Generator

P2₁/c, in the two diagrams the Tables print

P2₁/c, in the two diagrams the Tables print

P2₁/c, in the two diagrams the Tables print. Space group P2₁/c, number 14, projected down c on a primitive monoclinic cell. The symmetry elements drawn: 2 2₁ screw axes, 2 glide planes, 4 inversion centres. 4 general positions, the orbit of one point, each labelled with its height along c and marked with a comma where the operation that produced it reversed handedness.

Space group P2₁/c, number 14, projected down c on a primitive monoclinic cell. The symmetry elements drawn: 2 2₁ screw axes, 2 glide planes, 4 inversion centres. 4 general positions, the orbit of one point, each labelled with its height along c and marked with a comma where the operation that produced it reversed handedness.

19 essays call space-plan. 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.

P2₁/c, in the two diagrams the Tables print. Space group P2₁/c, number 14, projected down c on a primitive monoclinic cell. The symmetry elements drawn: 2 2₁ screw axes, 2 glide planes, 4 inversion centres. 4 general positions, the orbit of one point, each labelled with its height along c and marked with a comma where the operation that produced it reversed handedness.

The step a flat surface has no room for

Seventeen becomes two hundred and thirty, and the factor of thirteen is not a matter of having more directions to work in. It comes from an operation the plane cannot hold — a translation that no choice of origin will remove.

The round trip, on Pnma. 8 operations were generated from the standard generators of Pnma; the orbit of three points in general position was formed, the group was discarded, and 8 operations were rediscovered from the 24 points alone. The two sets are identical, which is what the figure asserts.

Forgetting a group in three dimensions

For three phases this site said its machinery was two-dimensional and decided nothing about a space group. That was true, and it was a limit rather than a principle — nothing in the decidability argument mentions the number two.

Sixteen candidates, ten groups

One point group, one lattice, and every consistent way of attaching translations to it — enumerated in full. The count comes out at sixteen, and then at ten, and the step between the two numbers is a decision about what "the same group" means rather than an arithmetic fact.

2 screw axes. 2 of the eleven screw axes a lattice permits, each drawn as the helix it is: 3₁, 3₂. A turn of 2π/n followed by an advance of m/n of the repeat, so that n turns land exactly m cells along. 0 of those drawn are its own mirror image; the rest come in left- and right-handed pairs.

Eleven groups that are their own reflection's rival

There are two hundred and thirty space groups, and there are two hundred and nineteen. Both numbers are correct and they answer different questions, and the eleven that separate them are the reason a crystal can be built one way round and not the other.

The 4₁ screw axis. 1 of the eleven screw axes a lattice permits, each drawn as the helix it is: 4₁. A turn of 2π/n followed by an advance of m/n of the repeat, so that n turns land exactly m cells along. 0 of those drawn are its own mirror image; the rest come in left- and right-handed pairs.

Turning and climbing at once

A rotation has a fixed point and a screw has none. That sounds like a small difference and it is the reason a space group is not a point group with extra letters, the reason two hundred and thirty is not seventy-three, and the reason a helix can be a crystal.

The general positions of P2₁/c. Space group P2₁/c, number 14, projected down c on a primitive monoclinic cell. 12 general positions, the orbit of a three-point asymmetric motif, each labelled with its height along c and marked with a comma where the operation that produced it reversed handedness.

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.

The five kinds of glide plane. All five glide letters: a, sliding by a/2; b, sliding by b/2; c, sliding by c/2; n, sliding by (a+b)/2, (b+c)/2 or (a+c)/2; d, sliding by (a+b)/4, (b+c)/4 or (a+c)/4. 3 of those drawn are axial — the slide is half of a single cell vector lying in the plane. A glide's square is a pure translation, so its slide can only be half of a lattice vector, and the quarter-cell d glide exists only where centring has already made the half-diagonal a lattice vector.

Reflect, then slide by half of something

A glide's slide must double to a lattice vector, which leaves three candidates in any plane and a fourth that exists only where centring has already made a half-diagonal into a lattice vector. Five letters, and the fifth is the one the enumeration explains.

