I think I have a good grasp of complex numbers and their applications. Quick Google searches show quarternions and octonions. I figured beyond complex numbers would be needed for higher dimensions, but why are the next more complex numbers seemingly 4 and 8 dimensional rather than 3, 4, 5, etc. TIA
In: Mathematics
Quaternions, Octonions, and other higher dimensional complex numbers are often used to mathematically represent rotations in 3 and 4 dimensions respectively. If you need to represent rotations in dimensions higher then 4 dimensions, you can create higher dimensional complex numbers that can represent those rotations as well.
The reason that 4 dimensional complex numbers are needed to represent 3D rotations, rather then 3 dimensional complex numbers, is because 3 dimensional complex numbers have a problem called gimbal lock, where certain orientations can’t be rotated in certain directions, which can make mathematical problems impossible to solve with 3 dimensional numbers.
Quaternions are often used in 3D applications, like Video Game Engines and Phone Apps to represent rotations, because compared to other number systems that can also represent rotations, they don’t suffer from gimbal lock (which could cause bugs in the 3D engine) and they’re faster to compute then other options, like Matrices. For example, both Unity and Unreal engine use Quaternions to represent 3D rotations.
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