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Crystals can be represented mathematically by infinite lattices of points occupied by atoms or ions.The symmetry properties of crystal lattices are physically very important and mathematically very fascinating. In this problem we investigate the symmetries of these lattices mathematically.

A crystal symmetry operation is a transformation which when applied to the vector positions of the ions causes the following:

1. The points in space occupied by atoms or ions before and after the transformation are identical.

2. Each atom or ion in the crystal shifts onto the position of an identical atom or ion.

3. The distance between any neighbouring pairs of atoms or ions is unchanged before and after the transformation.

Which of the following are sometimes always or never true? If always or never, give a proof. If sometimes, give an example where it works and an example where it does not. You might want to focus your attention on BCC or FCC packing, although feel free to invent mathematical lattices of your own.

a) A rotation about a given point is a symmetry.

b) A reflection through a plane which does not pass through any of the lattice points is a symmetry.

c) A shear which maps the lattice onto itself is a symmetry.

d) For a crystal lying on an integer lattice, the translation by $(l/2, m/2, n/2)$ is a symmetry, where $l, m, n$ are integers.

e) Repeated application of the same symmetry will eventually restore the crystal to its original state.

f) If $T_1({\bf v})$ and $T_2({\bf v})$ are both symmetry operations then the combination $T_1(T_2{\bf v}) $ is a symmetry operation.

g) If neither $T_1({\bf v})$ nor $T_2({\bf v})$ is a symmetry operation then $T_1(T_2({\bf v}))$ is not a symmetry operation.

h) Application of a symmetry operation leaves at least one point fixed.

i) Application of a symmetry operation leaves exactly $3$ points fixed.

Can you invent any mathematical lattices with unusual symmetry properties?