Clockwork orbifold notation

Conway's orbifold symbols, decorated with clocks.

Conway's orbifold notation names a wallpaper group by the features of its quotient orbifold: gyration points n, kaleidoscopic boundaries *n…, glide "miracles" ×, and the handle o. An animation group adds one datum to the same picture: every symmetry operation now carries a time offset — the fraction of the animation period by which it advances the animation — and possibly a time reversal. The notation records these offsets as decorations on the Conway symbol. The base symbol is the orbifold of the spatial projection of the group (the symmetry of the long-exposure photograph, with every frame overlaid); the decorations say how the projection lifts to spacetime.

The rules

1. Gyration subscripts — time screws

A gyration point of order n whose counterclockwise generator advances the animation by k/n of a period is written n_k (subscript omitted when k = 0). This is Xu–Wu's time-screw rotation (R_2π/n | T_t^k/n), and the subscript convention deliberately mirrors the crystallographic screw axis n_k: a spacetime group is a 3D space group with the screw direction along time.

Because the generator is fixed (counterclockwise, forward time), 4₁ and 4₃ are different groups — mirror twins, like left- and right-handed quartz. The subscripts at different gyration points of one group are not independent: in each relation of the orbifold group the offsets must sum to an integer, and on an undecorated (primitive) lattice pure spatial translations carry offset zero — any drift can be removed by a Galilean boost — which ties the phases of translation-related centres together. For the 442 family the two conditions force exactly the four possibilities 442, 4₁4₁2₁, 4₂4₂2, 4₃4₃2₁ — precisely the time-axis space groups P4, P4₁, P4₂, P4₃. (Assignments like 4₁4₃2 obey the relation sums but need a drifting lattice; they reappear on centred lattices as the c-prefixed groups.)

2. Tilde marks — time glides

A mirror can only carry a time offset of 0 or ½ (it is an involution). A mirror carrying ½ — Xu–Wu's time glide (m | T_t^1/2) — is marked with a tilde on its arc of the kaleidoscopic boundary: the pattern is mirror-symmetric only after waiting half a period. Arcs are the segments of a * boundary between corners; a tilde is written in the position of the arc it marks, e.g. *22… in print, *~22~2 … in ASCII. At a corner of odd order the two adjacent arcs must carry equal marks; at even corners they are free — the same parity rule as in two-colour (black–white) symmetry, with "colour swap" replaced by "wait half a period".

Spatial glides × may carry any offset q with 2q equal to the offset of the translation they square to: ×_½ is a glide-reflection that also waits half a period (a mixed space–time glide, Xu–Wu's (m_x | T_y^1/2T_t^1/2)); quarter offsets ×_¼ occur on time-centred lattices, the analogue of crystallographic d-glides.

Three points of grammar. A boundary with no corners has a single arc, and its tilde is written before the star (~*, ~*×_½); where the boundary word has a symmetry, marks are written in the first equivalent position. Note the convention switch between the two kinds of subscript: a subscript on a digit is the numerator of k/n (crystallographic screw style), while a subscript on × is the fraction itself — an involution has no order n to divide by. And on c-prefixed lattices tildes are omitted: the centring puts parallel mirror families half a period apart, so mirror phase is relative, and where the residual choice matters it is carried by the superscript letter and the generator list (for the chiral pair c4₁*2 / c4₃*2, the subscript is fixed by which screw sense pairs with the in-phase mirror family).

3. Stacking prefixes — time-centred lattices

Some spacetime groups contain a translation that moves half (or a third of) a cell and waits half (or a third of) a period — a centring of the spacetime lattice that no change of frame can remove once there is point symmetry to pin the frame. Prefix:

c — half-cell shift at half period (body/base-centred types; the checkerboard that swaps colours every half period);
r — third-cell shift at a third of a period (rhombohedral stacking, the ABC-stacked animation).

Example: c4 4₂ 2₁ is the time-axis I4: the centring forces the two kinds of 4-centres to run half a period out of phase, so one class is plain 4 and the other is a 4₂ screw.

A symbol with no prime clause — rules 1–3 only — names a clockwork group: a spacetime group whose every symmetry runs time forwards, equivalently a wallpaper group carrying a clock. There are 68 of them, and a clock of order N on such a group is exactly a perfect N-colouring of the underlying wallpaper.

