RANGEBLENDS · WANDER · THE MENU

Where do you want to go?

Every link opens Wander at exactly that: a world with a craft in it, a race on its grid, a game at its start line, missions with the clock running. The full picker with all sixty-seven worlds, garages and seeds is one page over.

The Ruler LandExplore◈ The SeamsRacesGamesFlyThe fleetMissionsPrizesDeliveriesPuzzlesCraftControls

The Ruler Land

Twenty ranges whose ground is arithmetic, not noise: the ruler sequence, the light-cone edge, Walsh functions, Gaussian primes, the Legendre symbol, a fluid's wave and solved Sudoku grids, each written twice (Python and JavaScript) and checked against one fixture. The Ruler Land page draws every law's profile and links each range; every one has prizes on its summits and a warp gate on its biggest hill, and the Tour (G) or the Flyover flies the six biggest.

🪨 The Light Cone, flownThe wall of rule 30's edge with the Flyover running: the six biggest hills, the gate, two prizes, home. The first place to visit. 🗺 The Ruler Land pageAll sixteen ranges with their laws drawn, their numbers, a Fly link and a warp to each gate. Start here. 🪨 The Light ConeRule 30's edge read as stone: three walls of overhangs and pockets from the exact bits, XOR and comb drivers on the outer two. 🟫 The Walsh MesasExact square waves as ground, mesas aligned to powers of two. 🏝 The Gaussian IslesAn island on every Gaussian prime; no hash. 🏯 The Ziggurat RangeNested plateaus wider with rank, rivers through the cuts, towns on the flats. 🔢 The Sudoku MesasSolved grids as ground, no two mesas in a line level. ⛰ The Ruler Hills and their riverA hill per column by rank; one basin per region, mouth to the west. 🏜 The Ruler DunesThe hills drawn with a fluid's steepened wave. 🌾 The Legendre MarchesNew: a fen from the Legendre symbol, the causeway of squares along the axis. 🎛 The Synth RangeNew: square waves synthesized to a wanted spectrum, terraced, at night. 🛤 The Prime RailsNew: every odd prime hangs off its octave's Mersenne ceiling or Fermat floor: two rails, a station at every prime, twin and sexy bridges, Cunningham towers, Proth ridges, the gap valley between. 🗝 The Fold MarchesNew: the XOR laws as ground. Relief from mask agreement between neighbouring primes, the two-squares realm and the Gaussian realm either side of the road, vault doors where the remainder folds to a nested square. 🌀 The Vortex SeaNew: an exact integer fluid, frozen. Two-dimensional vorticity on an integer stream function, divergence exactly zero, one seed's whirls as hills and channels as sea. 🕯 The Octave ValleysNew: one mirrored valley per octave of the number line, its two walls always summing to 2^k − 2, and a beacon pillar at every Mersenne prime; this region holds 524,287. 🌊 The Burgers ShoreNew: the Burgers equation alive as ground, the scarp sharpest at dusk; a hovercraft for the bays. 🌉 The Forge BridgesNew: the gap valley with a bridge at every twin-prime pair and a gorge at every record gap. 🧱 The Certificate WallNew: rule 30's edge as a wall, every course certified reachable or cut at its break bit. 3️⃣ The Ternary HillsNew: the Ruler Hills in base three, a hill wherever 3^k divides the column; a quad for the mounds. ⚅ The Sixfold HillsNew: two rulers on one ground, twos and threes summed; rolling downs by buggy. ⛰ The Sixfold AlpsNew: the same under the Alps' law, rock domes over green vales; fly the helicopter. 🕳 The Ternary CanyonsNew: a plateau pocked with bowls of four sizes, the deepest as lakes. 🏝 The Ternary IslesNew: an archipelago of three island sizes on six lanes; a boat world. 🛩 The SkywayNew, for the hypersonic jet: a new district every half minute, a beacon always on the horizon, a spire on a 1,020-block summit. 🛸 The Drone SlotsNew, for the quadcopter: slot canyons at drone scale, pools, sills to hop, towers to land on. 🎈 The Balloon ValeNew, for the balloon: broad hills, a lake in every hollow, a flat to land on at every crest. 🏔 The Sky MesasNew: tablelands of 120-block tiers to thread in the jet and land the Apache on. 🌍 The Grand TourNew: every law in the kit by district, a new country every half minute, towns in the meadows; an hour never repeats. 🛩 The Sky TourNew, flyers only: walls, alps, mesas and rails, a landmark three hundred tall in every district. 🚙 The Buggy TourNew: every district drivable and crossable, dunes to meadows to hills; towns on the way. 