Crystallography & Symmetry
patterns derived from their groups
Concept

Rotoreflection — where it appears

A rotation about an axis followed by a reflection in the plane perpendicular to it, taken as one operation. Schoenflies' symbols are built on it where Hermann–Mauguin's use the rotoinversion, so the group written S6 in one scheme is written 3̅ in the other.

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

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 The seven friezes rolled into cylinders are the seven axial families. Each of the seven frieze groups drawn on a strip 3 cells long, beside the same strip rolled into a cylinder so that its ends meet. A translation by one cell becomes a rotation by a 3th of a turn about the axis, a mirror across the strip a mirror containing the axis, the centre line a mirror perpendicular to it, a half-turn in the strip a half-turn about a horizontal axis, and a glide a rotation by half a cell's angle combined with that perpendicular mirror. Each cylinder's symmetry group was built from the rolled strip and again from the family's own generators, and the two agree. At n = 3 the orders are 3, 6, 6, 6, 6, 12, 12, and the last column names the crystal class each member is, coloured by whether it is proper, contains the centre, or is neither.

Seven friezes round a cylinder

A point group with one principal axis belongs to one of seven infinite families, and there are seven frieze groups. They are the same seven. Draw a frieze on a strip, roll the strip into a cylinder, and every translation becomes a turn about the axis and every glide a rotoreflection.

restriction · Finite groups

Named alongside it

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

Axial classCrystal classCrystallographic restrictionEnumerationFrieze groupGlide reflectionHermann–Mauguin notationHomomorphismInversion centreKernelLimiting groupMetric tensor

All concepts