# symmetry-so3-rotation-order.mp4

Chapter section: Commutators

Caption (verbatim):

Non-commuting operations of $SO_3$

Image description (verbatim):

Noncommuting 90-degree rotations in three dimensions

## Before the animation

Chapter lines 366–366:

In the case of a single continuous symmetry transformation, the generator combined with a parameter, like angle in the case of rotation, specifies an arbitrary transformation. However, when there are multiple independent transformations that can be composed to form a multi-dimensional group, the order of the composition may also need to be taken into account because it can produce different resultant states. For example, in the 3-dimensional rotation group, called $SO(3)$, rotating about the $x$-axis then the $y$-axis leaves a sphere in a different state than applying the same actions in the opposite order.

## After the animation

Chapter lines 374–374:

This order-dependence is expressed by the **commutator**:

Chapter lines 376–380:

```math
[X,Y]
=
XY-YX.
```