Symmetry Axes of a Cube

Let us now continue with the

in a cube. First we will look at the twofold rotation axes. Each of them is defined by the centers of two edges as it is shown in Fig. 23. So, the cube has a total of 6 twofold axes.

Figure 23

important crystal structures (especially in the description of their symmetry). A regular octahedron, contrary to the tetrahedron, has the same three mutually perpendicular fourfold rotation axes that the cube has (see

21b) with the difference that in the case of an octahedron a fourfold axis is defined by two vertices and in the case of the cube by the geometric centers of two faces (the number of octahedron vertices agrees with the number of cube faces and vice versa).

The solid figure which has a sixfold rotation axis takes on the shape of a regular hexagonal prism that represents the unit cell of the same symmetry as that of an infinite hexagonal lattice in three dimensions. will be considered in more details later.

Before continuing with the three-dimensional case, we will look shortly at the symmetry points of a superposition of plane figures. The superposition of two equilateral triangles with a common geometric center has a threefold rotation point. This is shown in Fig. 22a. A similar superposition of two squares has a fourfold rotation point (see Fig. 22b). Both examples will be helpful in farther consideration of the rotation axes in some three dimensional lattices.

Symmetry Axes of a Cube

Let us now continue with the consideration of the possible rotation axes in a cube. First we will look at the twofold rotation axes. Each of them is defined by the centers of two edges as it is shown in Fig. 23. So, the cube has a total of 6 twofold axes.

Figure 23 Six twofold rotation axes of a cube.

es (especially in the description of their symmetry). A regular octahedron, contrary to the tetrahedron, has the same three

mutually perpendicular fourfold rotation axes that the cube has (see e of an octahedron a fourfold axis is defined by two vertices and in the case of the cube by the geometric

centers of two faces (the number of octahedron vertices agrees with the a sixfold rotation axis takes on the shape of a regular hexagonal prism that represents the unit cell of the same point lattice in three dimensions. This dimensional case, we will look shortly at the symmetry points of a superposition of plane figures. The superposition of two equilateral triangles with a common geometric center has a threefold milar superposition of two squares has a fourfold rotation point (see Fig. 22b). Both examples will be helpful in farther consideration of the rotation axes in some three-

consideration of the possible rotation axes in a cube. First we will look at the twofold rotation axes. Each of them is defined by the centers of two edges as it is shown in Fig. 23. So, the cube

It is easy to show that the body diagonals of the cube represent its threefold axes. We can see in Fig. 24 that the

connects two opposite cube vertices. groups, with 3 vertices each

triangles. Each of the triangles is lying in a plane orthogonal to the diagonal and its geometric center overlaps with the point where the diagonal intersects the plane of the triangle. It is obvious that after rotating the by an angle 2π 3(or its multiples), the new positions of

(those out of the axis) overlap with some “old” positions of the vertices Therefore, this transformation leaves the cube invariant. Besides the axis shown in Fig. 24, there are 3 more threefold axes in the cube, that is, as

Figure 24 Each diagonal of a cube represents one of its threefold rotation axis.

It is easy to show that the body diagonals of the cube represent its threefold axes. We can see in Fig. 24 that the displayed body diagonal connects two opposite cube vertices. The remaining 6 vertices form two with 3 vertices each, that represent the vertices of two equilateral Each of the triangles is lying in a plane orthogonal to the diagonal and its geometric center overlaps with the point where the diagonal

the plane of the triangle. It is obvious that after rotating the (or its multiples), the new positions of the cube vertices (those out of the axis) overlap with some “old” positions of the vertices

this transformation leaves the cube invariant. Besides the axis n in Fig. 24, there are 3 more threefold axes in the cube, that is, as

Each diagonal of a cube represents one of its threefold rotation axis.

Figure 25 The 13 rotation axes of a cube.

It is easy to show that the body diagonals of the cube represent its body diagonal The remaining 6 vertices form two the vertices of two equilateral Each of the triangles is lying in a plane orthogonal to the diagonal and its geometric center overlaps with the point where the diagonal the plane of the triangle. It is obvious that after rotating the cube cube vertices (those out of the axis) overlap with some “old” positions of the vertices. this transformation leaves the cube invariant. Besides the axis n in Fig. 24, there are 3 more threefold axes in the cube, that is, as

many as the number of body diagonals. In conclusion, a cube has a total of 13 rotation axes. All of them are shown in Fig. 25.