# Clear as crystal

## Problem

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?

## Teachers' Resources

### Why do this problem

Many NRICH tasks have been designed with group work in mind. Here we have gathered together a collection of short articles that outline the merits of collaborative work, together with examples of teachers' classroom practice.

### Group working

Explain the task and make it clear that everyone needs to be ready to share what they did with the rest of the class at the end of the session.

You may want to make Zome, calculators, spreadsheets, graphing software, squared or graph paper, poster paper, and coloured pens available for the Resource Manager in each group to collect.

While groups are working, label each table with a number or letter on a post-it note, and divide the board up with the groups as headings. Listen in on what groups are saying, and use the board to jot down comments and feedback to the students about the way they are working together. This is a good way of highlighting the mathematical behaviours you want to promote, particularly with a
challenging task such as this.

You may choose to focus on the way the students are co-operating or the focus might be mathematical.

Make sure that while groups are working they are reminded of the need to be ready to present their findings at the end, and that all are aware of how long they have left.

We assume that each group will record their diagrams, reasoning and generalisations in preparation for reporting back. There are many ways that groups can report back. Here are just a few suggestions:

- Every group is given a couple of minutes to report back to the whole class. Students can seek clarification and ask questions. After each presentation, students are invited to offer positive feedback. Finally, students can suggest how the group could have improved their work on the task.
- Everyone's posters are put on display at the front of the room, but only a couple of groups are selected to report back to the whole class. Feedback and suggestions can be given in the same way as above. Additionally, students from the groups which don't present can be invited to share at the end anything they did differently.
- Two people from each group move to join an adjacent group. The two "hosts" explain their findings to the two "visitors". The "visitors" act as critical friends, requiring clear mathematical explanations and justifications. The "visitors" then comment on anything they did differently in their own group.

### Possible approach

### Key questions

### Possible extension

In considering statement g), challenge students to find an example where the two transformations are of different types, for example, combining a translation with a reflection to form a glide reflection.

### Possible support