Crystallography & Symmetry
patterns derived from their groups
Generator

p3, twinned

p3, twinned
p3, twinned. p3 twinned by a rotation. To the left of the composition line the motif sits where p3 puts it; to the right every copy has been carried over by the twin law, which is one of the 3 operations the hexagonal lattice has and p3 does not. 24 images on the left, 24 on the right, and the lattice runs through the line unbroken — which is exactly why a twinned crystal looks like a single one.

p3 twinned by a rotation. To the left of the composition line the motif sits where p3 puts it; to the right every copy has been carried over by the twin law, which is one of the 3 operations the hexagonal lattice has and p3 does not. 24 images on the left, 24 on the right, and the lattice runs through the line unbroken — which is exactly why a twinned crystal looks like a single one.

7 essays call twin-plate. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about. Every one of this site's 393 essays names its parameters at the call site, which the standard pass of 2026-08-09 established and param-floor holds.

Where it is called

Changing this generator changes every one of these figures.

p3, twinned. p3 twinned by a rotation. To the left of the composition line the motif sits where p3 puts it; to the right every copy has been carried over by the twin law, which is one of the 3 operations the hexagonal lattice has and p3 does not. 24 images on the left, 24 on the right, and the lattice runs through the line unbroken — which is exactly why a twinned crystal looks like a single one. Symmetry at work

A twin is a symmetry the lattice has and the crystal does not

Two orientations of one structure, grown together across a boundary the lattice runs straight through. The operation relating them cannot be a symmetry of the crystal, or there would be nothing to see, and it must be a symmetry of the lattice, or the boundary would be a crack — which leaves exactly a coset, and a short computable list.

 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. Symmetry at work

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.

 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. Symmetry at work

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.

 Σ5: two square lattices at 36.87°. Two square lattices, one turned through 36.87° about a shared point. At this angle one point in 5 lands exactly on a point of the other lattice — 29 of the 149 drawn — and those shared points are themselves a lattice, the coincidence site lattice, of index 5. The angle comes from tan(θ/2) = 1/3, and Σ is the odd part of 3² + 1² = 10. Nothing here is measured: whether a point is shared is decided by an integer congruence. Symmetry at work

Turn a lattice against itself and almost nothing lines up

Two copies of one lattice rotated about a shared point share that point and, at almost every angle, no other. At a discrete set of angles they share a whole sublattice — one point in three, or five, or seven — and a grain boundary built on such an orientation costs a fraction of what a general one costs, because a fraction of the atoms are already where both sides want them.

 5 units of 70.53°: 7.36° left. 5 tetrahedral units of face-centred cubic metal, each the mirror image of its neighbour in a {111} plane, arranged about a common ⟨110⟩ edge. The angle between two such planes is arccos(1/3) = 70.53°, computed from the plane normals rather than quoted, and 5 of them come to 352.64°. The shaded sector is what is left over: 7.36°, or 2.04 per cent of a full turn, which must be taken up by strain, by a gap, or by a defect along the axis. What a lattice forbids

Five copies, and the gap they leave

Gold, silver and silicon grow particles with a five-fold axis down the middle, out of a lattice that forbids one. Nothing is violated: five tetrahedral pieces of ordinary face-centred metal, each the mirror image of its neighbour, come to three hundred and fifty-two and a half degrees rather than three hundred and sixty — and the seven degrees left over have to go somewhere.

 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. Symmetry at work

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.

 The twin fraction, recovered from a moment and nothing else (12 atoms). A structure of 12 atoms twinned at each of 6 fractions, with the second moment of its intensity distribution measured and the fraction solved back out of it. The recovery is within a few hundredths as far as thirty per cent — 4 rows here — and 4 of the 6 fractions get a number at all. Beyond thirty per cent the relation flattens: the derivative of 2α(1−α) vanishes at a half, the two roots meet, and a small error in the moment becomes a large one in the fraction. Where the sampled moment falls below 1.5 the quadratic has no real root and the estimate refuses rather than clamping, which is why a nearly perfect twin is the hard case in practice rather than the easy one. How it is known

A twin hides in the statistics

A twinned crystal scatters as two orientations at once and the detector cannot separate them. What arrives is a sum of two intensities — and adding two independent quantities narrows a distribution, which is a signature no model of the structure is needed to read.

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