You may also like

problem icon

Ball Bearings

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

problem icon

Air Routes

Find the distance of the shortest air route at an altitude of 6000 metres between London and Cape Town given the latitudes and longitudes. A simple application of scalar products of vectors.

problem icon

Epidemic Modelling

Use the computer to model an epidemic. Try out public health policies to control the spread of the epidemic, to minimise the number of sick days and deaths.

Crystal Symmetry

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Why do this problem?

This problem gives students useful practice in working with vectors and matrices, in the context of operation which preserve the structure of a crystal. Students have the opportunity to use their geometrical and algebraic reasoning skills to answer the questions.

Possible approach

Start by sharing with students the coordinate definition of the crystal structure, and ask them to make sketches to show what the structure looks like.

Discuss what might be meant by "preserving the structure of the crystal" and ask students to come up with some suggestions of transformations represented by constant matrices or constant vectors which could be used to transform the crystal while preserving its structure.

Once students have had some opportunity to consider suitable transformations, set them the problem of working out which of the given matrices and vectors are suitable candidates. Challenge them to explain geometrically as well as algebraically what happens when each matrix is applied or each vector is added.

Key questions

What do we mean when we say the structure of the crystal was preserved by a transformation?
What transformations will always/sometimes/never preserve a crystal structure?

Possible extension

Clear as Crystal poses challenging questions about transformations and crystal symmetries.

Possible support

Coordinated Crystals gives an introduction to the idea of representing crystal lattices using coordinate geometry.