Life on a tiling that isn't the square grid

Conway's Game of Life is usually described as a rule, but half of it is a grid. "Any live cell with two or three live neighbours survives" only means something once you have said what a neighbour is, and on the square lattice that question has a single answer everybody agrees on: the eight cells touching it. Change the grid and the rule stops being well defined before it stops being interesting.

That is the whole subject of this page. The Atlas holds thousands of tilings whose tiles have three, four, six, twelve neighbours — often several of those at once, in the same tiling — and running B3/S23 on them forces a choice that the literature has never settled.

What is known

The regular non-square cases were worked out by hand, one tessellation at a time, mostly by Carter Bays. He set out three conditions for a rule to deserve the name Game of Life: every touching neighbour is counted the same way, the rule supports a glider, and that glider arises naturally from random soup, with random configurations showing bounded growth. Applying them he found a hexagonal rule with a period-5 glider and a rule on the Cairo pentagonal tiling with a period-48 one, having earlier done the triangular case. Three tessellations, published across 1994–2007. Nobody has run those conditions across the eleven uniform tilings, let alone the $k$ -uniform families.

The aperiodic case has had far more attention, and it is stuck on one question. Owens and Stepney ran B3/S23 on the Penrose kite-and-dart and rhomb tilings, catalogued still lifes and oscillators, showed that arbitrarily large snakes and chains exist, and found the two tilings behave statistically quite differently from each other. They did not find a glider. Goucher later built one — but in a purpose-made four-state automaton on generic quadrilateral tilings, explicitly not Life. Bailey and Lindsey get Life-isomorphic automata on quasiperiodic tilings from the multigrid construction, so gliders transfer there by design, not by playing B3/S23 against the tiling's own adjacency. Hong and Mei classified every four-cell still life on the Robinson triangle. Whether Conway's rule itself admits a glider on any aperiodic tiling is, as far as I can find, still open.

Hyperbolic tilings have been studied hard, but for a different question. Margenstern's programme constructs universal automata on the pentagrid, heptagrid and dodecagrid with as few states as possible. That is engineering a machine, not exploring a rule space; nobody seems to have asked Bays' question of ${7, 3}$ .

And on general graphs there is a body of work that turns out to be the relevant methodology: Marr and Hütt's outer-totalistic automata on arbitrary topologies, and more recently Rollier, de Oliveira, Bruno and Baetens' Essential metrics for Life on graphs, which proposes mean-field and Derrida curves as a rule's genotype against state and defect averages as its phenotype.

The problem nobody has settled

Take 3.4.6.4. Its triangles have three edge-neighbours, its squares four, its hexagons six. What is "born on exactly three"?

For the hexagon it is a mild condition. For the triangle it means every neighbour is alive. The same four characters describe two unrelated automata depending on which tile reads them, and any tiling that is not edge-transitive has this problem.

There are three defensible readings, and this Atlas implements all three instead of picking one silently:

Absolute. The count is the count. B3 means three live neighbours whatever the tile. This is what the Penrose work uses, and it has the virtue that a published rule string means the same arithmetic everywhere — but it makes low-degree tiles nearly inert and high-degree tiles nearly saturated.

Normalized. Rescale each tile's live count to a reference degree — here the busiest tile in the tiling — and test the rule there. A triangle and a hexagon then respond to the same fraction of live neighbours. When every tile already has the reference degree this is exactly the absolute reading, which is what makes it a generalization instead of a different rule. The cost is that it no longer agrees with any published rule string on a mixed tiling.

Per shape. Give each side count its own rule. The most expressive and the least constrained: the shapes become different automata sharing a board, and the rule space explodes accordingly.

None of these is the right answer. Which one you pick is a modelling decision, and it is the reason the control is in the sidebar instead of buried in the code.

The board is the plane, not a torus

A cellular automaton on a torus is a different dynamical system from one on the plane, and the difference is exactly the phenomenology you want to watch. Growth is capped. A glider travels until it re-enters its own wake. A methuselah that would run for a thousand generations saturates instead. The R-pentomino settles at 116 cells on the plane; on any board small enough to see, it does not.

