#quaternions — Public Fediverse posts

But wait, there's more! I also wrote a C++ program and Lua scripts to help you make animations of these higher-dimensional fractals. You can find these at https://codeberg.org/caten/RealAlgebraPlot

Not only can you see animations of the rock-paper-scissors fractal of central interest in my paper, you can also take slices of even higher-dimensional fractals coming from other real algebras, such as the octonions.

I wrote everything here from scratch as much as possible, so it has basically zero dependencies. I even opted to use Netpbm (https://en.wikipedia.org/wiki/Netpbm) so that I'd have an easy time writing image files directly without using a library. I am using #GraphicsMagick to produce animations. The same utility can be used to convert the output image files to a more familiar format like png.

#math #mathematics #algebra #analysis #ComplexAnalysis #fractals #Mandelbrot #MandelbrotSet #UniversalAlgebra #octonions #quaternions #cpp #Lua #LuaLang

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In this talk I show how to multiply quaternions AND ALSO OCTONIONS using the dot product and cross product of vectors in 3 dimensions:

https://www.youtube.com/watch?v=JI5xPGN_sWo

For quaternions this is well-known, but for octonions it's less so.

I sketched the proof that both these algebras have a norm obeying

|ab| = |a| |b|

To see the details, go to the proof of Theorem 2 here:

https://golem.ph.utexas.edu/category/2020/07/octonions_and_the_standard_mod_1.html

I do the proof for octonions, but the same argument also works for quaternions!

#quaternions #octonions