Catalogue A — the 2D crystals

The seventeen types of periodic symmetry in the plane — the ground floor of the suite: every spacetime group, clockwork crystal and colouring in the catalogues below is built over one of these.

1. The definition

A 2D crystal is a discrete subgroup Γ ≤ E(2) of the isometries of the plane whose translations form a lattice of full rank two. In the ambient-group recipe that organises this whole suite — Eq. (4) of the tutorial, Γ(r, t) = (Rr + u, st + τ) — this is the degenerate instance: delete the internal coordinate altogether, leaving bare isometries

r ⟼ Rr + u,   R ∈ O(2),

with no clock to advance and no colour to permute. A crystal type is a class of such subgroups under proper affine conjugacy: two crystals are the same when an orientation-preserving affine map of the plane carries one group onto the other. There are exactly seventeen types — the wallpaper groups (Fedorov 1891, rediscovered by Pólya 1924) — and this page is their catalogue.

Proper conjugacy is the suite's convention — a motion of space should never be a mirror — but in the plane the restriction costs nothing: allowing improper conjugation merges no classes, and the count is 17 either way. Every wallpaper group is carried onto its mirror image by a proper affine map. For the twelve groups containing a reflection or a glide this is immediate — the group's own improper elements do the mirroring — and for the five rotation-only groups p1, p2, p3, p4, p6, conjugating by a reflection sends each rotation to its inverse, which the group already contains, and sends the lattice to a mirrored lattice that a proper change of basis restores. The plane is the one floor of the suite on which chirality cannot occur. Not so upstairs: in E(3) the same relaxation merges the eleven enantiomorphic pairs of catalogue B (P4₁22 against P4₃22, and their kin), collapsing 230 proper types to 219 improper ones, and among the coloured crystals of catalogue D exactly one of the 269 — a five-colouring of p4 — splits into a chiral pair when the equivalence is tightened to proper maps.

Each entry below is drawn the way this site prefers to draw a wallpaper group: as its trivial clock — the spacetime product Γ × ℤ of catalogue C, the film in which every copy of the motif runs in phase — frozen at a single instant, which is the wallpaper pattern itself (click a plate to let it run). This is a recall page: around each plate sit the standard facts and every crosslink the suite makes to that base — its spacetime groups, its clockwork crystals, its colourings and the places it appears inside a colouring as colour-preserving subgroup or colour-fixing kernel, and its vertical stack in 3D.

2. The catalogue

Grouped by crystal system — the metric constraint the point group puts on the lattice. Headings give the Conway orbifold symbol first and the Hermann–Mauguin name second; the anchors are the HM names (#p4g), which is how every other catalogue links here.

Oblique — p1, p2

No constraint on the cell: a general parallelogram lattice, point group of order at most 2, nothing improper.

Rectangular — pm, pg, pmm, pmg, pgg

A primitive rectangular cell, forced by mirrors or glides along the axes.

Rhombic (centred) — cm, cmm

The centred rectangle: mirrors with glides interleaved between them, primitive cell a rhombus.

Square — p4, p4m, p4g

A square cell, forced by fourfold rotation.

Hexagonal — p3, p3m1, p31m, p6, p6m

The triangular lattice, forced by threefold or sixfold rotation.

References

E. S. Fedorov, Симметрія на плоскости (Symmetry in the plane), Zap. Imp. S.-Peterb. Mineral. Obshchestva 28 (1891) 345–390 — the first enumeration of the seventeen.
G. Pólya, Über die Analogie der Kristallsymmetrie in der Ebene, Z. Kristallogr. 60 (1924) 278–282.
J. H. Conway, D. H. Huson, The orbifold notation for two-dimensional groups, Structural Chemistry 13 (2002) 247–257.
J. H. Conway, H. Burgiel, C. Goodman-Strauss, The Symmetries of Things, A K Peters, 2008 — Chapters 2–3: the orbifold signatures used throughout this suite, and the Magic Theorem proof that seventeen is the count.
International Tables for Crystallography, Vol. A: Space-group symmetry, IUCr — the plane-group tables (Nos. 1–17) and the space-group tables cited in the Vertical stack rows.