Crystallography & Symmetry
patterns derived from their groups
Concept

Motif — where it appears

The object a symmetry group repeats, attached to one point of an orbit. Its own symmetry survives into the pattern only where that symmetry normalises the group, which is why a careful drawing uses a motif with none.

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

Why p4 cannot be drawn with dots. The same group applied to a single dot and to a motif with no symmetry of its own. The dot's orbit turns out to have more symmetries than the group it was made with, so a figure drawn that way illustrates a different group from the one in its caption.

The motif must be a comma

A dot is too symmetric to illustrate most wallpaper groups. Its orbit acquires mirrors nobody asked for, and the resulting figure is quietly of a different group from the one in its caption.

classification · Accidental symmetry Every group a symmetric motif can lift a plane group to. The seventeen plane groups arranged by lattice, with an arrow from each group to every group a motif of higher symmetry, placed at some position, turns it into, on the generic lattice of the first. Dashed arrows mark the lifts that also halve the cell, two of which return the same type on the smaller cell. p1 lifts only to p2, the rectangular groups to pmm, pmg and cmm, the square ones to p4g and p4m, and p3 to four different groups. Three groups, outlined, cannot be lifted at all without a special metric: p2, pmm and p6m.

Where a symmetric motif lifts the group

A motif with more symmetry than its site asks for can raise the group of the whole pattern — or not, and which one is decided by the site, not the motif. The criterion is one line: a symmetry of the motif survives into the pattern exactly when it normalises the group. Run over every Wyckoff position of the seventeen, it finds twenty-nine of seventy-two where a motif can lift the group, general positions only in the four groups with a direction to slide along, two sites where the same motif gives different groups depending on how it is turned, and seven lifts between groups that halve the cell.

classification · Accidental symmetry

Named alongside it

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

Accidental symmetryAsymmetric unitCentred latticeNormaliserPlane groupRound tripSite symmetrySpecial positionStabiliserSupergroupWyckoff positions

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