Patterns
Every pattern on this page is made from nuna tiles on the square lattice: a circle wherever i + j is even, a diamond wherever it is odd, and each tile one circle fused to the diamond beside it. The engines below work out what those tiles can do together.
Catalogue
Every grouping of one to six tiles joined on the lattice, sorted by how it repeats across the plane: by sliding alone, only after a turn, only with gaps left between, or not at all that any search has found. Beside them sit the boards: every way a repeating block from 2×2 up to 6×6 cells can be covered by tiles alone, and 998 more that leave gaps.
Six-tile groupings are still being classified. 2,469 of the 51,392 that exist are in the catalogue so far.
Click the board or press → for the next pattern, ← to go back, ↑ ↓ to change how a grouping is filled with tiles. Keep adds the pattern to a collection held in this browser.
Two-tone index
Orientation
Source
Checked
- Counts are read from the catalogue data when the page is built: 7,295 groupings, 862 gap-free coverings, 998 coverings with gaps.
- Every curve on the board is drawn as an SVG arc of radius r on a lattice of step r√2. The rendered board was scanned: its paths use only move, arc and close, with straight lines only on the frame’s outer rectangle.
- The Library palette is drawn in the page’s own tokens, so the board turns with the theme. Both themes were rendered and compared. Saved SVG and PNG files carry real colours.
Cousins
Cut a circle from the square around it and four offcuts are left, one in each corner. A cousin is that circle with some of its own offcuts put back in new places. Nothing is added and nothing thrown away, so a cousin with all four returned has the square’s area, 4r². Nuna is one of them.
Of the 73, 35 tile the plane on their own and 262 pairs tile together; 32 pairs are still open. A cousin with k pieces carries four minus k spare quarter discs that nothing can absorb, so anything under four pieces never tiles, alone or in any company.
Choose shapes from the shelf, then Find pattern searches for a repeat that uses every one. ← → step through shapes, , . through arrangements, ↑ ↓ through two-tone inkings.
How the cousins are made
Cut a circle from the square that circumscribes it and four offcuts are left, one at each corner. Every cousin is that circle with some of its own offcuts put back — one, two, three or four of them, in a new position. Nothing is added and nothing is thrown away, so a cousin with all four pieces returned has exactly the area of the square it came from: 4r².
Where a piece may go. An offcut has one concave arc and two straight legs. Its arc can lie back along the circle — which forces its arc-corner onto the circle’s centre, so there are exactly four such positions — or a leg can be fixed flat against the leg of a piece already placed. Those two moves generate all 73 cousins. The offcut is its own mirror image across its diagonal, so it is only ever turned, never flipped: four turns, no reflections.
Why some can never tile. A cousin with k pieces, h of them on the circle, has 4−h quarter discs and k−h offcuts. The difference is 4−k, so anything with fewer than four pieces carries spare quarter discs that no other cousin can absorb. Those shapes leave one offcut-shaped hole per missing piece, per tile — and the hole is a corner piece, the same thing that was cut out to make it.
Checked
- Shape, pair and open-pair counts are read from the cousins data when the page is built.
- The search runs off the page’s main thread and can be stopped at any time.
Living
The same tiles, handed to a rule and watched. Each generation a tile stays if the number of tiles touching it is on the rule’s stay list, and a tile is born in empty places whose count is on its born list. A pattern can hold still, pulse with a beat, slide across the cloth, heal a missing tile, scar around it, or melt away.
No pattern is ruled out: a pattern that dies under one rule lives under another. The gap-free coverings from the Catalogue show this most plainly. Every tile in them touches at least five others, so the default rule clears the whole cloth in one step. They open here under a dense rule instead, born with three to six and staying with five to eight, where a missing tile can heal.
Space plays and pauses, . steps one generation, F takes out a tile, R starts again, ← → move through the source.
Click a tile to take it out, or an empty place to set one down. The board wraps at its edges, so it behaves exactly like the endless cloth.
Source
Rule Default
Neighbours are the placements that touch a tile, 24 in all. The rule is yours to change: a cloth that melts under one rule may heal under another.
Board
Checked
- The step was written from the rule descriptions in the nuna life pattern pack and checked against everything the pack records: all 723 generations of its living and travelling cloths reproduced exactly, the beat of all 241 swatches, and the beat of 240 sampled ornaments across the four rules.
- The page’s own copy of the step was run against the same recorded generations in the browser, with the same result.
- Every cell in the pack was checked as a legal arrangement: no place used twice, every tail on a diamond place.
- Gap-free coverings: on every repeat up to 4×6 (about 6,900 coverings) the fewest neighbours any tile has is five. Of the 50 distinct 4×4 coverings with one tile taken out, under the dense rule 21 heal, 15 scar, 12 pulse and 2 die.
