respectively. That is, successive operations of S 4 cause similar effects to those of C 4
of O. Meanwhile, successive operations of σ d are related to those of 6C 2 of O.
Let us imagine in Fig. 17.15 that a regular tetrahedron is inscribed in a cube. As
an example of symmetry operations, suppose that three pairs of planes of σ d (six
planes in total) are given by equations of x ¼ Æ y and y ¼ Æ z and z ¼ Æ x. Their
Hesse’s normal forms are represented as
1
ffiffi ffi
2
p x Æ y
ð
Þ¼0,
ð17:49Þ
1
ffiffi ffi
2
p y Æ z
ð
Þ¼0,
ð17:50Þ
1
ffiffi ffi
2
p z Æ x
ð
Þ¼0:
ð17:51Þ
Then, a dihedral angle α of the two planes is given by
cos α ¼
1
ffiffi ffi
2
p ∙
1
ffiffi ffi
2
p ¼
1
2
or
ð17:52Þ
cos α ¼
1
ffiffi ffi
2
p ∙
1
ffiffi ffi
2
p À
1
ffiffi ffi
2
p ∙
1
ffiffi ffi
2
p ¼ 0:
ð17:53Þ
That is, α ¼ π/3 or α ¼ π/2. On the basis of the discussion of Sect. 17.2, the
intersection of the two planes must be a rotation axis of C 3 or C 2 . Once again, in the
case of C 3 the intersection is a straight line connecting the origin and a vertex of the
cube. This can readily be verified as follows: For instance, two planes given by x ¼ y
and y ¼ z make an angle π/3 and produce an intersection line x ¼ y ¼ z. This line, in
turn, connects the origin and a vertex of the cube.
If we choose, e.g., two planes x ¼ Æ y from the above, these planes make a right
angle and their intersection must be a C 2 axis. The three C 2 axes coincide with the x-,
y-, and z-axes. In this light, σ d functions similarly to 6C 2 of O in that their
combinations produce 8C 3 or 3C 2 . Thus, constitution and operation of T d and
O are related.
Let us more closely inspect the structure and constitution of O and T d . First we
construct mapping ρ between group elements of O and T d such that
ρ : g 2 O ⟷ g
0
2 T d , if g 2 T,
ρ : g 2 O ⟷ À g
0
Þ
À1
2 T d , if g =
2 T:
À
In the above relation, the minus sign indicates that with an inverse representation
matrix R must be replaced with ÀR. Then, ρ(g) ¼ g
0 is an isomorphic mapping. In
fact, comparing Fig. 17.12 and Table 17.6, ρ gives identical matrix representation for
O and T d . For example, taking the first matrix of S 4 of T d , we have
17.3 O and T d Groups
661
of O. Meanwhile, successive operations of σ d are related to those of 6C 2 of O.
Let us imagine in Fig. 17.15 that a regular tetrahedron is inscribed in a cube. As
an example of symmetry operations, suppose that three pairs of planes of σ d (six
planes in total) are given by equations of x ¼ Æ y and y ¼ Æ z and z ¼ Æ x. Their
Hesse’s normal forms are represented as
1
ffiffi ffi
2
p x Æ y
ð
Þ¼0,
ð17:49Þ
1
ffiffi ffi
2
p y Æ z
ð
Þ¼0,
ð17:50Þ
1
ffiffi ffi
2
p z Æ x
ð
Þ¼0:
ð17:51Þ
Then, a dihedral angle α of the two planes is given by
cos α ¼
1
ffiffi ffi
2
p ∙
1
ffiffi ffi
2
p ¼
1
2
or
ð17:52Þ
cos α ¼
1
ffiffi ffi
2
p ∙
1
ffiffi ffi
2
p À
1
ffiffi ffi
2
p ∙
1
ffiffi ffi
2
p ¼ 0:
ð17:53Þ
That is, α ¼ π/3 or α ¼ π/2. On the basis of the discussion of Sect. 17.2, the
intersection of the two planes must be a rotation axis of C 3 or C 2 . Once again, in the
case of C 3 the intersection is a straight line connecting the origin and a vertex of the
cube. This can readily be verified as follows: For instance, two planes given by x ¼ y
and y ¼ z make an angle π/3 and produce an intersection line x ¼ y ¼ z. This line, in
turn, connects the origin and a vertex of the cube.
If we choose, e.g., two planes x ¼ Æ y from the above, these planes make a right
angle and their intersection must be a C 2 axis. The three C 2 axes coincide with the x-,
y-, and z-axes. In this light, σ d functions similarly to 6C 2 of O in that their
combinations produce 8C 3 or 3C 2 . Thus, constitution and operation of T d and
O are related.
Let us more closely inspect the structure and constitution of O and T d . First we
construct mapping ρ between group elements of O and T d such that
ρ : g 2 O ⟷ g
0
2 T d , if g 2 T,
ρ : g 2 O ⟷ À g
0
Þ
À1
2 T d , if g =
2 T:
À
In the above relation, the minus sign indicates that with an inverse representation
matrix R must be replaced with ÀR. Then, ρ(g) ¼ g
0 is an isomorphic mapping. In
fact, comparing Fig. 17.12 and Table 17.6, ρ gives identical matrix representation for
O and T d . For example, taking the first matrix of S 4 of T d , we have
17.3 O and T d Groups
661