https://en.wikipedia.org/wiki/Algebraic_operation

[2] https://en.wikipedia.org/wiki/Analytic_function

[ogogmad]

If you replace the complex numbers with split-complex numbers, then the set of holomorphic functions becomes uninteresting. You therefore need to use * (meaning the split-complex analogue of complex conjugation) to get anything interesting. It follows that the obvious analogue of complex analysis over the split-complex numbers is pretty sterile, but I think that the split-complex numbers themselves remain interesting.

The operation * (meaning conjugation) isn't totally ugly. The reason for that is because it shows up in matrix theory. A lot of families of matrices (like the unitary and the self-adjoint matrices) are defined using *. Therefore, while the function theory of the split-complex numbers might be boring, their matrix theory is still somewhat interesting because of its dependence on *.

The relevance to generating fractals using the split-complex numbers is that you need to use non-holomorphic functions to get interesting results. Some results here: https://news.ycombinator.com/item?id=32211495

[goldenkey]

Complex conjugation is non-holomorphic and yes, it's ugly. I still don't see why it being useful as a construction makes it a beautiful operation in analysis given that it isn't algebra.

There are holomorphic split-complex functions and they aren't uninteresting at all - they are quite beautiful and complex, relating to wave equations: https://en.wikipedia.org/wiki/Motor_variable#D-holomorphic_f...

They're not something often looked into in the fractal scene but if anyone has spare time, I'd love to see a D-holomorphic fractal analog of the Mandelbrot :-)

[ogogmad]

A holomorphic function over the split-complex numbers is the direct sum of two holomorphic functions over R. In that sense, it's trivial.

You will be very disappointed once you see the split-complex Mandelbrot set. It will be an instructive exercise for you to see why it looks the way it does.

My point about complex conjugation is illustrated in [1]. Linear algebra over the complex numbers does not limit itself to the four arithmetic operations {+,-,*,/}, but uses complex conjugation as the 5th and final operation. Look at the definition of a unitary matrix, or a self-adjoint matrix, or a normal matrix. Also, look at the definitions of the QR decomposition and SVD. Complex conjugation is everywhere in linear algebra. Consequently, matrix theory over the split-complex numbers is actually somewhat interesting.

[1] - https://en.wikipedia.org/wiki/Matrix_decomposition

To some extent, this seems like a weird discussion about mathematical aesthetics.

[goldenkey]

Linear algebra uses the Hermitian adjoint [1], sure, but how often do we graph or build functions out of it for analysis purposes? I understand that it is useful in formulating equations, relations, etc.

Linear algebra does a lot of things, ie. transpose, that aren't algebraic. I think it's important we understand what is computational versus algebraic. In that vain, linear algebra proves to be immensely useful, but we will trip ourselves if we start thinking certain aspects of it are analytic.

Anyhow, point taken, that image hardly looks interesting.

I think we agree and like you said, are just talking about aesthetics. I agree that the utility of math doesn't end at algebra or analyticity.

https://en.wikipedia.org/wiki/Hermitian_adjoint

[ogogmad]

Here's the split-complex Mandelbrot (and Mandelbar for comparison): https://imgur.com/a/j2c7qkB

Read my other reply.

[goldenkey]

Thank you.