[empath-nirvana]

> 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]

For 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_ma...

[jasomill]

The 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).