So the default here is unbounded. The board is a sparse set of blocks allocated as the pattern reaches them and freed once it has left, which is what makes "infinite plane" mean genuinely unbounded instead of a large fixed array with the walls out of shot.

A bounded board is still worth having, for the opposite reason: its state space is finite, so every orbit closes.

Five surfaces, and no sixth

Gluing a board's edges is quotienting the plane by a group $Γ$ of isometries acting freely, so the board is a two-dimensional Euclidean space form, and there are exactly five of those. The plane ( $Γ$ trivial), the cylinder (one translation), the Möbius band (one glide reflection), the torus (two translations) and the Klein bottle (a translation and a glide). The projective plane is the one that looks like it belongs and cannot: a closed flat surface has Euler characteristic $0$ by Gauss–Bonnet, and $χ (RP^{2}) = 1$ .

Which of the five a given tiling can be glued into is not a free choice. A seam glued by TRANSLATION costs nothing: translating by $W v_{1}$ is a symmetry of every periodic tiling by construction, so the cylinder and the torus always exist. A seam glued with a FLIP folds through a glide reflection, and that is a quotient of the tiling only when the tiling admits that glide. A chiral tiling admits none, and gluing it anyway would join tiles whose edges do not meet, inventing adjacency that is not there. So the surface picker tests the map itself and disables what it cannot honestly offer. Of the eleven uniform tilings, ten pass; the one rejection is the snub trihexagonal tiling, which is the only chiral one. Across a five-hundred-record sample of the Atlas's Euclidean corpus, $96%$ pass.

Reading the wallpaper group is not a substitute for testing the map. The group says a mirror exists somewhere; it does not say the axis lines up with the seam being glued, and where the axes fall is not something the name records. The snub square tiling is $p 4 g$ , which is $4 * 2$ in orbifold notation, so the mirrors are there: two perpendicular families, running through the triangles and missing every square, because the square centres are its four-fold points and in $p 4 g$ those lie off the mirror lines. Which of its seams can be flipped is settled by mapping the polygons, not by reading the three letters.

Running a board that has no consistent side

The flip makes two demands a flat array of fixed offsets cannot meet. The reflection's $- 1$ eigenvector is perpendicular to the seam, and the lattice vector along it is generally not $v_{2}$ : on the hexagonal lattice the reflection across $v_{1}$ sends $v_{2}$ to $v_{1} - v_{2}$ , so the perpendicular lattice direction is $2 v_{2} - v_{1}$ , generating a sublattice of index $2$ . And the glide maps tiles onto tiles while permuting the slots inside a cell, which no fixed offset can express.

Both are paid for once, before the first generation. The adjacency is rewritten on the sublattice $⟨ v_{1}, w ⟩$ , whose cell holds one or two of the tiling's own and in which the reflection is exactly $diag (1, - 1)$ . Then the board is run on its ORIENTATION DOUBLE COVER: a cylinder for the Möbius band, a torus for the Klein bottle, twice as wide, carrying a state invariant under the deck transformation $ι$ .

That last step is not an approximation. $ι$ is an automorphism of the adjacency graph and maps every tile to a congruent one, so it commutes with the rule; an $ι$ -invariant configuration stays $ι$ -invariant exactly, in integer arithmetic, forever. Invariant configurations on the cover are in bijection with configurations on the quotient, and a covering map preserves each cell's neighbour multiset. Running the cover is running the quotient, and the simulation kernel never learns that the board has no sides.

One visible consequence: on a tiling whose only glide shifts by half a cell, the board's width is a half integer, and the sidebar reports it as such. Six and a half cells is what the seam actually closes up over.

