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> This make me wonder on the connection between 2, 3 and 4 dimensions, and Hamilton found out the hard that you need 4 numbering system or quaternion, in order
by empath-nirvana 2y ago
> This make me wonder on the connection between 2, 3 and 4 dimensions, and Hamilton found out the hard that you need 4 numbering system or quaternion, in order to properly represent 3 dimensions [1].
Sort of. Quarternions have some nice properties as representations of _rotations_ in 3d space. 3d space works just fine with 3 dimensions.
- guyomes 2y agoFor rotations in 3D space, another nice representation is based on antisymmetric matrices [1]. All rotations can be represented as exp(A), where A has the form: 0 -z y z 0 -x -y x 0 Moreover, this generalizes to other dimensions. This allows to see that in 2D, rotations have one degree of freedom, in 3D they have three, and in 4D they have six. [1]: https://en.wikipedia.org/wiki/Rotation_matrix#Exponential_map https://en.wikipedia.org/wiki/Rotation_matrix#Exponential_ma...
- jasomill 2y agoThe two dimensional case is my favorite: let I be the identity matrix and J the "imaginary unit matrix" 0 -1 1 0, then JJ = -I = exp(Jπ) and exp(Jθ) = cos(θ) -sin(θ) sin(θ) cos(θ).