Inspired by https://www.reddit.com/r/desmos/comments/1x56t7n/complex_numbers_without_complex_numbers/

Image Link: https://www.desmos.com/calculator/vkzxtnwbx0

i was bored so i decided to try to graph one of my ocs into desmos

https://www.desmos.com/calculator/jpc5yhro9g

https://www.desmos.com/calculator/vk9qruga5p

In one discussion related to the introduction of matrices in Desmos, I wrote that complex matrices were missing. I was told that a complex number can be implemented by a real 2x2 matrix, and a complex matrix can be implemented by an extended block matrix where each block is an isomorphism of a complex number. That is, we expand a 2x2 complex matrix into a 4x4 real matrix.

On one hand, this is a solution; on the other hand, we can compute everything on paper without any implementations.

I would like to add that there is not only the classical representation of the imaginary unit as a 2x2 skew-symmetric matrix. In my Desmos graph, I show an entire class of matrices that perfectly supports complex numbers, being an isomorphism. It includes the classical skew-symmetric matrix as well.

The advantage of the classical matrix is that it supports the classical geometric representation of the complex plane and the multiplication operation on this plane as a 90-degree counterclockwise rotation. In the extended class, this is just a special case.

This is what I tried to illustrate.

Like i want it to take the shape of a sphere. How do I do it?

https://www.desmos.com/calculator/2ocbaizaut?lang=en

The above is a minimal toy example to illustrate what I'm trying to do.

This is my first time attempting to make a full on notebook (and actually finishing it) so idk how good it turned out, but I've made so many projects I ended up not even posting semi recently, I felt like I had to at least post this

https://www.desmos.com/calculator/8zeukdcdj0

Derived from

https://www.desmos.com/calculator/3nc5dmzuu4

I graphed 2 circles, a X, and graphed the intersect points of the blue circle, the red circle is a bit smaller then the blue circle since they are the landings of sin(x) from pi to 2pi