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Crystal Symmetry

Age 16 to 18
Challenge Level

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.