48
3 Groups
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
x(t) = a cos
nt
a
y(t) = a sin
nt
a
z(t) = t
Here a is the radius. Is this helix left- or right-handed? Write down the parameterization 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 = 2πk/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 Publications, 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)
3 Groups
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
x(t) = a cos
nt
a
y(t) = a sin
nt
a
z(t) = t
Here a is the radius. Is this helix left- or right-handed? Write down the parameterization 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 = 2πk/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 Publications, 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)