# symmetry-d3-rotations-vs-flips.mp4

Chapter section: Representations

Caption (verbatim):

$D_3$ in 3-dimensional representation

Image description (verbatim):

D3 rotations and flips with labeled component axes and a fixed equal-components axis

## Before the animation

Chapter lines 118–118:

The matrices that provide a set of transformations from which all other operations can be constructed are the symmetry's **generators**. For example, in the 3-dimensional representation of $D_3$, all operations can be constructed from one rotation matrix and 3 flip matrices, one for each axis. 

Chapter lines 120–120:

Visually, permutations that correspond to rotations of the triangle are 120-degree rotations about a diagonal axis in this vector space, while those that correspond to flips are flips about planes that contain the diagonal axis.

## After the animation

Chapter lines 128–128:

Now, we can notice something about these visualizations. The transformation of any vector lies in a plane, and all such planes are parallel to one another. Thus, if we subtract the average of a vector's components from each component, that is, if we move the point where the plane intersects the axis of rotation to the origin, we preserve the permutation structure of the transformations. We thus see that the $D_3$ symmetry is just as well represented as 120-degree rotations in the subspace of a 2-dimensional plane.

Chapter lines 130–130:

For our earlier vector, the average of the components is 3. We can separate it into a common part and a zero-sum part.