Crystallography is a rich yet confusing source of ideas in mathematics - confusing because there are at least 10 mathematically distinct ways to classify crystals, not even counting the many different notation systems and terminologies.

But let's dip our toe in the water! It's all about lattices and their symmetry groups.

A 'lattice' is a discrete nonempty subset L ⊂ ℝⁿ closed under addition and subtraction. This is the same as a subgroup ℝⁿ of isomorphic to ℤᵏ for some k. k is called the 'rank'. If k = n, we say our lattice has 'full rank'. Henceforth when I say 'lattice', I always mean a lattice of full rank. That's pretty common in math, though crystallographers call it a 'Bravais lattice'.

The 'Bravais group' of a lattice L ⊂ ℝⁿ is the group of all maps f: L → L that preserve distances. It's also the group of all maps f: ℝⁿ → ℝⁿ that preserve distances and map L to itself. When we talk about 'a Bravais group', we mean the Bravais group of some lattice.

In 3 dimensions, there are 14 Bravais groups - up to isomorphism, as abstract groups. You can see how they work in the picture, or in more detail here:

https://en.wikipedia.org/wiki/Bravais_lattice#In_3_dimensions

But this is just the start of classifying crystals, because there's more to an actual crystal than a mere lattice. Some *subgroups* of Bravais groups are also important!

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In the town of Quartzsite I picked up a beautiful chunk of labradorite. This mineral creates an eerie blue shimmer in the sunlight - a phenomenon called 'labradorescence'. Reading up on it, I discovered it's a form of feldspar. 60% of the Earth's crust is feldspar, and I know so little about this stuff!

Turns out there are 3 fundamental kinds of feldspar:

• orthoclase is potassium aluminosilicate
• albite is sodium aluminosilicate
• anorthite is calcium aluminosilicate

Then there are lots of feldspars that contain different amounts of potassium, sodium and calcium. We get a triangle of feldspars with orthoclase, albite and anorthite at the corners.

But not all points in this triangle are possible! There's a big region called the 'miscibility gap', where as you cool the molten mix it separates out!

And there are also subtler problems. When you cool down the feldspar called labradorite, it separates out a little, forming tiny layers of two different kinds of stuff. When the thickness of these layers is the wavelength of visible light, you get a weird optical effect: labradorescence!

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Food for mathematicians.

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