Chemistry Reference
In-Depth Information
3.3 There are only three regular tesselations of the plane: triangles, squares, and
hexagons.
3.4 The rotation generates points that are lying on a circle, perpendicular to the ro-
tation. If the rotational angle is not a rational fraction of a full angle, every time
the rotation is repeated, a new point will be generated. To obtain an integer or-
der, the additional requirement is to be added that the original point is retrieved
after one full turn.
3.5 Consider a subgroup
H
⊂
G
such that
|
G
|
/
|
H
|=
2. Then the coset expansion
of
G
will be limited to only two cosets:
G
=
H
+ˆ
gH
Here
g
is a coset generator outside
H
. The subgroup is normal if the right and
left cosets coincide, Since there is only one coset outside
H
, it is required that
ˆ
ˆ
gH
=
H
g
ˆ
Suppose that this equation does not hold. Then this can only mean that there are
elements in
H
such that
h
x
ˆ
=
h
y
g
But then the coset generator must be an element of
H
, which contradicts the
staring assumption.
3.6 Soccer ball:
I
h
. Tennis ball:
D
2
d
. Basketball:
D
2
h
. Trefoil knot:
D
3
.
3.7 The figure (from Wikipedia) shows the helix function for
n
=
1. One full turn
is realized for
t/a
=
2
π
≈
6
.
283. This is a right-handed helix.
