1
5 The groups and 2D. The 10 point groups.
The groups!
We have now come to a study of the variables as functions in space. Their
permutations under the constraints of symmetry and translation give the groups
that are so important in the foundations of mathematics, in the natural sciences
and also in daily life. The illumination of these symmetry properties using the
exponential scale also lays the ground for a study using finite periodic
functions. The concept of lines as building principles in 2D mathematics is
again found to be useful.
5.1 Permutations in 2D
We have regularly touched the subject of symmetry to make it familiar to you.
Now it comes with full power – the mirror, the rotation and the glide all under
the constraint of periodic translation. You will learn during the journey below.
We show the simple possible permutations and also the symmetry symbol that
describes the ten point groups. The numbering is after [1].
I
x,y
p1
II
x,y; -x,-y;
p2
III
x,y; -x,y;
pm
VI
x,y; -x,-y; -x,y; x,-y;
pmm
X
x,y; -x,-y; y,-x; -y,x;
p4
XI
x,y; -x,-y; y,-x; -y,x; y,x; -y,-x; x,-y; -x,y;
XIII
x,y; -y,x-y: y-x,-x;
XIV
x,y; -y,x-y; y-x,-x; -y,-x; x,x-y; y-x,y;
p3m1
XV1
x,y; -y,x-y: y-x,-x; -x,-y; y,y-x: x-y,x;
p6
XV11 x,y; -y,x-y; y-x,-x; y,x; -x,y-x; x-y,-y;
-x,-y; y,y-x; x-y,x; -y,-x; x,x-y; y-x,y;
p6m
p4m
p3
2
In order to build a structure unit translations are added to these positions. If
half such a unit is added as translation the complete symmetries of the repeated
patterns in the plane are obtained. This symmetry operation belongs to
periodicity and is called ’glide’. Walking is a typical example as shown in
figure 5.1.1. Another is centering as shown in figure 5.1.2. The so called unit
cell is obvious-the translation is unity- and there are the half unit translations
that gives the centering. That is that every position is repeated by a fraction of
the unit-or exactly a 1/2 and a 1/2.We give the possibilities below.
IV
x,y; -x,1/2+y;
pg
V
x,y; -x,y; x+1/2,y+1/2; -x+1/2,y+1/2;
cm
VII x,y; -x,-y; 1/2+x,-y; 1/2-x,y;
VIII
pmg
x,y; -x,-y; 1/2+x,1/2-y; 1/2-x,1/2+y;
pgg
IX x,y; -x,-y; 1/2+x,1/2-y; 1/2-x,1/2+y;
- x,y; x,-y; 1/2-x,1/2+y; 1/2+x,1/2-y;
XII
cmm
x,y; -x,-y; y,-x; -y,x; 1/2 -x,1/2+y;
1/2+ x,1/2-y; 1/2 -y,1/2-x; 1/2+ y, 1/2+x;
p4g
XV x,y; -y,x-y; y-x,-x; y,x; -x,y-x; x-y,-y;
p31m
p1 and p2 are described in a general coordinate system with an oblique angelin descriptions here and in chapter 6 we take the special case when the angel is
right.
The five groups p3-p6m are described above in a hexagonal system. As we use
a cartesian system we give below the permutations accordingly.
XIII
x,y; -
1
2
x-
x, y; x, -y;-
XIV
-
1
2
x+
3
2
1
2
3
2
2
3
x-
y, -
3
y,
2
3
2
y,
x-
x-
3
2
1
2
1
1
3
3
1
y; - x +
y, x - y;
2
2
2
2
2
x1
1
3
3
1
y; - x y, x + y;
2
2
2
2
2
p3
1
3
3
1
y; - x +
y,
x + y;
2
2
2
2
p3m1
3
1
x, y;- x, y;-
2
XV
-
1
2
3
x+
y, -
2
x, y; -x, -y;-
XV1
1
2
x-
3
1
3
2
1
x, y; -x, -y;-
2
1
XVII
2
x-
3
2
3
2
2
1
x+
2
3
1
y;
2
1
1
3
3
1
y; x +
y,
x - y;
2 2
2
2
2
x-
2
2
x-
1
1
2
3
2
3
y, -
1
y; -
2
x+
2
3
3
x-
3
1
2
y;
3
y, -
2
p31m
2
x-
1
2
y;
p6
1
y; x +
y, x + y;
2
2
2
2
3
y,
2
1
1
3
y,
2
x+
x-
2
1
x-
2
x3
y,
3
3
y,
2
x-
2
y,
2
3
x-
1
y;
2
x+
2
3
x-
3
1
y;-
2
3
2
1
1
2
y, -
3
x+
3
2
2
x+
1
1
y, -
3
2
x-
1
y;
2
y;
2
3
p6m
3
1
-x, y;x, -y; x y, x - y; x +
y,
x - y;
2
2
2
2 2
2
2
2
-
1
2
x-
3
2
y, -
3
2
x+
1
y; -
2
Figure 5.1.1 Walking-a glide operation.
