I was just watching this video
Group theory 1: Introduction with Borcherds
and he mentions how a Dodecahedron with 12 faces can be put into a 4th dimension that will allow you to flip it and then you'd have an order of 120 instead of 60 with only rotations. I guess I can imagine flipping a 2D rectangle in 3D like a card however I guess we can't "flip" in the same dimension? Can someone give me some additional intuition?
Thank you.
In $\mathbf{R}^{4}$ with coordinates $(x, y, z, w)$, rotations fixing the $(x, y)$-plane have matrices of the form $$ R_{\theta} = \left[\begin{array}{@{}cccc@{}} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \cos\theta & -\sin\theta \\ 0 & 0 & \sin\theta & \phantom{-}\cos\theta \\ \end{array}\right]. $$ The rotation with $\theta = \pi$ effects the reflection $(x, y, z) \mapsto (x, y, -z)$ on the three-dimensional subspace $w = 0$.
A similar trick with the last two coordinates shows that a reflection of $n$-dimensional Cartesian space may be effected by a half-turn of $(n+1)$-dimensional space.