Ornaments
An exhibit from nuna life. Each ornament is the form a rule comes to rest in: a few seed tiles go in, the rule decides everything after, and what is left when it stops is the ornament. Nothing here is a stored picture. Every piece is regrown from its seed in your browser, so what you watch is the rule at work.
Choose a piece in any room to grow it in that room’s case. A piece grown under a 24-neighbour rule can be sent to the Living board above, settled, with its own rule, to be damaged and watched.
The ethos of nuna life, in ten principles
What the Game of Life on the nuna lattice has taught, written to be carried into whatever comes next.
1. The rule is the author
Nothing in nuna life is drawn. A seed of a few tiles goes in, and the rule decides everything after. The ornaments, cloths, gliders and fires are found, not designed, and that is where their authority comes from: each one is the only thing those tiles could have become. To make something new, change the seed or change the rule, never the result.
2. Symmetry in, symmetry out
The rules treat every direction alike, so a symmetric seed can only grow a symmetric shape. Beauty is chosen at the seed: an 8-way seed grows an 8-way ornament, a single mirror grows a face or a letter, a turning seed grows a pinwheel.
3. An ornament is where a rule comes to rest
Most growth ends. A seed under the default rule churns for a few generations or nearly a thousand, then stops, or settles into a steady beat. That resting form is the ornament: a kind of equilibrium, a negotiation the rule finished. Watching it grow is watching the negotiation.
4. Rules are materials
The same seed grown under another rule is the same idea in another material:
- Default is stone. Both fail is lace cut in stone. Island is bold and round, and Labyrinth dense.
- Lace blooms. Woven draws lines, and Lines is chunky.
- Traffic moves. Wildfire never stops. Direction order cuts gems.
So a rule never disqualifies a pattern; it gives it a material. If a rule rules out something interesting, change the rule, or give the changed rule its own section.
5. Families grow; nothing is alone
- Tiers: an ornament with a ring of new seed tiles round it grows the next tier of its family.
- Crosses: the inside of one seed with the outside of another breeds a new one.
- Echoes: a copy drawn larger is thinned by the rule to what holds, and keeps its likeness.
- Mutations: damage one, and about one time in six it heals exactly. Otherwise it becomes a close variant, like an alternate letterform.
Every shape has relatives, and its family tree is part of what it is.
6. Touching is negotiating
Pieces close enough to touch negotiate their borders:
- Two ornaments edge to edge settle into something new.
- A seed planted on a grid becomes a cloth.
- A field of pieces placed too close never settles at all.
Room is a material too: three places of it lets each piece stay itself.
7. Patterns live
Cloths can hold still, beat, or travel, and a missing tile in one heals, scars, fuses or melts. A pattern is a behaviour as much as an image, so file it with its motion, its period, and what it does when damaged.
8. Every shape has an identity, so every find counts
- A fingerprint per shape: a shape's canonical form, with its turns and mirrors folded together, is its fingerprint. The catalogue holds 5,050.
- A find: anything settled that you make can be checked against them and called new to nuna, new to you, or seen.
Making something the catalogue doesn't have is the point. Anyone can find it, and it can be filed.
9. Measure, and correct in the open
- Measure properly: at the same phase each period, over whole cycles and every alignment.
- Prove it by brute force: a puzzle's answer counts only once every move has been tried.
- Correct out loud: when a measurement turns out wrong (the glider lanes needed 21 places, not 12), the correction is filed beside the claim, not hidden.
10. Everything travels
Every piece is a code that opens in every edition, both books, the games and the puzzle maker. What is found in one place seeds the next. The oracle is that seeding made deliberate: each discovery filed with its evidence, its rules, its examples and the projects it could carry to.
Families grow
An ornament with a ring of new seed tiles round it grows the next tier of its family. Each row is one family: the base ornament, then one ring out, then two.
Rules are materials
Finished volume-one ornaments used as seeds under a different rule, so the same piece is regrown in a new material. Life calls rules other than Conway’s Life-like; here a rule is a material.
Breathers and pulses
Oscillators that grow from a single combination shape and keep returning to it, among them the breather: four tiles that bloom to thirty and come back every 41 generations. In Conway’s Life oscillators are the clocks of every machine; here they are motion pixels, marks that pulse at their own tempo. Symmetric ornaments that settle into a steady beat instead of holding still. None is a phoenix, where every tile changes each step: at least two-thirds of each stays put, a still frame (Life’s stator) around a moving heart (its rotor). The most restless, with up to a third of themselves moving, are named Restless ornament.
Glyphs and masks
Small seeds with a single mirror that settle into one compact piece with enclosed spaces, like a letter, a sign or a mask. A script for nuna: 72 from the default rule, 36 bolder ones from Island and 36 more letter-like ones from Both fail, chosen from about 35,000 found. Mirror-symmetric seeds under the default rule: faces, masks and totems.
