Crystallography & Symmetry
patterns derived from their groups
Theme

The thread: The motif must be asymmetric

A dot is too symmetric to illustrate most groups — its orbit acquires operations the group never had. The comma on a nineteenth-century wallpaper plate is there for a computable reason.
1 of 43 of 44 of 4the pattern is grown from the group, never drawnp4 Operations

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.

 a dot at (1/12, 1/12)8 symmetries per cella motif with no symmetry4 symmetries per cell57 of 121 dot positions give more symmetry than p4both generated with the 4 operations of p4the dot gains 4 The classification

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.

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