> Poincaré once said that math isn't about numbers, but relationships between numbers
He described math more poetically and profoundly: "the art of giving the same name to different things"
> Quaternions? Well that's about 4D
Quaternions are about 3D. You need 4 numbers for it, but the space that they operate on is 3D.
Re:Quaternion Knowledge Graph Embeddings
by raincole on Thursday April 25, 2024 @05:11AM
(#40155191)
He's talking from a mathematical perspective, not a compute graphics one.
(Unit) quaternions happens to work as rotations in 3D space. But quaternions' algebraic structure is indeed 4D, just like imaginary numbers' is 2D.
Re:Quaternion Knowledge Graph Embeddings
by ducttapecrown on Thursday April 25, 2024 @01:42PM
(#40160689)
Since you can multiply a quaternion by a quaternion, the quaternions act on themselves, so they indeed act on a 4D space. They can also act on a couple 3D spaces: the unit sphere in 4D, called the three sphere, or the imaginary quaternions.
In math, this is called a representation of a Lie group, and there are representations in all dimensions.
Imaginary numbers aren't "imaginary" they are how to do consistent math in a 2D framework. Quaternions? Well that's about 4D. Poincaré once said that math isn't about numbers, but relationships between numbers. For some reason we don't talk about mathematical structure (explicitly) until the late stages. I wouldn't necessarily call these concepts "abstruse" but they are a bit more abstract. A big part of the problem though, we often ignore the ground we are building upon and so when you finally look at it, it is new and confusing. But then again, the success of Bourbaki's New Math is arguable[0]
[0] https://en.wikipedia.org/wiki/New_Math#In_other_countries