Which of these 14 groups split, and which do not
14 space groups tested for whether an origin exists at which every operation is a rotation, a mirror or an inversion with nothing added: 7 split and 7 do not. The verdict only; the reason is not in this drawing.
2 essays call
symmorphic-split. 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.
Where it is called
Changing this generator changes every one of these figures.
Operations
One part in however many, and why it is never quite that
A crystal's contents are the asymmetric unit repeated by the group. The unit's volume is the cell's divided by the order of the group — except that it is always a little more, and the excess is exactly the special positions counted whole.
What a lattice forbids
Thirteen ways to hold a lattice
The crystallographic restriction is about one matrix. A crystal has a whole group of them acting on one lattice at once, and asking how many such groups there are gives thirteen — not the ten of the plane point groups, and not the seventeen of the plane groups.