s
. .
s
a
Figure 4.1: Rotational symmetry of an isosceles, non-equilateral
triangle. The operation a is rotation through 180◦ about the
indicated axis. The point s belongs to the orbit of s.
From: Modern Mathematical Methods for Physicists and Engineers, by C. D. Cantrell © 2000 Cambridge University Press
R2π/3 r
R2π/3
r
R4π/3 r
Figure 4.2: A G-set for the group C3 .
From: Modern Mathematical Methods for Physicists and Engineers, by C. D. Cantrell © 2000 Cambridge University Press
2
1
R2π/6
3
4
0
5
Figure 4.3: Generation of the cyclic group C6 .
From: Modern Mathematical Methods for Physicists and Engineers, by C. D. Cantrell © 2000 Cambridge University Press
C2
C2
C2
Figure 4.4: Rotational symmetries of a rectangular parallelepiped.
From: Modern Mathematical Methods for Physicists and Engineers, by C. D. Cantrell © 2000 Cambridge University Press
b
c
f
d
a
Figure 4.5: Rotational symmetries of an equilateral triangle.
From: Modern Mathematical Methods for Physicists and Engineers, by C. D. Cantrell © 2000 Cambridge University Press
C2
C3
Figure 4.6: Rotational symmetries of a regular tetrahedron.
From: Modern Mathematical Methods for Physicists and Engineers, by C. D. Cantrell © 2000 Cambridge University Press
C4
C2
C3
Figure 4.7: Rotational symmetries of a regular octahedron.
From: Modern Mathematical Methods for Physicists and Engineers, by C. D. Cantrell © 2000 Cambridge University Press
b
3
2
a
.
c
4
1
Figure 4.8: Rotational symmetry operations of a rectangle.
From: Modern Mathematical Methods for Physicists and Engineers, by C. D. Cantrell © 2000 Cambridge University Press
C4,C2= C42
C'2
C''2
Figure 4.9: Rotational symmetry of a square prism. The operations
shown generate the group D4.
From: Modern Mathematical Methods for Physicists and Engineers, by C. D. Cantrell © 2000 Cambridge University Press
X'
Y
Y'
X
Figure 4.10: Transformation of a function on R2 through rotation,
illustrating Eq. (4.175).
From: Modern Mathematical Methods for Physicists and Engineers, by C. D. Cantrell © 2000 Cambridge University Press
2
b
c
f
d
3
1
a
Figure 4.11: Rotational and permutational symmetries of an
equilateral triangle. The vertexes have been labeled 1, 2 and 3.
From: Modern Mathematical Methods for Physicists and Engineers, by C. D. Cantrell © 2000 Cambridge University Press
3
2
c.
a
1
4
b
Figure 4.12: Rotational and permutational symmetries of a
rectangle. The vertexes have been labeled 1, 2, 3 and 4.
From: Modern Mathematical Methods for Physicists and Engineers, by C. D. Cantrell © 2000 Cambridge University Press