1
2
x+
3
2
y,
3
2
x+
1
y;
2
Figure 5.1.2 Centering.
5.2 The point groups and the exponential scale. Geometrical illustration of
symmetry.
Regarding symmetry as consisting of lines for the planar groups, or planes in
3D, we demonstrate the geometrical meaning of these permutations by using
the exponential scale and formulate the equations below. We repeat each
symmetry operation for each group.
We use as variables the pair of x,2y to describe the 2D geometrical pictures.
4
We start with
the oblique groups.
p1 No.1 x,y;
The structure from equation 5.2.1 is a line as in figure 5.2.1 and this first group
has no symmetry at all.
ex+ 2y = 100
5.2.1
Figure 5.2.1 Group p1.
p2 No.2 x,y; -x,-y;
This group has only one element of symmetry, that is a 2-fold axis of rotation.
And from the symmetry operation the equation in 5.2.2 gives two lines as in
figure 5.2.2, which via origin are related by a rotation.
ex+ 2y + e-x- 2y = 100
5.2.2
Figure 5.2.2 Group p2.
The rectangular groups
pm No.3 x,y; -x,y;
The equation in 5.2.3 contains the mirror needed from the symmetry operation
which we also see in figure 5.2.3 in form of two lines, meeting and bending to
make the mirror.
5
ex+ 2y + e-x+ 2y = 3000
5.2.3
Figure 5.2.3 Group pm
pmm No.6 x,y; -x,-y; -x,y; x,-y;
As Mathematica uses a cartesian system the reason for the the combination
x,2y is now obvious. There are four lines that meet in a rhomb in figure 5.2.4
and with the exponential scale in equation 5.2.4 they form the two mirror
planes.
ex+ 2y + e-x- 2y + ex-2y + e- x+ 2y = 3000
5.2.4
Figure 5.2.4 Group pmm
The square groups.
p4 No.10 x,y; -x,-y; y,-x; -y,x;
We have the equation 5.2.5 and the four lines form a square as shown in figure
5.2.5. And there is a 4-fold axis of rotation.
ex+ 2y + e-x- 2y + e2x- y + e-2x + y = 106
Figure 5.2.5 Group p4.
5.2.5
6
p4m No.11 x,y; -x,-y; y,-x; -y,x; y,x; -y,-x; x,-y; -x,y;
The equation in 5.2.6 gives eight planes in the figure 5.2.6 that contains the
four fold axis as well as mirror planes.
ex+ 2 y + e-x - 2y + e2x - y + e-2x + y +
e2x + y + e-2x - y + ex -2 y + e- x +2 y = 1012
5.2.6
Figure 5.2.6 Group p4m.
The hexagonal groups.
p3 No.13 x,y;-
1
3
3
1
1
3
3
1
xy,
x - y;- x +
y,x - y;
2
2
2
2
2
2
2
2
And the equation in 5.2.7 gives three lines that meet and form a triangle in
figure 5.2.7. With a 3-fold axis of rotation.
e
x+ y
1
3
1
3
1
3
1
3
(- x - y)+( - y+ x)
(- x + y)+(- y - x)
4
2
2
2
2
2
2
+e 2
+e 2
= 10
Figure 5.2.7 Group p3.
