# Symmetries/Isometries/Cube/Introduction/Section
## Definition

Proper symmetry/Isometry on a geometric object/Definition
The set of all symmetries of a geometric object $T$ form a group, with the composition of mappings as binary operation. This group is called the (proper) symmetry group of $T$. It is a subgroup of the group of all (proper) isometries of the Euclidean vector space. If $T$ is sufficiently complicated, then there are no isometries beside the identity. In the cases $T=\emptyset$ and $T=\{0\}$, every isometry is a symmetry. The set of all symmetries allowing the improper isometries form also a group. The proper isometries are the symmetries that can be performed in the physical world; they are the mappings that preserve the orientation. As an introductory example, we discuss the symmetries of a cube.

## Example

This implies that, for the basis vector $(1,0,0)$, there are six possible image vectors, for the second basis vector $(0,1,0)$, there are four possibilities, and that these choices determine the mapping uniquely. All together, there are $24$ transformations of the cube. These motions may be described by a $3\times 3$-matrix, where the columns describe the image vectors of the basis vectors. If the first basis vector is fixed, then the four possible motions are given by the matrices
    ${\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}},\,{\begin{pmatrix}1&0&0\\0&0&-1\\0&1&0\end{pmatrix}},\,{\begin{pmatrix}1&0&0\\0&-1&0\\0&0&-1\end{pmatrix}},\,{\begin{pmatrix}1&0&0\\0&0&1\\0&-1&0\end{pmatrix}}.$
This can be seen as follows: if a center of a face is mapped to itself, then this also holds for opposite center; therefore, also the corresponding axis is fixed. A motion that fixes such an axis through the center of faces is a rotation around this axis, by a multiple of $90$ degree. Such a motion is a motion in the plane (namely, of the plane perpendicular to this axis), and this observation yields the given matrices.
An important property of this motion is that it is a rotation of the space around a fixed axis. This property holds in general for space motions due to fact. Since the rotations just mentioned are multiples of a quarter rotation, we get the identity if we perform this rotation four times in a row. In the case of a half rotation, already the second execution gives the identity. The order, in the sense the group theory, of such a motion is $4$ or $2$.
We consider now in the example of a cube the space diagonal $D$ that runs through $(1,1,1)$ and durch $(-1,-1,-1)$. We can also rotate the cube around this axis, by a multiple of $120$ degree. Have a look on the cube so that a vertex is in the center of the view. The rotation sending $(1,0,0)$ to $(0,0,1)$, sends $(0,0,1)$ to $(0,1,0)$. Hence, these two rotations around the space diagonal have the matrix representation
    ${\begin{pmatrix}0&0&1\\1&0&0\\0&1&0\end{pmatrix}}\,\,{\text{ and }}\,\,{\begin{pmatrix}0&1&0\\0&0&1\\1&0&0\end{pmatrix}}$
(and they are inverse to each other). A further possible rotation is a half rotation around the axis through the center of the edges, that is, an axis given by the centers of two opposite edges.
A strategy to understand the motions of a cube is to understand what happens to the centers of the faces under a motion, which of these six points are fixed, which are moved to which point. A motion defines always a bijection of this point set to itself, that is, a permutation. However, there are further characteristic points, and, more generally geometric subobjects of the cube, that are transformed under a motion of the cube to itself, like set of vertices, the set of the centers of the edges, the set of edges, the set of faces, the set of all space diagonals, etc. Every motion acts on these objects in a characteristic way. In mathematical terms, the precise concept of this observation is the concept of a group action and of a group homomorphism. If we use the notions for the vertices from the picture above, then the quarter rotation and the rotation by $120$ degree mentioned above induce the following action on the vertices.
Quarter rotation around the axis through face centers.
Rotation by $120$ degree around the space diagonal.