The symmetry elements of Ccmm. Space group Ccmm, number 63, projected down c on a C-centred orthorhombic cell. The symmetry elements drawn: 4 2-fold rotation axes, 6 glide planes, 12 2₁ screw axes, 2 mirror planes, 8 inversion centres. 2 kind(s) of element in this group have no line or mark in a projection down c — an axis at an angle to the page, or a plane lying parallel to it — and are not in the picture.

The operations nobody put in

A group is not a list of generators. Compose two of them and something arrives that neither contained — a screw where there were only mirrors, a glide where there was only a mirror and a centring vector — and in three dimensions most of a group's operations get there this way.

P2₁2₁2₁: the sections its symmetry forces. The Patterson cell of P2₁2₁2₁ with the sections marked. Each operation (M, t) sends an atom at x to Mx + t, so the vector between them is (I − M)x − t; where I − M is singular that vector cannot leave a plane, and the plane's equation comes from the left null space in integers. This group has 3 such operations, giving the sections w = 0.5, u = 0.5, v = 0.5. A heavy atom's vector to its own image is somewhere on one of them, which is what made structure solution possible before computers: a plane can be searched by eye and a volume cannot.

How it is known

Where symmetry stacks the vectors

A Patterson map of a real structure is a blur with thousands of overlapping peaks. A screw axis rescues it: the vectors between symmetry-related atoms cannot leave a plane, so the search for a heavy atom is a search of a section rather than of a volume — and which plane it is falls out of the operation's matrix in integers.

P2₁/c under every cell choice. One group, described in each of the 6 bases that keep its cell the shape its system requires, with the symbol derived from the operations each time. The distinct symbols are P2₁/c, P2₁/a, P2₁/n — 3 names for one group. Nothing about the crystal has changed: the basis changes all have determinant ±1, so the lattice is untouched, and the census of rotations, screws, mirrors and glides is identical in every row, since conjugation cannot turn one kind into another. What changes is which lattice vector a glide's translation is half of, and the glide letter names exactly that.

One group, three symbols

P2₁/c, P2₁/a and P2₁/n are the same space group written on three choices of axes, and the literature contains all three as though they were different. Deriving a symbol from a group's own operations shows why — and found an entry on this site that had been carried under another setting's name for two phases.

3 lattices. 3 lattices: hexagonal P, with 24 symmetries; rhombohedral P, with 12 symmetries; hexagonal R, with 12 symmetries. The corner points are the conventional cell; the points in the second colour are the centring translations, drawn at every position inside the cell rather than one per face. The cell shapes are the picture's, chosen so no two systems look alike; only the angles a system is defined by mean anything.

A lattice described on somebody else's axes

R-centring a hexagonal cell does lower its symmetry, from twenty-four to twelve — and the lattice that results is the fourteenth, the rhombohedral one, which already appears on the list under its own axes. It is the only row in the enumeration where losing symmetry and being a duplicate are the same verdict.

Each arithmetic class holds exactly one symmorphic group. The arithmetic crystal classes this site enumerates in full, with the number of space groups each produces and the number of those that are symmorphic — that is, that have an origin at which every operation's translation part vanishes. The right-hand column is one in every row, over 25 groups in all, and the figure asserts it rather than reporting it. That is the bijection behind the number 73: there are seventy-three arithmetic crystal classes in three dimensions and seventy-three symmorphic space groups, and the correspondence is this one, class by class.

One symmorphic group per class

Every arithmetic crystal class holds exactly one space group in which some origin clears every translation part at once. That bijection is why there are seventy-three symmorphic space groups and seventy-three arithmetic classes, and it is checked here class by class rather than counted.

Aem2: one plane, two glides. The plane of Aem2 that carries two operations, drawn edge-on with each slide beside it. The two differ by the A-centring translation, which lies inside this plane — that is the whole condition for a plane to carry two, and it is why no primitive group has one. Both slides are axial, b and c, so neither letter has a claim on the symbol, and before 1992 the Tables simply chose. The letter e is the choice being refused.

The plane that carries two glides

A plane can hold two glide operations at once, with slides that have no claim on each other. Every symbol printed before 1992 chose one of them, so the name recorded a convention rather than a group — and the International Tables invented a letter to stop it.

