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2-1 The orientation of the triangular base of the pyramid is shown for each of the indistinguishable configurations. 2-3 d ðC3þ f Þ ¼ d e ¼ C3þ and ðd C3þ Þf ¼ e f ¼ C3þ . 3 shows that the point group C3v (or 3m ) is isomorphous with the permutation group S(3). 3 Symmetry operators and point groups The symmetry point groups We first describe the proper point groups, P, that is the point groups that contain the identity and proper rotations only. (i) In the cyclic groups, denoted by n or Cn, with n > 1, there is only one axis of rotation Æk and the group elements (symmetry operators) are E and Cnz , or R( n) with ¼ Æ2pk=n, Æk Æ n Àp < p.
8, S42 ¼ C2 . 7. 1-1). 8. Projection diagram showing the operations connected with an S4 axis. È É S4þ C2 S4À E . The identity operator E is always present (whether there is an axis of symmetry or not) and it must always be included once in any list of symmetry operators. The following convention is used in drawing up a list of symmetry operators: where the same configuration may be generated by equivalent symmetry operators we list only the ‘‘simplest form,’’ that is the one of lowest n, with Àp < p, avoiding redundancies.
Iv) The octahedral or cubic group, named 432 or O, consists of the proper rotations that transform a cube or an octahedron into itself. 12 . 13). 12. Alternate vertices of the cube (marked 1, 2, 3, and 4) are the apices of a regular tetrahedron. O1, O2, O3, and O4 are three-fold axes of symmetry. Small crosses show where the C4 axes, OX, OY, and OZ, intersect the cube faces. Oa, Ob, Oc, Od, Oe, and Of are six binary axes. 13. The dodecahedron and the icosahedron are two of the five Platonic solids (regular polyhedra), the others being the tetrahedron, the cube, and the octahedron.