4. The prime clause — time reversal

If the animation also has time-reversing symmetries (as a spacetime pattern: the loop equals its own reverse, possibly displaced), the group G contains its forward part G⁺ with index 2. Write the forward symbol, a slash, and one canonical representative of the reversing coset:

/1′ — plain palindrome (time mirror m_t);
/g′ — glide time-reversal (m_t | T_x^1/2): the reversed animation equals the original shifted half a cell;
/m′, /2′, /4′, … — reversal composed with a spatial reflection (a mirror, or — when the bracket shows only glides, as in o/m′[××] — a glide reflection), half-turn, quarter-turn, ….

The descriptor names the simplest operation the reversing coset contains, in the order 1′, g′, then m′/2′/…; the coset usually contains several kinds, and the symbol records the first realised. The slash clause is close kin to both colour notations: crystallographers prime the reversing operations (Belov–Tarkhova's p4′g′m), while The Symmetries of Things names a two-colouring by the orbifold pair full group / colour-preserving kernel. A bracketed clockwork symbol is that pair read backwards — the kernel with its clocks, then the full projection — and the clause names the reversing coset just once, because reversal negates every offset: decorations on the reversing features themselves would carry no invariant content.

Consistency forces the reversal to negate all forward offsets (conjugation by m_t maps a k/n screw to a −k/n screw), which is why, for instance, 4₁4₁2₁/1′ does not exist but 4₁4₁2₁/m′ does: a mirror flips the screw's handedness and the reversal flips it back. When the reversal changes the spatial projection's wallpaper type (reversal mirrors add directions), the enriched projection's orbifold is appended in brackets, e.g. o/m′[*×] — an animation with no forward symmetry beyond translations whose photograph is nonetheless cm = *×, entirely by grace of the reversal.

Reading a symbol

One dimension down: the thirteen chronofriezes

The same rules name the 13 spacetime groups of a looping strip (tutorial §2), with base symbols one dimension lower: the spatial projection of a strip group is a line group, whose quotient 1-orbifold is either the circle o (translations only) or the mirrored interval ** (two mirror points). Every decoration carries over verbatim — tildes for half-period mirrors, the c prefix for the centred spacetime lattice, the reversal clause and its bracket:

Two features are instructive. The time glide ~*~* survives here as its own class: in 2+1D it re-bases to ×× by trading the time axis for the spare spatial direction, but a strip has no spare direction — the smallest setting where the crystallographic and frame-preserving classifications already agree on keeping the decoration. And o/m′[**] — flip space and reverse time, the 2-fold spacetime rotation — shows the bracket at its purest: the photograph's entire mirror symmetry is contributed by the reversal coset.

The atlas: 17 wallpapers, 275 animations

The base symbol organises the whole catalog: every spacetime group projects to one of the 17 wallpaper groups, and the wallpaper atlas arranges all 275 by this projection — each wallpaper group rendered as an animation in its own right (the product with a trivial clock), followed by every spacetime group whose long-exposure photograph it is.

Beyond: coloured groups

Spacetime groups are wallpaper groups coloured by a circle of phases, and the discrete theory of colour symmetry — the 46 black–white Shubnikov types, perfect ℤ_N colourings — runs alongside every rule above: time reversal is a two-colouring, time screws are cyclic colourings, centrings are coloured lattices. The dictionary runs both ways: a spacetime group with a clock of order N is a perfect N-colouring of its own projection.

Why this works: the Seifert picture

Quotient a looping animation by its group G: the result is a 3-orbifold fibred in circles (the time circles) over the 2-orbifold of the spatial projection. The clockwork decorations are exactly the Seifert data of this fibration: gyration subscripts are the local Seifert invariants β/α at cone points, tildes and primes describe the behaviour of the fibre over mirror and reversal loci, and the stacking prefix records the holonomy of the fibration — the fractional phases carried by lattice translations, a class in H¹ of the base with ℝ/ℤ coefficients. (The Euler number itself vanishes identically for these flat fibrations; its vanishing is the offsets-sum-to-an-integer rule.) The notation is thus a spacetime-group counterpart of the Conway–Delgado‑Friedrichs–Huson–Thurston fibrifold names for space groups — specialised to the fibration along time that every looping animation carries canonically. A spacetime group is a product (wallpaper × clock) exactly when all decorations vanish: trivial Seifert data means a trivial circle bundle.

Symbol ↔ operators

Every catalog entry also lists its characteristic nonsymmorphic operations in Xu–Wu's Seitz-style form — (R_2π/4 | T_t^1/4) for a time screw, (m | T_t^1/2) for a time glide — together with the complete generator list, so every decoration of the clockwork symbol can be cross-checked against explicit operations. See the catalog.