Catalogue B — the 3D crystals

The 230 space groups of ordinary three-dimensional space, each with a worked crystal structure — and with the 17 wallpaper stacks and the 51 clockwork lifts that tie them to the rest of this site marked.

1. The definition

This catalogue runs the suite's one recipe in the ambient group E(3). A 3D crystal — a space group, in the standard vocabulary — is a discrete subgroup G ≤ E(3) whose translations form a rank-3 lattice, and two space groups are of the same type when they are conjugate by a proper affine map of ℝ³. There are exactly 230 types. The count was reached independently by Fedorov and by Schoenflies in 1891–92 — each correcting the other's list by letter until the two agreed — and it is the ancestral classification of this whole subject: every other catalogue on this site borrows its vocabulary, and its licence to call discrete subgroups "crystals", from these 230.

2. 230, or 219

The word proper in the definition is load-bearing. A screw axis climbs as it turns: in P4₁ a quarter-turn lifts the structure a quarter of the cell. Viewed in a mirror it turns the other way — the reflected crystal is of type P4₃, and while an improper affine map carries P4₁ onto P4₃, no proper one does. Proper equivalence therefore keeps such left- and right-handed partners apart; allowing improper conjugacy merges exactly eleven enantiomorphic pairs,

P4₁ ⇄ P4₃ (76 · 78)   P4₁22 ⇄ P4₃22 (91 · 95)   P4₁2₁2 ⇄ P4₃2₁2 (92 · 96)   P3₁ ⇄ P3₂ (144 · 145)   P3₁12 ⇄ P3₂12 (151 · 153)   P3₁21 ⇄ P3₂21 (152 · 154)   P6₁ ⇄ P6₅ (169 · 170)   P6₂ ⇄ P6₄ (171 · 172)   P6₁22 ⇄ P6₅22 (178 · 179)   P6₂22 ⇄ P6₄22 (180 · 181)   P4₃32 ⇄ P4₁32 (212 · 213)

and the count drops to 219. This site counts properly throughout: the spacetime census of 275 uses the equivalence with det A · c > 0 — introduced in the tutorial precisely as "the convention under which the crystallographic screw pairs 4₁/4₃ count as two groups" — and the one chiral coloured crystal of catalogue D, the five-colouring of p4, is the two-dimensional shadow of the same phenomenon. Each entry below links its partner with a ⇄ badge.

3. Crystal systems and crystal classes

Every catalogue in the suite sorts its entries under the symmetry a reader can see at a glance — catalogue D groups its colourings by the wallpaper group being coloured. Here the visible thing is the crystal class: quotient a space group by its translation lattice and a finite subgroup of O(3) remains, the point group, which is the macroscopic symmetry of the crystal — the symmetry of its faces, its optics, its response tensors. Exactly 32 classes arise (Hessel, 1830 — six decades before the space groups, because crystal habit was measurable long before X-rays saw the lattice), and they gather into seven crystal systems according to the constraint the class forces on the lattice, from triclinic (none) to cubic (three equal axes). The catalogue below is therefore grouped by system, then by class, with each class's groups in the standard numbering — the 3D analogue of sorting colourings by wallpaper group, and the same two-tier scheme as the printed International Tables.

4. Two bridges from the plane

Seventeen of the 230 already live on this site in lower dimension. Take a wallpaper group Γ and extrude it: stack copies of the plane at all integer heights, allowing no symmetry that mixes height into the plane. The result is the vertical stack Γ × ℤ, a symmorphic space group — p1 gives P1, p2 gives P2, and so on through p4g → P4bm and p6m → P6mm. In this site's terms the stack is the phase-as-height lift of the trivial clock over Γ: film a motionless wallpaper pattern, reread the loop of time as a vertical coordinate, and the stacked film frames are the crystal. These seventeen carry a "stack of Γ" badge below, linking to catalogue A.

The second bridge is the same lift applied to a clock that actually runs. A nontrivial clockwork crystal pairs each wallpaper symmetry with an advance of a k-hour clock; reading phase as height stacks the plane at k levels per period and turns a rotation-that-advances-the-clock into a genuine screw axis, a glide-that-advances into a diagonal glide or a centring — Belov's 1956 stacking construction, the one his school used to draw coloured mosaics, run forwards. All 51 nontrivial clockwork crystals lift this way, and they land on 51 distinct space groups — each carries a ⏱ badge below, linking back to catalogue C. Together with the seventeen stacks, 68 of the 230 point back into the plane. Two subtleties are visible in the badges: an enantiomorphic pair of clocks (a clock and the same clock run backwards) lifts to an enantiomorphic pair of space groups — the four ⏱-marked ⇄ pairs 76·78, 144·145, 169·170, 171·172 — and a nontrivial lift is never itself a stack, so the two badges never share a card.

5. The catalogue

Grouped by crystal system, then crystal class. Each entry shows one worked structure — a mineral where a mineral exists, a synthetic or molecular crystal otherwise — hot-linked from the source collection described under Sources; the picture links to the full-size original. Symbols are printed in the plain-text convention of that source: subscripts flattened (P21/c for P2₁/c) and a minus sign where the International Tables print an overbar (Fm-3m). Badges: ⇄ the enantiomorphic partner, "stack of Γ" the wallpaper stack of catalogue A, ⏱ the clockwork crystal of catalogue C lifting there.

Sources

The structure images and worked examples are hot-linked from Frank Hoffmann's space-group list project at crystalsymmetry.wordpress.com/230-2 (companion site of the MOOC The Fascination of Crystals and Symmetry, University of Hamburg; with contributions by Barbara Mayer, Lois Johnson, Daniel Wyllie Lacerda Rodrigues and Vladimir Vasiliev) — a growing collection of worked examples offering, in its own words, "at least one crystal structure for all of the 230 space groups". Each image above links to its original there; the drawings are made with VESTA or Mercury and remain the property of that site's author. Two errata are handled: No. 67 is printed there as Cccm (the symbol of No. 66) and is corrected to Cmma here, and No. 221's plate on the source page repeats No. 220's.

E. S. Fedorov, Симметрія правильныхъ системъ фигуръ (The symmetry of regular systems of figures), Zap. Mineral. Obshch. 28 (1891) 1–146.
A. Schoenflies, Krystallsysteme und Krystallstructur, Teubner, Leipzig, 1891.
International Tables for Crystallography, Vol. A: Space-group symmetry, IUCr/Wiley — the standard reference for symbols, numbering and the 230/219 distinction.
N. V. Belov, T. N. Tarkhova, Colour symmetry groups, Kristallografiya 1 (1956) 4–13 — the stacking construction of §4.