Chemistry Reference
In-Depth Information
number of atoms! Explain your reasoning. What is the smallest molecule with
no symmetry at all?
3.3 The 2D analogue of a polyhedron is a polygon. In a regular polygon all vertices,
edges, and angles between adjacent edges are identical. A 2D plane can be
tessellated in identical regular polygons, which then form a covering of the
plane. In how many ways can this be performed?
3.4 Why is the order of a rotational axis of a polyhedral object always an integer?
3.5 Prove that a halving subgroup is always a normal subgroup.
3.6 Determine the point group of a soccer ball, a tennis ball, a basketball, and a
trefoil knot.
3.7 The parameter equations defining a helix in Cartesian space are given by
a
cos
nt
a
x(t)
=
a
sin
nt
a
y(t)
=
z(t)
=
t
Here
a
is the radius. Is this helix left- or right-handed? Write down the param-
eterization of its enantiomer. The symmetry of a helix is based on a screw axis,
which corresponds to a translation in
t
. It is composed of a translation along the
z
-direction with a concomitant rotation in the
xy
-plane. Now decorate the helix
with atoms at points
t
k
/a
2
πk/m
, where
k
and
m
are integers. Determine
the screw symmetry of this molecular helix. If
n/m
is irrational, the helix is
noncommensurate
. Will it still have a symmetry in this case?
=
References
1. Benfey, O.T. (ed.): Classics in the Theory of Chemical Combination, pp. 151-171. Dover Pub-
lications, New York (1963)
2. Wunderlich, J.A., Lipscomb, W.N.: The structure of
B
12
H
2
−
12
ion. J. Am. Chem. Soc.
82
, 4427
(1960)
