Crystallography & Symmetry
patterns derived from their groups
Concept

Tolerance — where it appears

A threshold below which two numbers are treated as equal, which every claim decided in integers avoids needing. The moment one is introduced the answer stops being a fact about the structure and becomes a fact about the threshold.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

p4m, displaced and then measured. Every atom of a p4m pattern moved by up to 1.2 per cent of a cell edge, and the resulting positions examined at 26 tolerances. At zero tolerance only the identity survives, so the structure has no exact symmetry whatever. Between 0.021 and 0.094 the count sits at 8, which is the group that was displaced. Above that it climbs to 15, accepting operations no version of this pattern has. The climb is not even steady: at 4 of the 25 steps the count falls as the tolerance is loosened, because operations accepted separately at one threshold merge into one at the next. The correct answer is a step on a staircase and nothing in the coordinates says which step.

Near-symmetry, and the tolerance that is not here

Every claim on this site is decided by integer arithmetic, so no threshold is ever chosen. Measured coordinates do not arrive that way, and the moment a tolerance is introduced the answer stops being a fact about the structure and becomes a fact about the threshold.

diffraction · Accidental symmetry How often a dot gives the wrong group. Every position on a grid inside the cell, tried as a single-dot motif for each of the seventeen groups. The bar is how often the resulting pattern turned out to have more symmetry than the group it was made with — so the caption would have been wrong and nothing about the picture would have shown it.

The Alhambra question

Textbooks say the Alhambra contains all seventeen wallpaper groups. Careful analysts of the same building have counted eleven, thirteen, fourteen and seventeen — and the disagreement is not about the mathematics but about what counts as an instance.

classification · Ornament 13 of one on 12 of the other. Two rows of atoms whose spacings are in the ratio 1.042. Every 12 cells of the substrate come to within 3.97 per cent of 13 cells of the film, so the two are nearly in register at those points and out of register between them. There is no exact coincidence anywhere, and there cannot be: exact coincidence needs the ratio to be rational, and no measured ratio is.

Two different lattices never coincide, and the question becomes how nearly

Grow one crystal on another and their spacings are in a ratio that no measurement ever makes rational, so exact coincidence is unavailable in principle. What is left is the best rational approximation inside a tolerable repeat — a quantity that jumps rather than drifts as the ratio changes, and whose acceptability is decided by elasticity rather than by arithmetic.

applied · Interfaces The figure of merit a supercell always beats. One line list, indexed on the true cell and on five multiples of it. The mean discrepancy is not merely similar down the column, it is identical to every digit: the supercell's grid of allowed Q values contains the true cell's grid exactly, so each line lands on precisely the same place and misses by precisely the same amount. Any figure of merit built on agreement alone therefore returns one number for the whole family, and cannot prefer the true cell. What falls is the last column, and the only thing in it that the fit does not already contain is the count of lines the cell says should have been seen.

The figure of merit a supercell always beats

Indexing a powder pattern returns a ranked list rather than an answer, and the ranking needs a number. The obvious number — how well the cell accounts for the lines — is exactly the number a supercell cannot lose on, because the supercell's grid contains the true cell's grid and the discrepancies are identical to every digit. What has to be paid for is the lines nobody saw.

applied · Indexing

Named alongside it

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

Accidental symmetryThe AlhambraCoincidence site latticeColour symmetryContinued fractionEpitaxyFigure of meritGolden ratioGroup frequencyIncommensurateIndexingMisfit

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