Los estudios sobre la teoría de la traducción

Now, we will concentrate our attention on a 8 points (or atoms) located

this set of points are the same as the symmetry axes of the cube. If we add one additional point in the middle of the cube, then the symmetry of the resulting system will remain the same

point of all the axes

Figure 26 The system of 14 points placed

faces of a cube have the same threefold rotation axes as the cube.

Figure 27

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.

Symmetry Axes of a Set of Points

Now, we will concentrate our attention on a system consisting of

8 points (or atoms) located at the vertices of a cube. The symmetry axes of are the same as the symmetry axes of the cube. If we add one additional point in the middle of the cube, then the symmetry of the resulting system will remain the same, since this point will be a common point of all the axes and also other symmetry elements. Also, if we add

The system of 14 points placed at the vertices and in the geometric centers of the faces of a cube have the same threefold rotation axes as the cube.

Figure 27 Axial view of one of the triangles from Fig. 26.

many as the number of body diagonals. In conclusion, a cube has a total of

system consisting of a set of of a cube. The symmetry axes of are the same as the symmetry axes of the cube. If we add one additional point in the middle of the cube, then the symmetry of the since this point will be a common . Also, if we add

points in the middle of the faces of the cube, then the symmetry of this new 14-point system (shown in Fig.

system consisting of only 8 points.

Figs. 24 and 26, that the threefold axes are present in this 14

The six new points will form two groups of three points each, which are located in the middle of the triangle edges, as appears in Fig. 26.

view of one of the triangles from Fig

Let us now consider the fourfold rotation axes in the case of the 14-point system in

total of six points in the middle of the faces of the cube, two are on the axis and the remaining four represent vertices of a square lying in a plane orthogonal to the axis. If we project the 14 poin

the axis, then we will obtain a superposition of two squares shown on the

Figure 28 Fourfold rotation axis of a system consisting of 14 points located

and centers of the faces of a cube.

Figure 29 Three systems consisting of:

the vertices and the geometric center of a cube, and

centers of a cube. Each set of points has the same 13 rotation axes as a cube.

points in the middle of the faces of the cube, then the symmetry of this new point system (shown in Fig. 26) will still remain the same as in the system consisting of only 8 points. For example, it is easy to see, comparing that the threefold axes are present in this 14-point system. The six new points will form two groups of three points each, which are located in the middle of the triangle edges, as appears in Fig. 26. The axial view of one of the triangles from Fig. 26 is shown in Fig. 27.

Let us now consider the fourfold rotation axes in the case of the consideration. We can observe in Fig. 28 that, of the total of six points in the middle of the faces of the cube, two are on the axis and the remaining four represent vertices of a square lying in a plane orthogonal to the axis. If we project the 14 points on a plane orthogonal to the axis, then we will obtain a superposition of two squares shown on the

Fourfold rotation axis of a system consisting of 14 points located at the vertices and centers of the faces of a cube.

Three systems consisting of: (a) 8 points at the vertices of a cube, (b) 9 points and the geometric center of a cube, and (c) 14 points at the vertices centers of a cube. Each set of points has the same 13 rotation axes as a cube.

points in the middle of the faces of the cube, then the symmetry of this new 26) will still remain the same as in the comparing point system. The six new points will form two groups of three points each, which are The axial Let us now consider the fourfold rotation axes in the case of the

consideration. We can observe in Fig. 28 that, of the total of six points in the middle of the faces of the cube, two are on the axis and the remaining four represent vertices of a square lying in a plane ts on a plane orthogonal to the axis, then we will obtain a superposition of two squares shown on the

at the vertices

9 points at at the vertices and face

right of Fig. 28. Thus we can say that the 14-point system has the same three fourfold rotation axes as the cube. Besides that, the system of points has six twofold axes. Finally, we can conclude that in the three cases described above, and shown in Fig. 29, we have the same 13 symmetry axes as were identified before in the cube.