![]() Because a tetrahedron has six edges, and each C 2 axis go through two edges there are 6/2=3 C 2 axes. You can see that a C 2 axis goes through two opposite edges in the tetrahedron. Figure 2.2.8 The C 2 axes in a tetrahedron belonging to the point group T d (Attribution: /gallery) In addition to the C 3 axes there are C 2 axes (Fig. Figure 2.2.7 Symmetry operations associated with the C 3 axes in the point group T d We can express this by writing the respective numbers as coefficients in front of the Schoenflies symbol for the operations (Fig. Because there are four C 3 axes, there are four C 3 1 and four C 3 2 operations and eight C 3 operations overall. Therefore C 3 3=E, and we only need to consider the C 3 1 and the C 3 2 rotation about 120 and 240° respectively. How many unique C 3 operations are associated with these axes? After three rotations around 120° we reach the identity. Because each C 3 axis goes through one vertex, there are four vertices, and we know that in a platonic solid all vertices are symmetry-equivalent, we can understand that there are four C 3 axes. The C 3 axes go through the vertices of the tetrahedron. It is a property of the high-symmetry point groups that they have more than one principal axis. The tetrahedron has four principal C 3 axes (Fig. Figure 2.2.6 The C 3 axes in a tetrahedron (Attribution: /gallery) Next, it is useful to look for the principal axes. First, we should not forget the identity operation, E. Let us find the symmetry elements and symmetry operations that belong to the point group T d. The tetrahedron, as well as tetrahedral molecules and anions such as CH 4 and BF 4- belong to the high symmetry point group T d. Figure 2.2.5 Icosahedron at Allentown Cedar Beach Park If you would like to see and study an icosahedron from the outside and inside, there is one for study on the playground of the Allentown Cedar Beach Park, in Allentown, Pennsylvania. The icosahedron is the most complex of all platonic solids. There are no possibilities to connect other regular polygons like hexagons to make a platonic solid. In addition, six squares can be connected to form a cube, and twelve pentagons can be connected to form a dodecahedron. The third possibility is the icosahedron made of twenty triangles. The second platonic solid is the octahedron made of eight regular triangles. The first possibility is to construct a tetrahedron from four regular triangles. Figure 2.2.4 The platonic solids (Attribution: Drummyfish ) There are only five possibilities to make platonic solids from regular polygons (Fig. We will see that this is a property that can be used to understand the symmetry elements in high symmetry point groups. In a platonic solid all faces, edges, and vertices (corners) are symmetry-equivalent. Platonic solids are polyhedra made of regular polygons. The symmetry elements of the high symmetry point groups can be more easily understood when the properties of platonic solids are understood first.
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