Crystallography & Symmetry
patterns derived from their groups
Concept

Homotopy — where it appears

A classification of maps by whether one can be deformed continuously into another. The defects a medium admits are the classes of maps from a loop or a sphere into its space of order parameter values, so a question about defects and a question about homotopy are the same question asked twice.

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

The kinds of line defect each breaking allows. The eleven proper crystal classes, each with the order of the binary group that covers it, the number of conjugacy classes of that group other than the identity — which is the number of kinds of line defect — and whether the group commutes. 6 of the eleven do not, and in those media two defect lines cannot pass through each other without leaving a third line behind.

The defect that needs two laps

Which defects a medium can have is not a fact about the medium. It is a fact about the space its order parameter lives in, and for a rotational symmetry broken down to a point group that space has a fundamental group twice the size of the point group. The kinds of line defect are its conjugacy classes — and in six of the eleven cases they do not commute, which means two defect lines cannot pass through each other.

applied · Defects Seven fields and the number that counts each one. The degree of each field, computed by triangulating the sphere drawn round the defect, mapping every vertex, and adding the signed areas of the image triangles. The total is 4π times the degree, and the integral column is that total divided by 4π before rounding. Each is read on three successively finer meshes and required to give the same integer on all three, because a mesh too coarse for its field does not produce a noisy answer — it produces a confident wrong one.

The point defect whose charge has no sign

A line defect is read on a loop; a point defect is read on a sphere, and the number that comes off the sphere is a degree. In a nematic that degree is an integer whose sign depends on a choice nobody can make — and the media where no such number exists at all are exactly the ones whose residual symmetry is a crystal class.

applied · Defects

Named alongside it

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

DefectDisclinationOrder parameterConjugacy classDouble groupLocal symmetry

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