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'Coordinated Crystals' printed from http://nrich.maths.org/
This problem is a useful exercise in visualisation and 3D vectors.
It provides a natural context for the mathematics and has many
extension possibilities. It is an example of a problem where a
clear geometric image really facilitates the work with vectors and
where the use of the scalar product really facilitates the
calculation of the angles.
Initially focus on trying to understand the atomic structure.
Encourage discussion and the drawing of diagrams? Share these.
Which ways of thinking about the atomic structure are the simplest
To understand how close the various atoms are to each other
requires clear thinking. It will be easiest to think in terms of
each atom $A$ surrounded by a 'box' of $B$ atoms, in which case it
will be easier to see which distances, and therefore angles, are
The extension concerning the other crystal configurations is
mathematically very interesting.
You could consider structures well known from chemistry or
encourage students to research the idea following the link from the
What sort of atom lies at the origin?
What is the configuration of all of the $A$ atoms or all of
the $B$ atoms?
What angle is formed between the atom at the origin and its
two closest neighbours?
Consider creating a version of the problem with face-centred
cubic packing, where the first challenge is to determine an
algebraic form of the location of the different atoms.
Focus on the central atom and its nearest neighbours only.