Echoes and giants
Ornaments redrawn 2 to 5 times larger and handed to the rule. A big solid shape is overcrowded, so the rule thins it to the parts that hold, the way it keeps the edges of a photo; what settles is the same ornament at a larger size with new fine detail. The formula: draw the outline larger, fill it with tiles that keep each original tile’s direction and the ornament’s symmetry, try four fills, and keep whichever settled result looks most like the original. Doubling in 2× steps beats one big jump: Four crowns doubled twice scores 0.95 at 4×, and a third doubling reaches 8× (4,644 tiles, too large for this page) at 0.93. Likeness is measured from mid-distance; 1.00 would be identical. The largest ornaments: tiers grown ring by ring around the named families, up to a fifth tier of 5,456 tiles whose centre survived every tier unchanged, like growth rings, and seeds twice the usual size. In Life the biggest settled patterns come from methuselahs.
Crosses and duets
The inside of one seed joined to the outside of another, where at least one parent was itself a cross: breeding, the way Life enthusiasts grow new patterns from old ones. Two different named ornaments set edge to edge and left to negotiate where they touch. 44 of the 78 possible pairs settled into something new.
Mutations
Volume-one ornaments with one symmetric group of tiles taken away, left to regrow. About one time in six they heal back exactly; otherwise they settle into a close variant, like an alternate form of a letter. Mutations of mutations: each family drifts one step further from its original. Second-generation variants heal slightly more often (19% against 16%).
Seeds that never settle
Seeds that never stop: the opposite of the ornaments. Gems are crystals that grow under direction order from as few as 4 tiles, faceted like cut stones, until they fill the board; 513 of the 902 direction-order rules grow one, in 209 distinct cuts, and the 30 most common are here. Starbursts and garlands draw outward forever in the 16-neighbour Lines and Woven rules; garlands only ever appear in the Lines rule. Kaleidoscope, pinwheel and Rorschach fires are the wildfire rule started from symmetric seeds, and fireworks throw symmetric showers of gliders. No seed under the default rule ever does this.
Wallpapers
The same seed planted on a grid, close enough that neighbours collide as they grow. They settle their borders into a repeat, like Conway’s agars, the patterns that fill the whole plane. Mixed seeds never agree, and keep boiling. A finished ornament repeated edge to edge in a square grid or a half-drop, so neighbours touch on every side and negotiate their borders. Many grow linking bridges and become chains, laces and trims.
The complete alphabets
Every ornament an 8-way symmetric seed can grow, for each of the four 24-neighbour rules, with every circle of the seed within 5 or 6 places of the centre. These were found by trying every seed, not by sampling, so each alphabet is closed: there is no other ornament at that size under that rule. A pulsing ornament is counted once, whichever phase it is in. Radius 7 has been closed too, for the Default rule: 377,560 ornaments from 5,298,236 seeds, too many to show here, kept as a data file.
The oracle
What nuna life has found, filed so it can be sorted and carried into future projects. Every entry has a kind, tags, the rules it concerns, examples by reference into the volumes, and the kinds of project it carries to.
Checked
- Every piece on show was regrown from its seed with the nuna life engine before it was let in, and matched the recorded settled form tile for tile. Pieces that beat were checked by their beat: the settled state returns exactly one period later. Three pieces failed and were left out.
- The complete alphabets were found by enumeration: every legal 8-way seed tried once (2,119 at radius 5, 96,242 at radius 6, a count confirmed by a separate method; 5,298,236 at radius 7 for the Default rule, where all 11,786 radius-6 ornaments reappear as they must). At radius 5 they contain every shape in the published sampled alphabets for all four rules; Default and Island match exactly, and enumeration adds 20 Labyrinth and 2 Both fail ornaments the sample missed.
- Tiers carry no seed of their own: each one regrew exactly from its parent’s settled form with its ring added.
- Every piece’s game code is rebuilt from its rule and settled tiles. That matched the volume’s own code for all 300 pieces tested.
- The 41-generation breathers record their seed as their form; the seed is not part of the beat. They are shown settled into the beat, which returns every 41 generations as the oracle says.
Still to come
- Plaid nuna-tartan.html
- A flannel plaid where the nuna wave takes the place of the square, so the checks become nuna shapes. Sett notation, twill and nap, seamless tile export. Built · to be lifted
- Linework nuna-linework.html
- Lines drawn around the tile and its circle and diamond combinations, the wave among them, with every subset of edges of the circle, diamond, tile and 2×2 block. Built · to be lifted
- Fractal tile no file yet
- A nuna outline filled with a smaller lattice of circles and midlines, each point inked by the parity of the circles that hold it. Construction verified · not built
- Illusion nuna-illusion.html
- A tile-based optical-illusion sandbox, driven by touch. In progress
- Bandana bandana generator
- Bandanas composed from fields, borders, wave bands, corners and a centre boss. Its field can already draw from this page’s catalogue. Built · to be lifted
A note
For Aaron to write
What the pattern work is for, in your words. Where it started (the tile sheets, the book), what you look for in a pattern, and which of these you return to.