What three dimensions can and cannot hold

The 3D view draws the quotient itself, and each surface is a different amount of honest. The cylinder embeds isometrically, so its tiles keep their true shape and size. The torus and the Möbius band embed but not isometrically, so their tiles stretch where the surface curves; that distortion is the price of seeing the identification at all. The Klein bottle does not embed in $R^{3}$ at all, because every closed surface in $R^{3}$ is orientable, so it is drawn as an immersion and passes through itself. Both standard ones are offered. The bottle is the shape everyone recognises, with the neck puncturing the wall; its proportions are fixed, so it cannot take the board's aspect ratio, and the length of neck inside the body is hidden. The bagel is the figure-8 immersion, a lemniscate cross-section swept round a circle with a half turn; its tube is near enough uniform that every cell stays visible. In both cases the self-intersection is an artefact of three dimensions: no cell there is adjacent to the one it appears to touch, and the automaton knows nothing about it.

Why the tiling being periodic is what makes this fast

Every Euclidean record in the Atlas is a fundamental cell of $n$ tiles plus a lattice basis. So a tile in the plane is addressed by $(i, j, t)$ — lattice cell, slot within it — and the adjacency is translation-invariant: slot $t$ 's neighbours are always the same fixed list of $(Δ i, Δ j, t^{'})$ . That list is the tiling's analogue of the eight Moore offsets, and computing it once is what makes everything else cheap.

Concretely: copy a block of the board into a scratch buffer with a one-cell border of its neighbours, and a neighbour's position in that buffer becomes a constant byte offset from the cell's own. The update loop is then an array read per neighbour with no hash lookup, no bounds test and no per-cell adjacency list — the tile-with-border idea from Rokicki's survey of Life algorithms, carried across to a tiling.

Two well-known tricks do not carry across. The bit-packing that takes square-grid Life past $1 0^{9}$ cell updates per second works by shifting a machine word so every cell sees its neighbour at once; on a tiling the offsets differ per slot, so a shift does not align. And HashLife generalizes in principle — its quadtree would sit on the $Z^{2}$ lattice index — but it advances $2^{k - 2}$ generations per node, which is useless for a viewer whose entire purpose is showing you every generation.

Where this could go

The gap in the literature is not subtle. Bays did three tessellations by hand; the Atlas holds thousands, across three geometries, with exact adjacency already computed. Running his conditions — bounded growth, and a glider that appears on its own from soup — across the uniform tilings is a search nobody appears to have done.

What would make that search work is not another density-and-complexity scatter plot. Collapsing a rule's whole spatio-temporal evolution into two scalars cannot distinguish a chaotic burst from a fleet of drifting gliders, which is exactly the failure mode a previous attempt of mine ran into. Detecting localized moving structures directly is the harder and more honest instrument, and Rollier et al.'s graph-native descriptors are the current best starting point.

Sources

Carter Bays, "A Note on the Game of Life in Hexagonal and Pentagonal Tessellations", Complex Systems 15(3), 2005; and "Candidates for the Game of Life in Three Dimensions", Complex Systems 1, 1987 (the criteria).
Nick Owens and Susan Stepney, "Investigations of Game of Life cellular automata rules on Penrose Tilings: lifetime, ash and oscillator statistics", Journal of Cellular Automata 5(3), 2010, 207–225.
Adam P. Goucher, "Gliders in cellular automata on Penrose tilings", Journal of Cellular Automata, 2012.
Duane A. Bailey and Kathryn A. Lindsey, "A Game of Life on Penrose tilings", arXiv:1708.09301, 2017.
Seung Hyeon Mandy Hong and May Mei, "The Game of Life on the Robinson Triangle Penrose Tiling: Still Life", arXiv:2302.10157, 2023.
Maurice Margenstern, "Cellular Automata in Hyperbolic Spaces" and the pentagrid/heptagrid universality series (e.g. arXiv:1606.09488, arXiv:2306.06728).
Carsten Marr and Marc-Thorsten Hütt, "Outer-totalistic cellular automata on graphs", arXiv:0812.2408, 2009.
Michiel Rollier, Lucas Caldeira de Oliveira, Odemir M. Bruno and Jan M. Baetens, "Essential metrics for Life on graphs", Physica D 134950, 2025 (arXiv:2506.21226).
Eric Peña and Hiroki Sayama, "Life Worth Mentioning: Complexity in Life-Like Cellular Automata", Artificial Life 27(2), 2021, 105–112.
Tomas Rokicki, "Life Algorithms", G4G13 gift exchange, 2018.