Symmetries/Isometries/Cube/Introduction/Section

Definition

Let $T\subseteq V$ be a subset of a Euclidean vector space. A proper isometry

$f:V⟶V$

fulfilling $f(T)=T$

is called a proper symmetry or a motion of $T$.

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 $T=\mathrm{\varnothing }$ 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

We consider a cube $W\subset {ℝ}^3$, with sides of length $2$, and the origin as its center. Thus, the vertices are

$(\pm 1,\pm 1,\pm 1).$

We ask what are the possibilities to transform the cube into itself. By this, we mean that the cube shall not be deformed in any way, the cube is considered rigid, and the motion should be realizable in the physical world. We talk about a (proper) motion of the cube. During such a motion, the center of the cube does not change its position, and faces are mapped to faces, edges to edges, vertices to vertices. Also, the centers of the faces are mapped to centers of faces, and opposite centers of faces are mapped to opposite centers of faces. The centers of faces are the six points

$(\pm 1,0,0),(0,\pm 1,0),(0,0,\pm 1).$

If the point $(1,0,0)$ is mapped to the face center $S$, then $(-1,0,0)$ is mapped to the opposite point, that is, $-S$. Every face center $S$ is allowed, but this does not determine the motion. For the face center $(0,1,0)$, there are four possible image points (only $S$ and $-S$ are excluded), because the cube can be rotated around the axis given by $S$ by a multiple of $90$ degree. These rotations correspond to the possibilities to move the point $(0,1,0)$ to one of the four remaining face centers. The choice of the second face center $T$ determines the motion uniquely. How clear is that?

To understand this, the following observations are useful.

1. Motions can be performed after another, that is, if we have two motions of the cube ${\displaystyle }$ \varphi ${\displaystyle }$, then the composition $\psi \circ \phi$ is defined, which performs first $\phi$ and then $\psi$.
2. The identical motion, which moves nothing, is a motion. If we compose an arbitrary motion with the identical motion, the motion does not change.
3. For a motion $\phi$, there exists the opposite motion (or "inverse motion“) ${\phi }^{-1}$, which has the property that the compositions ${\phi }^{-1}\circ \phi$ and $\phi \circ {\phi }^{-1}$ are just the identity.

With these observation, we can understand the principle mentioned above: assume that there are two motions $\phi$ and $\psi$, both sending $(1,0,0)$ to $S$ and $(0,1,0)$ to $T$. Let ${\psi }^{-1}$ denote the inverse motion of $\psi$. Then we consider the composed motion

$\theta ={\psi }^{-1}\circ \phi .$

This motion hat the property that $(1,0,0)$ is mapped to $(1,0,0)$ and that $(0,1,0)$ is mapped to $(0,1,0)$, as $\phi$ sends $(1,0,0)$ to $S$, and ${\psi }^{-1}$ sends $S$ back to $(1,0,0)$ (and accordingly for $(0,1,0)$). This means that $\theta$ has the property that $(1,0,0)$ as well as $(0,1,0)$ are sent to themselves, that is, these points are fixed points of the motion. But this implies that the $x,y$-plane is fixed. The only physical motion of the cube that fixes this plane is the identical motion. Therefore, ${\psi }^{-1}\circ \phi =\mathrm{Id}$, and $\phi =\psi$. Note that the reflection an the $x,y$-plane swaps the points $(0,0,1)$ and $(0,0,-1)$, but this is a so-called improper motion; it can not be performed in the physical world.

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

$\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right),\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& -1\\ 0& 1& 0\end{array}\right),\left(\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& -1\end{array}\right),\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& -1& 0\end{array}\right).$

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

$\left(\begin{array}{ccc}0& 0& 1\\ 1& 0& 0\\ 0& 1& 0\end{array}\right)\text{ and }\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right)$

(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.

Point	$A$	$B$	$C$	$D$	$E$	$F$	$G$	$H$
Image point	$E$	$A$	$D$	$H$	$F$	$B$	$C$	$G$

Rotation by $120$ degree around the space diagonal.

Point	$A$	$B$	$C$	$D$	$E$	$F$	$G$	$H$
Image point	$H$	$D$	$C$	$G$	$E$	$A$	$B$	$F$