5.2.7
7
x, y; -x, y;-
p3m1 No.14
1
2
x+
3
2
1
2
3
y,
3
x-
2
2
y,
3
2
x-
1
2
y; -
1
2
x+
3
2
y, -
3
2
x-
1
2
y;
1
1
3
3
1
y; x y, x - y;
2 2
2
2
2
x-
And figure 5.2.8 formed from equation 5.2.8 shows the symmetry of the group.
†
e
e
x+ y
(- 1 x- 3 y)+(- 1 y+ 3 x)
(- 1 x+ 3 y)+(- 1 y- 3 x)
2
2
2
2
2
2
2
+e
+e 2
+
-x+ y
( 1 x+ 3 y)+(- 1 y+ 3 x)
( 1 x- 3 y)+(- 1 y- 3 x)
8
2
2
2
2
2
2
2
+e
+e 2
= 10
†
x, y; x, -y;-
p31m No.15
-
1
2
x+
3
2
1
2
3
x-
y, -
2
3
2
y,
x-
3
2
x-
1
1
3
3
1
y; - x y, x + y;
2
2
2
2
2
1
1
3
3
1
y; - x +
y,
x + y;
2
2
2
2
2
And figure 5.2.8 formed from equation 5.2.8 shows the symmetry of the group.
e
x+ y
1
3
1
3
1
3
1
3
(- x - y)+( - y+ x)
(- x - y)+( y - x)
2
2
2
2
2
2
+e 2
+e 2
+
1
3
1
3
1
3
1
3
(- x + y)+( - y- x)
(- x + y)+( y + x)
xy
2
2
2
2
2
2
e
+e 2
+e 2
= 108
5.2.8
8
Figure 5.2.8 Group p3m1.
x, y;-x, -y; -
p6 No.16
1
2
x-
3
2
y,
1
x-
2
3
2
x+
3
2
y,
3
2
x-
1
2
y; -
1
2
x+
3
2
y, -
3
2
x-
1
2
y;
1
1
3
3
1
y; x +
y,x + y;
2 2
2
2
2
The equation 5.2.9 form a hexagon after the exponential scale in figure 5.2.9.
e
x+y
1
3
1
3
1
3
1
3
( - xy )+( - y+
x)
( - x+
y )+(- yx)
2
2
2
2
2
2
+e 2
+e 2
+
1
e
- x- y
3
1
3
1
3
1
3
( xy )+( y+
x)
( x+
y) +( y x)
8
2
2
2
2
2
2
+e 2
+e 2
= 10
Figure 5.2.9 Group p6
5.2.9
9
x, y; -x, -y;1
p6m No.16
2
x-
3
2
y,
1
2
3
x3
2
2
x+
y,
3
2
x-
1
2
y; -
1
2
x+
3
2
y, -
3
2
x-
1
2
y;
1
1
3
3
1
y; x +
y, x + y;
2 2
2
2
2
1
3
3
1 1
3
3
1
-x, y;x, -y; x y, x - y; x +
y,
x - y;
2
2
2
2 2
2
2
2
-
1
2
x-
3
2
y, -
3
2
x+
1
2
y;-
1
2
x+
3
2
y,
3
2
x+
1
2
y;
This gives a lengthy equation in 5.2.10 that gives an impressive polygon
in figure 5.2.10. That also contains the symmetry of this group.
e
e
e
e
x+y
1
3
1
3
1
3
1
3
( - xy )+( - y+
x)
( - x+
y )+(- yx)
2
2
2
2
2
2
2
2
+e
+e
+
- x- y
1
3
1
3
1
3
1
3
( xy )+( y+
x)
( x+
y) +( y x)
2
2
2
2
2
2
2
2
+e
+e
+
- x+ y
1
3
1
3
1
3
1
3
( xy )+(- yx)
( x+
y )+(- y +
x)
2
2
2
2
2
2
+e 2
+e 2
+
x- y
= 10
5.2.10
1
3
1
3
1
3
1
3
(- xy) +( yx)
(- x+
y) +( y+
x)
2
2
2
2
2
2
+e 2
+e 2
30
Figure 5.2.10 Group p6m
References 5
1 INTERNATIONAL TABLES for X-RAY CRYSTALLOGRAPHY, The
Kynoch Press, 1969.