Generator

Special positions in p4m

Every point of a 12×12 grid inside the cell of p4m, drawn at a size set by how many operations fix it. 80 of the 144 are general — nothing but the identity leaves them alone, so their orbit is the full 8 points. The other 64 are special, and fall into 3 kinds: 60 points fixed by 2 operations, with orbits of 4; 2 points fixed by 4 operations, with orbits of 2; 2 points fixed by 8 operations, with orbits of 1.

5 essays call wyckoff. 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

The points a group treats differently

Almost every point of a cell has an orbit as long as the group. The exceptions are the points some operation leaves alone, and they are where atoms sit, where a structure's formula comes from, and where a careless motif destroys the group it was meant to illustrate.

p4 inside p4m, by area. A fundamental domain for p4m beside one for p4, drawn by the same construction on the same grid. p4 sits inside p4m with index 2: it has 8 ÷ 4 = 2 times as few operations to rebuild the pattern with, so it needs 2 times as much of the cell to rebuild it from — measured here at 1.51 times as many samples. The two domains are one region and 2 copies of it, and the ratio between them is the index rather than a coincidence of shape.

Operations

Domains of a subgroup

A group with half the operations needs twice as much of the cell to rebuild the pattern from. That single sentence is the index arithmetic of the whole classification, and it turns the containments among the seventeen into a statement about area.

Dropping one invariant of 3m makes two orbits agree. Every lattice point within four cells of the origin, coloured by the values a proper subset of 3m's invariants takes on it — the 2 generators with the first one removed, over a window of 4 cells. With the full set, the 25 orbits of the group take 25 distinct sets of values, one each, so the invariants are a complete set of coordinates on the quotient. With one removed, the two circled points — in different orbits, so no operation of the group carries one to the other — take the same values and become indistinguishable. That is the whole content of the statement that a complete set of invariants separates orbits: the completeness is what is doing the work.

Operations

An orbit is what the invariants cannot tell apart

Two points of the plane lie in the same orbit of a group exactly when every invariant polynomial takes the same value on both. One direction of that is a definition; the other is a theorem, and it is checked here by comparing every pair of points in a window both ways.

p4m: which distortions the atoms of each position can make. Every Wyckoff position of p4m, with the number of independent displacement patterns of each symmetry its atoms supply. A zero is a distortion those atoms cannot make however their amplitudes are chosen — an atom pinned at a rotation centre cannot move in a way that keeps less symmetry than the centre has. The row sums, weighted by the dimensions along the top, come to twice the number of atoms in the cell, which is the check that nothing has been lost. The last column is the number of free coordinates the position has, and it equals the multiplicity of the identity representation in that row: a displacement that keeps every symmetry is exactly a move of the position within its own Wyckoff set.

Operations

Which modes a site can carry

An atom on a mirror cannot move in a way that breaks the mirror while its images move with it: the displacements of a Wyckoff orbit carry a representation, and some of its pieces have multiplicity zero. The count of those pieces is a character, and the one that breaks nothing is the position's own freedom.

One orbit of p4m, two of p4. The general position of p4m — 8 points in a cell, all equivalent under that group — with each point coloured by which orbit of p4 it belongs to. Losing half the operations does not move a single point; it changes which of them are related, and the one orbit becomes 2. An atom sitting on this position in the parent becomes 2 crystallographically distinct atoms in the child, which may then be different elements, or move independently, or order.

Operations

What a position becomes on the way down

Cool a crystal through a transition and it loses operations. Nothing moves — and one crystallographic site becomes two, which is how an ordering transition finds somewhere to put a second kind of atom.

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