Below is a photo of labradorite - not mine. You really need a movie to see the strange shimmer as you turn a piece of labradorite in the sunlight.

In fact there are 3 kinds of feldspar that separate out slightly as they cool and harden, forming thin alternating layers of two substances:

• The 'peristerite gap' produces layers in feldspars with 2-16% anorthite and the rest albite - these layers create the beauty of moonstone!

• The 'Bøggild gap' produces layers in feldspars with 47-58% anorthite and the rest albite - these are labradorites!

• The 'Huttenlocher gap' produces layers in feldspars with 67-90% anorthite and the rest albite - these are called bytownites.

The physics and math of all this stuff is fascinating.

(2/n)

What I'd like to understand better:

Feldspar crystals are complicated structures built from tetrahedra, and aluminum and silicon have to be distributed among these tetrahedra. This distribution is determined by the relative amounts of potassium, sodium and calcium, and it in turn controls the symmetry of the crystal, which can be either 'monoclinic' or the less symmetrical 'triclinic'.

There's a whole body of work - by Salje, Carpenter, and others - applying Landau's theory of symmetry-breaking phase transitions to map out the space of different possible feldspar crystals!

(3/n)

In case you're wondering about 'monoclinic' versus 'triclinic' crystals, the picture here shows the difference. But the picture doesn't fully capture the symmetry group of an actual crystal - because there's more to a crystal than just a shape of a parallelipiped!

Okay, let's get into the math.

The symmetry group G of a crystal, called a 'space group', fits into an exact sequence

0 → ℤ³ → G → P → 1

where ℤ³ is the group of translational symmetries and P is the group of symmetries that fix a point: the 'point group'. This sequence may or may not split! It splits iff G is a semidirect product of P and ℤ³.

For a triclinic crystal, there are only two possible space groups G, and both are semidirect products. P is either trivial or Z/2, acting by negation.

For monoclinic, there are 3 choices of the point group P as a subgroup of O(3):

• P = ℤ/2 (a single 2-fold rotation)
• P = ℤ/2 (a single reflection)
• P = ℤ/2 × ℤ/2 (generated by a 2-fold rotation and inversion — their product is a reflection).

For each choice of P there are 2 fundamentally different choices of lattice ≅ ℤ³ it can act on. One is made up of copies of the parallelipiped I showed you. The other is twice as dense. So we get 3 × 2 = 6 space groups G that are semidirect products.

But there are 7 other non-split extensions! These other 7 give nontrivial elements of the cohomology group H²(P, ℤ³). Thus, the hardest part of the classification of all 13 monoclinic space groups is essentially the computation of H²(P, ℤ³) for all 6 choices of P and its action on ℤ³.

I knew cohomology rocks. But it turns out cohomology helps classify rocks!

Now, which of these various groups are symmetry groups of feldspars?

(4/n)

Apparently feldspars have just two different symmetry groups:

For the monoclinic feldspars (including sanidine, orthoclase, high-temperature albite), the crystal has a 2-fold rotational symmetry, a mirror plane, and inversion symmetry (𝑥,𝑦,𝑧)↦(−𝑥,−𝑦,−𝑧). The point group is the Klein four-group ℤ/2 × ℤ/2. The lattice has an extra translational symmetry shifting by half a cell diagonal across one face.

For the triclinic feldspars (including microcline, low-temperature albite, anorthite), the only symmetry beyond translation is inversion: the map (𝑥,𝑦,𝑧)↦(−𝑥,−𝑦,−𝑧). So the point group is just ℤ/2. And the lattice is primitive: there are no extra generators of translation symmetry beyond the three edges of the parallelipiped I showed you.

Alas, each of these space groups G is the semidirect products of their point group P and their translation symmetry group ℤ³. So, no interesting cohomology classes show up! (Those show up in crystals with 'screw axes' or 'glide planes'.)

Thus, for you mathematical physicists out there, the main fun thing about feldspars seems to be the phase transitions, especially from the more symmetrical monoclinic feldspars to the less symmetrical triclinic ones! Like in this paper here:

• Ekhard Salje, Application of Landau theory for the analysis of phase transitions in minerals, https://www.researchgate.net/publication/222119979_Application_of_Landau_theory_for_the_analysis_of_phase_transitions_in_minerals

(5/n)