p4m, displaced and then measured

Every atom of a p4m pattern moved by up to 1.2 per cent of a cell edge, and the resulting positions examined at 26 tolerances. At zero tolerance only the identity survives, so the structure has no exact symmetry whatever. Between 0.021 and 0.094 the count sits at 8, which is the group that was displaced. Above that it climbs to 15, accepting operations no version of this pattern has. The climb is not even steady: at 4 of the 25 steps the count falls as the tolerance is loosened, because operations accepted separately at one threshold merge into one at the next. The correct answer is a step on a staircase and nothing in the coordinates says which step.

1 essay calls tolerance. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about. Every one of this site's 393 essays names its parameters at the call site, which the standard pass of 2026-08-09 established and param-floor holds.