Crystallography & Symmetry
patterns derived from their groups
Concept

Notation — where it appears

A convention for writing a group as a string of characters, of which crystallography keeps two that answer different questions. Hermann–Mauguin names directions and Schoenflies names a construction, and neither can be obtained from the other without going back to the group.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

The wallpaper group p4m. A pattern with the symmetry of p4m, generated by applying the group's 8 operations to an asymmetric motif and repeating across the lattice. The symmetries of the result were then found independently and match the group exactly.

Reading Hermann–Mauguin

p4g looks like a licence plate and is in fact a set of instructions. Half an hour with the rules turns the seventeen from a list to be memorised into a notation that can be read.

classification · Seventeen Reading 4/mmm off its own directions. Each position of 4/m2/m2/m reports one symmetry direction of the tetragonal system: the highest-order axis lying along it, and whether a mirror is perpendicular to it. Nothing is looked up — every row is computed from the group's own matrices.

Reading a class off its own axes

A Hermann–Mauguin symbol is not a name that was assigned. It is a report on three directions, read in order, and the whole of it can be derived from the group's matrices — with one genuine convention and one exception, and the exception is orthorhombic.

point-groups · Crystal classes The thirty-two, in both notations. Each class with the symbol crystallography uses and the symbol spectroscopy uses, both derived from the class's own matrices. The Hermann–Mauguin symbol is a report on three families of directions, read in an order the lattice system fixes. The Schoenflies symbol is a report on a construction: a principal axis of order n, whether there are n twofold axes across it, and which mirrors were added. Neither can be computed from the other without going back to the group, which is why the two lists are set beside each other rather than one derived from the other.

One class, two names

Hermann–Mauguin names directions and Schoenflies names a construction, and the two are derived here from the same integer matrices by computations that share no step. Neither can be obtained from the other without going back to the group — which is why a molecule has one kind of symbol and a crystal has both.

point-groups · Crystal classes

Named alongside it

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

Hermann–Mauguin notationCrystal classSettingCentringGeneratorsHolohedryMetric tensorPoint groupPrincipal axisRotoreflectionSchoenflies notation

All concepts