By the way, if any actual mathematicians are reading this, I’d be really curious to know if this way of thinking about the Mandelbrot Set would be of any practical benefit (besides educational and aesthetic value of course). For example, I could imagine a formalization of this being used to pose non-trivial questions which wouldn’t have made much sense to talk about previously, but I’m not sure if that would actually be the case for a trained mathematician.
I usually think of the field of complex numbers algebraically, but one can also think of the real numbers, complex numbers, and quaternions geometrically. The real numbers are good with dealing with 1 dimensional space, and the complex numbers are good for dealing with 2 dimensional space geometrically. While the division ring of quaternions is a 4 dimensional algebra over the field of real numbers, the quaternions are best used for dealing with 3 dimensional space geometrically.
For example, if U,V are open subsets of some Euclidean space, then a function f:U→V is said to be a conformal mapping when it preserves angles and the orientation. We can associate the 2-dimensional Euclidean space with the field of complex numbers, and the conformal mappings between open subsets of 2-dimensional spaces are just the complex differentiable mappings. For the Mandelbrot set, we need this conformality because we want the Mandelbrot set to look pretty. If the complex differentiable maps were not conformal, then the functions that we iterate in complex dynamics would stretch subsets of the complex plane in one dimension and expand them in the other dimension and this would result in a fractal that looks quite stretched in one real dimension and squashed in another dimension (the fractals would look like spaghetti; oh wait, I just looked at a 3D fractal and it looks like some vegetable like broccoli). This stretching and squashing is illustrated by 3D fractals that try to mimic the Mandelbrot set but without any conformality. The conformality is why the Julia sets are sensible (mathematicians have proven theorems about these sets) for any complex polynomial of degree 2 or greater.
For the quaternions, it is well-known that the dot product and the cross product operations on 3 dimensional space can be described in terms of the quaternionic multiplication operation between purely imaginary quaternions.
I would be very surprised if it did: I think complex numbers are simply just that good, with no downside that a framing that avoids them, such as this, sidesteps.
To be quite frank, you’re avoiding complex numbers only in the sense that you spell out the operations involved in handling complex numbers explicitly—so of course there’s no added benefit, you’re simply lifting the lid of the box...
That being said, as you discover by decomposing complex multiplication into it’s parts (rotation and scaling), you get to play with them separately, which already leads you to discover interesting new variations on the theme.
By the way, if any actual mathematicians are reading this, I’d be really curious to know if this way of thinking about the Mandelbrot Set would be of any practical benefit (besides educational and aesthetic value of course). For example, I could imagine a formalization of this being used to pose non-trivial questions which wouldn’t have made much sense to talk about previously, but I’m not sure if that would actually be the case for a trained mathematician.
I usually think of the field of complex numbers algebraically, but one can also think of the real numbers, complex numbers, and quaternions geometrically. The real numbers are good with dealing with 1 dimensional space, and the complex numbers are good for dealing with 2 dimensional space geometrically. While the division ring of quaternions is a 4 dimensional algebra over the field of real numbers, the quaternions are best used for dealing with 3 dimensional space geometrically.
For example, if U,V are open subsets of some Euclidean space, then a function f:U→V is said to be a conformal mapping when it preserves angles and the orientation. We can associate the 2-dimensional Euclidean space with the field of complex numbers, and the conformal mappings between open subsets of 2-dimensional spaces are just the complex differentiable mappings. For the Mandelbrot set, we need this conformality because we want the Mandelbrot set to look pretty. If the complex differentiable maps were not conformal, then the functions that we iterate in complex dynamics would stretch subsets of the complex plane in one dimension and expand them in the other dimension and this would result in a fractal that looks quite stretched in one real dimension and squashed in another dimension (the fractals would look like spaghetti; oh wait, I just looked at a 3D fractal and it looks like some vegetable like broccoli). This stretching and squashing is illustrated by 3D fractals that try to mimic the Mandelbrot set but without any conformality. The conformality is why the Julia sets are sensible (mathematicians have proven theorems about these sets) for any complex polynomial of degree 2 or greater.
For the quaternions, it is well-known that the dot product and the cross product operations on 3 dimensional space can be described in terms of the quaternionic multiplication operation between purely imaginary quaternions.
I would be very surprised if it did: I think complex numbers are simply just that good, with no downside that a framing that avoids them, such as this, sidesteps.
To be quite frank, you’re avoiding complex numbers only in the sense that you spell out the operations involved in handling complex numbers explicitly—so of course there’s no added benefit, you’re simply lifting the lid of the box...
That being said, as you discover by decomposing complex multiplication into it’s parts (rotation and scaling), you get to play with them separately, which already leads you to discover interesting new variations on the theme.