Crystallography & Symmetry
patterns derived from their groups
Concept

Merohedry — where it appears

How much of its own lattice's symmetry a crystal class is missing, measured as the index of the class in the holohedry. That index is exactly the number of twin orientations available, so a class using all its lattice's symmetry cannot twin this way at all.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

The thirty-two, by crystal system. 32 classes in 7 crystal systems. Each column is one crystal system and each cell one class, ordered by the number of operations it holds. Nothing here is tabulated: the classes come from the enumeration and the marking from a character sum over each group.

The holohedry is the ceiling

A crystal never has more point symmetry than its lattice. That single containment decides which system a class belongs to, why there are seven systems and not thirty-two, and why a lattice can be more symmetric than the crystal sitting on it — which is the usual case rather than the exception.

point-groups · Crystal classes Which classes can twin by merohedry, and how many ways. Every crystal class, with the number of twin laws its own lattice offers it. The index of the class in the point group of its lattice is the number of orientations available; 25 of the thirty-two have more than one, and the 7 holohedral classes have exactly one — their crystal already has every symmetry their lattice has, so there is nothing left over to twin by. The names along the right are the old mineralogical ones: hemihedral for half, tetartohedral for a quarter.

Twenty-five of the thirty-two can twin, and seven cannot

The number of twin laws available to a crystal is the index of its class in the point group of its lattice, minus one. Doing that arithmetic for all thirty-two classes takes a moment and produces a census with a sharp edge on it — the seven classes that cannot twin this way are exactly the seven that already use everything their lattice has.

applied · Twinning The twin laws of class 32. The point group of the lattice of class 32 has 24 operations and the class has 6, so it splits into 4 cosets: the crystal itself, and 3 twin laws. Every operation in a block produces the identical second orientation, which is why the block and not the operation is the law.

Quartz has exactly three twin laws, and its lattice is why

Class 32 on a hexagonal lattice has index four, so three twin laws and no more. They turn out to be the three the mineralogists named — Dauphiné, Brazil and the combination of the two — and reading quartz's lattice off its class instead of measuring it would have produced one law where there are three.

applied · Twinning monoclinic: 9 twin laws, 1 of them exact. The twin laws of a monoclinic lattice: a two-fold about the row [uvw] paired with the plane (hkl) it is meant to be a mirror in, kept when the twin index is at most six and the obliquity at most six degrees, which are Friedel's own limits and are a convention rather than a theorem. 9 of the 9 are listed. The index n is how many lattice nodes there are per node the operation restores, computed from the integers and checked against the sublattice built from the plane and the row. The obliquity is the angle between the row and the plane's normal: 1 of these laws have none, and for those the operation restores a sublattice exactly.

The index and the angle a twin misses by

Whether a crystal will twin on a given operation is decided by its lattice, not by its structure. Two numbers decide it: how many lattice nodes there are per node the operation restores, and how far the operation is from being a symmetry at all. Both are computed from integers, and one of them is a fiction that has to be labelled as one.

applied · Twinning

Named alongside it

The objects these essays reach for when they reach for this one.

HolohedryTwin lawCosetCrystal classEnantiomorphismFriedel lawIndexLatticeLattice automorphismObliquityOptical activityPhase transition

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