Crystallography & Symmetry
patterns derived from their groups
Series

What symmetry is — the series

8 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A rotation. The motif in the first colour, its images under a single rotation in the second, and the symmetry element marked where the operation itself says it lies.

    What a symmetry actually is

    Not a property of a shape but a motion that leaves it alone. Once symmetry is a verb rather than an adjective, everything else in the subject follows — including why there can only ever be seventeen wallpapers.

    part 1 · operations
  2. A glide. The motif in the first colour, its images under a single glide in the second, and the symmetry element marked where the operation itself says it lies.

    The four motions of the plane

    Slide, turn, flip, and the odd fourth thing that is a flip and a slide together but neither on its own. Every symmetry of every flat pattern that has ever been made is one of these.

    part 2 · operations
  3. Doing one after another. Two symmetries of a pattern, and the one that doing both lands on. The third picture is not a new operation drawn to fit — it is the composition, and it was already in the group.

    Why it is a group and not a list

    The symmetries of a pattern cannot be chosen independently. Do two of them in succession and the result is forced to be a third, which is why there is no eighteenth wallpaper for anybody to invent.

    part 3 · operations
  4. Growing the p4 orbit. One motif, then more of the group's operations applied to it, until applying another produces nothing new. The pattern is the orbit; the drawing is only its shadow.

    The orbit is the pattern

    A wallpaper is not designed and then found to have symmetry. It is the set of places a group sends a single mark, and once that is taken literally the pattern can be grown, checked, and caught out.

    part 4 · operations
  5. A conjugacy class of p4m. One conjugacy class of p4m drawn in place: every copy of the same symmetry that the group can carry onto every other. Conjugation was applied to each of the 8 operations by each of them in turn, and the kind and order of the result was checked to match every time.

    The same symmetry, somewhere else

    Two mirrors in a pattern can be the same symmetry or two different ones, and looking will not settle it. Conjugation is the operation that decides, and it turns an intuition about sameness into arithmetic.

    part 5 · operations
  6. 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.

    part 6 · operations
  7. p2's symmetries, sorted into classes by the group itself. A pattern with the symmetry of p2 over 2 by 2 cells, with its rotation centres and mirror lines marked in the International Tables' shapes and coloured by conjugacy class in the infinite group: two marks share a colour exactly when some operation of the group carries one element onto the other. Where rotations of several orders share a centre, the mark is the highest order's and so is its colour. Glides are not drawn. Classes counted: half-turns: 1 in the quotient, 4 in the group.

    Two mirrors a coset cannot tell apart

    Taken modulo its lattice a wallpaper group is finite, and its conjugacy classes are easy to list. But a coset holds every mirror of one direction at once, and the group itself keeps apart mirrors the list merges: pm has two classes of mirror, p2 four classes of half-turn, p3 six classes of rotation. Deciding which is which is Dehn's conjugacy problem, and for these groups it comes down to whether one vector lies in one lattice.

    part 7 · operations
  8. Three kinds of three-fold axis in one coset. Two space groups of the primitive cubic lattice, each seen down a body diagonal, with one cell of the lattice projected and the three-fold axes parallel to that diagonal marked by type and coloured by class in the group. Every axis has the same linear part, and the lattice places a rotation, a 3₁ screw and a 3₂ screw in one coset of it. P23 keeps all three apart; Pm3̄, whose inversion reverses a screw's hand, joins the 3₁ and the 3₂.

    Where the plane had places, space has pitches

    Conjugacy in a plane group came down to one lattice, and it counted four families of half-turn to a cell, three of three-fold rotation, two of quarter-turn and one of sixth. In space the same lattice gives the same four numbers for every one of the 219 space-group types. What changes is what the families are. In a centred lattice, and along some directions of a primitive one, they are not rotations in different places but screws of different pitch, and a right-hand screw and a left-hand one in the same coset are one symmetry exactly when the group contains a reflection.

    part 8 · operations

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