Crystallography & Symmetry
patterns derived from their groups
Generator

The origins of p2 that change nothing

The origins of p2 that change nothing
The origins of p2 that change nothing. One cell of p2 with its pattern, and every point marked to which the origin may be moved without a single operation of the group changing its translation part. There are 4 of them per cell, and the count does not change when the search grid is refined, so it is a fact about the group rather than about the grid. Two coordinate lists differing by one of these vectors describe the identical arrangement, which is why no structure's coordinates are ever unique.

One cell of p2 with its pattern, and every point marked to which the origin may be moved without a single operation of the group changing its translation part. There are 4 of them per cell, and the count does not change when the search grid is refined, so it is a fact about the group rather than about the grid. Two coordinate lists differing by one of these vectors describe the identical arrangement, which is why no structure's coordinates are ever unique.

6 essays call normaliser. 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 522 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.

The origins of p2 that change nothing. One cell of p2 with its pattern, and every point marked to which the origin may be moved without a single operation of the group changing its translation part. There are 4 of them per cell, and the count does not change when the search grid is refined, so it is a fact about the group rather than about the grid. Two coordinate lists differing by one of these vectors describe the identical arrangement, which is why no structure's coordinates are ever unique. Operations

The same pattern, described twice

Two coordinate lists for one structure can disagree in every number and describe exactly the same arrangement, because a group does not fix its own origin. The operations that may be applied to a description without changing what it describes are its normaliser, and they can be found by looking at pictures rather than at matrices.

 Four groups whose description count a metric can raise. Every plane group, the number of ways of writing one arrangement down on the lattice the group requires, and the number on the most symmetric lattice it may sit on. Eight groups already occupy the most symmetric lattice available to them and have nowhere to go. Four have a metric that raises the count, by two and in one case by six. The starred rows belong to groups whose normaliser has a free direction, where the quantity is a count of grid points rather than an index and cannot be compared. Operations

The normaliser is not a function of the group

How many ways there are of writing one structure down is computed from the group and printed in a table beside its name. It is not a property of the group. Draw a p2 pattern on a hexagonal cell rather than an oblique one and the number goes from four to twenty-four, with nothing done to the group at all.

 Seventy-two positions and the sets they fall into. Every plane group with the number of its Wyckoff positions, and the number of sets those positions fall into when the positions a normaliser exchanges are counted once: on the cell the group requires, on the most symmetric cell it may sit on, and under every change of basis carrying the group onto itself. 72 positions become 53 sets on the required cells and 51 on the best ones, and the last column never goes lower. The two groups a special cell changes are pmm and cmm. Operations

The same site under two names

A structure report puts each atom on a Wyckoff position, and two correct reports of one crystal can name different positions. The positions that can trade places are exactly the ones the normaliser exchanges — and which those are depends on the cell as measured, not only on the group.

 Subgroups, the classes a group sorts them into, and the sets its normaliser does. For every plane group, the number of subgroups of index two and of index three, the number of conjugacy classes those fall into under the group's own operations, and the number of sets they fall into under its Euclidean normaliser. Over the seventeen there are 74 subgroups of index two in 74 classes and 56 sets, and 82 of index three in 36 classes and 32 sets. 9 of the thirty-four rows have fewer sets than classes, which is where the tables' "equivalent" entries come from. Counts of subgroups and of classes agree with an independent count from transitive actions on n points. Operations

Three of them, and they are equivalent

The subgroup tables print a count and sometimes a word beside it. Three subgroups of one type may be three copies the group itself shuffles, or three the group holds firmly apart and only a change of description exchanges. p3 has three copies of itself at index three, no operation of p3 moves any of them, and one shift by a third of a cell exchanges all three.

 1,694 positions, 1,182 sets on the general cell, and 1,128 on the best. Every affine type of space group by crystal system: its Wyckoff positions, the sets those fall into under the normaliser on the cell the group requires, on the best cell it may sit on, and under the whole affine normaliser. The total is 1694 positions, 1731 with enantiomorphs counted twice, 1182 sets on the general cell and 1128 on the best, which equals the affine 1128 in every system. The two columns differ only in the orthorhombic system and in one monoclinic type. Operations

One cell reaches every Wyckoff set

In the plane, every identification the affine normaliser makes between two positions is made by an isometry of some cell, and the essay that found this warned against carrying it into space, where changes of basis can mix axes without limit. It carries. The 1,731 positions of the 230 groups were found from stabilisers and the sets from normalisers, and on one cell for each type the two partitions agree. Past the monoclinic system that is a one-line average. Below it, the infinite normaliser turns out to have nothing left to join.

 The subgroups of index two, and what joins them. Every affine type of space group by crystal system: its subgroups of index two, 1813 in all; classes under the origin shifts alone, 1392; classes under the whole affine normaliser, 1159; and the fewest classes one cell can give, 1161. Best cell and affine agree in every system except the triclinic. Operations

No cell joins the seven halves

Every identification the normaliser makes between Wyckoff positions is made by an isometry of some cell, and it was natural to expect the same of subgroups. It nearly is. Across all 219 types and their 1,813 subgroups of index two, one cell does everything the normaliser does, except in the two triclinic groups. P1 has seven ways to be halved, and its normaliser makes them all one. No cell can, because seven is the number the crystallographic restriction forbids.

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