The complete symmetry of an infinite crystal
By now you carry two separate catalogues up this ladder. One is the 32 point groups — every way to rotate, mirror or invert a finite object about a single fixed point that leaves it looking the same. The other is the 14 Bravais lattices — the translational skeletons on which crystals hang. And in guide 1 of this rung you met two new hybrid moves, the screw axis and the glide plane, each of which fuses a point operation with a fractional slide. This guide asks the question those three ingredients demand: put them all together, and how many genuinely different crystal symmetries can exist? The answer is one of the crown jewels of the whole subject — exactly 230.
A space group is the complete list of every symmetry operation that leaves an infinite periodic crystal unchanged. Where a point group describes the symmetry of a finite shape and a lattice describes only the repetition, the space group is the whole story at once — the local symmetry around a point AND the global repetition of the lattice AND the screw-and-glide moves that stitch the two together. Every periodic crystal ever grown, from copper to quartz to a protein, belongs to exactly one of the 230. It is the single most complete label crystallography can pin on a material.
The recipe: plant a point group on a lattice
The simplest way to build a space group is almost cartoonishly direct: take one of the 32 point groups and plant it, unchanged, at every lattice point of a compatible Bravais lattice. No screws, no glides — just point symmetry sitting on repeating points. A space group built this plain way is called symmorphic. You might guess the count is 32 times 14 = 448, but most of those pairings are illegal or redundant: a point group must belong to the same crystal system as its lattice (you cannot bolt full cubic symmetry onto a tetragonal box), and different centerings of the same system sometimes give the same group. When you keep only the legal, distinct ones, exactly 73 survive — the 73 symmorphic space groups.
Take rock salt (NaCl) and copper metal — both wear the symbol Fm-3m. The F says the underlying Bravais lattice is all-face-centred cubic; the m-3m is the full cubic point group, the richest symmetry a crystal can have. Plant that point group on that lattice and you are done — no operation in Fm-3m ever asks for a fractional slide, so it is symmorphic. This is why the same three characters describe a soft metal and a brittle salt: the space group fixes the symmetry scaffold, while the motif you hang on it (one Cu atom, versus a Na+ and Cl- pair) decides the chemistry.
BUILDING THE 230 SPACE GROUPS
32 point groups (rotate / mirror / invert about a fixed point)
+ 14 Bravais lattices (the translational skeleton: P I F C R centering)
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= 73 SYMMORPHIC groups point group planted on a lattice point;
no screw axes, no glide planes
+ swap a rotation for a SCREW axis (rotate, THEN slide along it)
swap a mirror for a GLIDE plane (reflect, THEN slide sideways)
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= 157 more, NONSYMMORPHIC groups
73 + 157 = 230 exact, complete, proven in the 1890sScrew and glide: the other 157
Nature does not stop at the plain 73. Very often the tidiest way for atoms to pack forces a translation right into a symmetry operation. Replace a pure rotation axis with a screw axis — rotate, then slide a fraction of a cell along the axis — or replace a pure mirror with a glide plane — reflect, then slide half a lattice vector sideways. A space group that needs at least one screw or glide to describe itself is nonsymmorphic, and there are 157 of them. They are not 'more symmetric' than the symmorphic ones; they simply weave translation into some of the operations, so no single point of the crystal carries the group's full symmetry.
Hexagonal close packing — magnesium, zinc, alpha-titanium — wears the symbol P6_3/mmc, and its heart is a screw axis. The 6_3 means: rotate 60 degrees and slide half the c-axis; do it six times and you have spun once around while climbing one full c, exactly the motion of a spiral car-park ramp. That single screw is what generates the ABAB stacking of HCP. The trailing c is a glide plane. Many minerals — olivine, and a great many distorted perovskites — sit in Pnma, whose n and a are both glide planes. Add the plain 73 to these translation-laced 157 and you land on the total that closes the catalogue: 73 + 157 = 230.
Reading the label, in one breath
Every one of the 230 wears a compact name — its space-group symbol, written in Hermann-Mauguin notation. The grammar is beautifully economical: the very first character is the centering letter of the Bravais lattice (P primitive, I body-centred, F all-face, A/B/C one-face, R rhombohedral), and everything after it lists the symmetry element found along each of the crystal system's characteristic directions, in a fixed, agreed order. The next guide is devoted entirely to decoding these; here is just enough to read one.
- Read the leading letter first: it names the Bravais centering (P, I, F, A/B/C or R) — the translational skeleton the whole crystal rides on.
- Read the following positions in order; the crystal system fixes which direction each position refers to (for cubic they run along <100>, then <111>, then <110>).
- At each position, a number is a rotation or screw AXIS along that direction; a letter (m, a, b, c, n, d) is a mirror or glide PLANE perpendicular to it.
- Flag the family: a subscript on an axis (like 6_3) marks a screw, and any plane letter other than m marks a glide — either one means the group is nonsymmorphic.
Try it on the two we have met. Fm-3m reads as: F, all-face-centred cubic; then m perpendicular to <100>, a -3 rotoinversion along <111>, and m perpendicular to <110> — pure mirrors and bare axes, so symmorphic. Pnma reads as: P, primitive orthorhombic; then an n-glide perpendicular to a, a plain mirror m perpendicular to b, and an a-glide perpendicular to c — two glides, so nonsymmorphic. Notice how the symbol quietly hands you both the lattice AND the presence of screws or glides at a single glance.
What the space group hands you
Naming the space group is not the end — it is a key that unlocks the rest of the crystal. For free, it hands you the complete map of where atoms are allowed to sit: the Wyckoff positions, each stamped with a multiplicity (how many equivalent copies the symmetry forces into one cell) and a site symmetry (the point symmetry an atom parked there must obey). The general position carries the full multiplicity and site symmetry 1; special positions sit on a symmetry element with fewer copies but higher site symmetry. You only ever list the atoms in the asymmetric unit, and symmetry regrows all the rest — and the formula units per cell Z drops straight out (NaCl in Fm-3m has Z = 4). Guide 4 is built entirely around this.
The master reference for all of this is the International Tables for Crystallography — one authoritative page per space group, carrying its symbol, its diagram, its generators, and its full Wyckoff list. And the payoff that closes the loop back to diffraction is the set of systematic absences: centering, screw axes and glide planes each silence a predictable family of reflections. All-face centering (the F in Fm-3m) allows a reflection (hkl) only when h, k, l are all even or all odd; a 6_3 screw kills every 00l with l odd; a c-glide wipes out its own family too. Those missing spots are a fingerprint — read which reflections are absent and you can read off the symmetry that erased them. Guide 5 is all about that.