rotation axis of C 3 . That is, this axis trisects a solid angle π/2 shaped by three
intersecting sides.
It is sometimes hard to visualize or envisage the positional relationship among
planes and straight lines in three-dimensional space. It will therefore be useful to
make a simple kit to help visualize it. Figure 17.14 gives an illustration.
Another typical example having 24 group elements is T d . A molecule of methane
belongs to this symmetry. Table 17.6 collects the relevant symmetry operations and
their (3, 3) matrix representations. As in the case of Fig. 17.12, the matrices show
how a set of vectors (x y z) are transformed according to the symmetry operations.
Comparing it with Fig. 17.12, we immediately recognize that the close relationship
between T d and O exists and that these point groups share notable characteristics.
(i) Both T d and O consist of five conjugacy classes, each of which contains the
same number of symmetry species. (ii) Both T d and O contain a pure rotation group
T as a subgroup. The subgroup T consists of 12 group elements E, 8C 3 , and 3C 2 .
Other remaining twelve group elements of T d are symmetry species related to
reflection; S 4 and σ d . The elements 6S 4 and 6σ d correspond to 6C 4 and 6C 2 of O,
y
x
z
π/3
(1, 1, 0)
(0, 1, 1)
(1, ‒1, 1)
2
2
3
O
Fig. 17.13 Rotation axes of C 2 and C
0
2 along with another rotation axis C 3 in a point group O
6KHHW
6KHHW
6KHHW
Fig. 17.14 Simple kit that helps visualize the positional relationship among planes and straight
lines in three-dimensional space. To make it, follow next procedures: (i) Take three thick sheets of
paper and make slits (dashed lines) as shown. (ii) Insert Sheet 2 into Sheet 1 so that the two sheets
can make a right angle. (iii) Insert Sheet 3 into combined Sheets 1 and 2
17.3 O and T d Groups
659
intersecting sides.
It is sometimes hard to visualize or envisage the positional relationship among
planes and straight lines in three-dimensional space. It will therefore be useful to
make a simple kit to help visualize it. Figure 17.14 gives an illustration.
Another typical example having 24 group elements is T d . A molecule of methane
belongs to this symmetry. Table 17.6 collects the relevant symmetry operations and
their (3, 3) matrix representations. As in the case of Fig. 17.12, the matrices show
how a set of vectors (x y z) are transformed according to the symmetry operations.
Comparing it with Fig. 17.12, we immediately recognize that the close relationship
between T d and O exists and that these point groups share notable characteristics.
(i) Both T d and O consist of five conjugacy classes, each of which contains the
same number of symmetry species. (ii) Both T d and O contain a pure rotation group
T as a subgroup. The subgroup T consists of 12 group elements E, 8C 3 , and 3C 2 .
Other remaining twelve group elements of T d are symmetry species related to
reflection; S 4 and σ d . The elements 6S 4 and 6σ d correspond to 6C 4 and 6C 2 of O,
y
x
z
π/3
(1, 1, 0)
(0, 1, 1)
(1, ‒1, 1)
2
2
3
O
Fig. 17.13 Rotation axes of C 2 and C
0
2 along with another rotation axis C 3 in a point group O
6KHHW
6KHHW
6KHHW
Fig. 17.14 Simple kit that helps visualize the positional relationship among planes and straight
lines in three-dimensional space. To make it, follow next procedures: (i) Take three thick sheets of
paper and make slits (dashed lines) as shown. (ii) Insert Sheet 2 into Sheet 1 so that the two sheets
can make a right angle. (iii) Insert Sheet 3 into combined Sheets 1 and 2
17.3 O and T d Groups
659