Crystallography & Symmetry
patterns derived from their groups
Concept

Cartan dieudonne — where it appears

The theorem that every isometry of n-dimensional space is a product of at most n + 1 reflections. The plane's answer is three, attained by the glide, and space's is four, attained by the screw — so the awkward motion in each dimension is the expensive one.

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

glide: 3 mirrors. A glide of the plane, drawn together with the mirrors it is a product of. The first shape is the motif; the pale ones are what each mirror in turn produces; the last is the image the motion itself gives. There are 3 mirrors, which is the smallest number that can produce this kind of motion, and their product was formed and compared with the motion before the figure was drawn.

Three reflections, and never four

Every motion of the plane is a product of mirrors, and the number needed is never more than three. That count is not a curiosity about mirrors — it is the classification of the four motions written as an integer, with the parity of the number deciding handedness and the geometry of the last two mirrors deciding everything else.

operations · Composition P4_1: a screw of 90°. One operation of P4_1, reduced to Chasles' three numbers: an axis, an angle of 90°, and a pitch of 1.25 along it. The points are the orbit of one position under repeated application, which climbs because the pitch is not zero — and it is not zero for any choice of origin, which is what makes this a screw rather than a rotation. It needs 4 mirrors, and their product was checked against the operation before this was drawn.

Every motion of space is a screw

A rigid motion of space that preserves handedness turns about some axis and slides along that same axis, and there is nothing else it can do. Rotations and translations are the two ends of that one description, the axis and the pitch are computed rather than recognised, and the operations a space group is made of stop being a list of kinds.

operations · What symmetry is

Named alongside it

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

CompositionFixed pointSymmetry operationGlide reflectionHalf-turnIntrinsic translationOrientationRotoinversionScrew axis

All concepts