P222: 16 descriptions of one structure. P222 has 4 operations in a cell. 8 origins leave every one of them exactly where it was, and 8 linear parts of the lattice's holohedry normalise the group, so its Euclidean normaliser has 64 elements per cell and the index is 16. That index is the number of coordinate lists that describe one and the same arrangement of atoms. Each was applied to a motif and the resulting point sets compared: the numbers differ and the sets are identical, which is the check that makes the count mean anything.

One crystal, and sixteen coordinate lists

Two structure reports can disagree in every number and describe the same arrangement of atoms, because a space group does not fix its own origin or its own axes. How many genuinely different lists there are is the index of the group in its Euclidean normaliser — a number, computable, and the thing a structural database has to divide out before it can say two entries are the same compound.

P2₁/c from 27 marks. The marks of P2₁/c's plan, counted by kind, and what they rebuild to. Each mark is reduced to what a reader can see and handed to a closure with the matrices withheld: an axis gives its direction, its position and how far one turn advances along it; a plane gives its normal, its position and its slide. The lattice supplies the candidate matrices, the closure supplies the rest, and what comes back is the group — 4 operations against 4, with nothing missing and nothing extra.

The plan contains the group

A space-group diagram has always been treated here as a picture of the group. It is more than that: hand back the marks alone — no matrices, no operations, not even the centring — and the group comes out exactly, forty-five times out of forty-five.

Ten ways for space to be flat. The thirteen groups, with each mirror-image pair counted once, because a shape and its mirror image are the same shape. 3 of the ten arrive that way — the three-fold, four-fold and six-fold screws, which are the enantiomorphic pairs this collection already counts among the two hundred and thirty. Six of the ten are orientable and four are one-sided.

Ten ways for space to be flat

Thirteen of the two hundred and thirty space groups hold no point still, and folding space along one of them gives a shape with no curvature anywhere. There are ten such shapes, not thirteen, and the difference is the same eleven pairs that separate 230 from 219.

p4m: 4 and 4. The standard motif — three points in no particular arrangement — repeated by p4m, with each copy coloured by the sign of the area of the triangle it makes. 4 copies have one sign and 4 the other, because the group contains an operation that reverses orientation. A structure built from one enantiomer cannot sit here: the group would put its mirror image in the same crystal. The colours were computed from the coordinates rather than assigned.

The groups a single hand may sit in

A protein is built from one enantiomer of every amino acid, and a crystal of it contains nothing else. That single fact deletes most of the classification at a stroke: any operation reversing orientation would put the other hand in the same crystal. The criterion is one line of arithmetic, and in the plane the enumeration is complete — five of the seventeen.

4_1: which index gives which group. The isomorphic subgroups of a 4₍1₎ screw group, index by index. An index sharing a factor with 4 gives nothing — the translation cannot be written on the new cell at all — and the rest give a screw whose index is the old one times the inverse of p modulo the axis order. So the answer alternates: some indices give the group back and others give its mirror image, and which is which is decided by p modulo the order of the axis.

A bigger cell, and sometimes the mirror

An isomorphic subgroup gives up nothing but scale — the same group again on a coarser lattice. In space the screw axes sharpen the question, and the answer contains a surprise: a cell three times taller holds the group's enantiomorphic partner, so a left-handed screw contains a right-handed one with nothing done to the crystal but a change of description.

Two dimensions to be ambiguous in, and one. Why a plane can carry two operations and a line cannot, side by side. Two reflections sharing a plane differ by a translation lying in that plane, and their slides are vectors in the plane — a two-dimensional space, in which a centring vector need not be a multiple of the slide. So the two slides can be genuinely different glides, b against c, and in 1992 the International Tables invented the letter e for the case where neither has a claim. Two rotations sharing an axis differ by a translation along that axis, because anything across it would move the line; their intrinsic parts are vectors along the line, a one-dimensional space in which every lattice vector is a whole multiple of the shortest. So the two differ by a whole number of repeats and are the same screw. The plane has one dimension of freedom left over and the line has none.

A line carries one screw

A plane can hold two glide operations at once, and in 1992 the International Tables invented a letter for the case where neither has a claim. The same question put to an axis has the opposite answer: across 3,388 axes, not one line carries two — and the reason is that a slide has two dimensions to be ambiguous in and an intrinsic translation has one.

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