🛻 The Monster TourNew: rough ground for trucks, mechs and tanks, canyon rims to ziggurats to ramps. 🛟 The Hover TourNew: land and water alike, never all sea, never all land; for hovercraft, landspeeders and jetskis. 🌿 The Sixfold ValeNew, in the valley look: the Sixfold Hills as a rolling green vale with a river, sand banks, reeds and villages; a buggy world. 🍃 The Ternary DaleNew, in the valley look: the Ternary Hills gentle, a river down the dale; a quad bike world. 💧 The Fold Water MeadowsNew, in the valley look: the Fold Marches kept low, domes among flooded meadows; hovercraft country. 🌅 The Shore ValeNew, in the valley look: the Burgers Shore's wave at dawn as a gentle riverside with lakes in the troughs; speedboat or hovercraft. 🏞 The Grand ValeNew, world-scale: the river cut hundreds of blocks deep with a green floor, villages and reeds, the walls in terraces, summits 600 to 780 over the floor; start on the floor by the trunk, walls towering both sides. Helicopter or plane, a buggy on the floor. 🦇 Blackhood over the Ember RiftNew hover-jet: a muscle wedge that flies at 200 a second, hangs like a helicopter (Space, C) and slides like a saucer (Q, Z); Shift is the burner. 🗡 Razorline over the SkywayNew hover-jet: the quickest slide and the tightest turns, swept wings lit at the tips. ◼ Nightslab, ultraviolet, at nightNew hover-jet in an electric paint: silent at the hover, violet edge lines, all burner at speed. Every craft takes the new electric paints (Vehicles ▾). 🏁 The Emberline: a pod raceNew: seven pods, three laps through the Walsh Mesas: a slot canyon, the Needles, a plateau and its drop, a kicker over a gorge. Straight onto the grid. ☠ Last Pod StandingNew: the Emberline in the Twin-ram pod; at each of the first two lines the last two pods are out. ⏱ Pod time trialNew: three laps alone in the Tri-engine pod against the ghost of your best lap; any word on the link lays a new circuit. 🏜 The Glasswind RunNew: the long fast circuit through the Ruler Dunes, in the Drifter pod. 🔤 A word's own trackNew: the Glasswind with the word heron; change the word on the link (seed=) and the course law lays another circuit. 🏎 The Podracer DownsNew, for the pod racer at three times speed: long rolling swells with no cliffs, a river to skim with banked bends and one-block fords, villages on the crests to steer by; start on a crest with a 250-cell descent east. 🌊 The Twin RiversNew, in the valley look: two rivers meeting, reedbeds on both, a town just downstream of the fork. 🪜 The Ladder ValeNew, in the valley look: the Ladder's terraces at a gentle step, the lower shelves water meadows, a river cutting down through them; every ramp a buggy climb. 🧱 The Light Cone ValeNew, in the valley look: the Light Cone's three walls kept low with a river on the plain between them; the hoverboard climbs the walls. 🕯 The Octave DaleNew, in the valley look: the Octave Valleys' mirrored walls with a river the length of the dale and the beacon pillar at 524,287. 🏔 The Sky ValeNew: the Grand Vale's law at stature 1.6, for the glider and the jet; start on the rim at dawn facing along the canyon, snow on the tallest summit. 🌾 The Marsh MeadowsNew, in the valley look: the Legendre Marches as a marsh with sand banks and reeds, water on two fifths of the ground, villages on the dry rises. 🏜 The Ternary GorgeNew, in the Sky Vale's recipe: the Ternary Hills at stature 1.5 cut by a gorge 330 deep, a green floor, domes to 1,099; start on the rim at dawn. 🗝 The Fold CanyonNew, in the Sky Vale's recipe: the Fold Marches scaled up, the domes and spires along the north rim, the gorge on your right; evening. 🪨 The Light Cone GorgeNew, in the Sky Vale's recipe: a canyon running 1,800 columns between rule 30's walls standing as ramparts on its rims. 🕯 The Octave CanyonNew, in the Sky Vale's recipe: the Octave Valleys' walls deepened into a canyon, the beacon of 2^19 − 1 standing 912 high, snow on its cap. ⚡ The Sixfold ChasmNew, in the Sky Vale's recipe: the Sky Vale's own law at stature 2, narrower and deeper, a chasm to thread at speed; summits to 1,380. 🏔 The Ruler AlpsMassifs under the max law; snow above 120. 🏜 The Ruler CanyonsThe hill law turned downward, lakes in the deepest cuts.
🧱 Light Cone 2: build your own walls

Every wall is the register x_j ← x_j XOR G(x_{j-1}, x_{j-2}) under a driver of your choice, read as stone one generation per cell: OR is rule 30's edge, XOR the Sierpinski lace, the comb freezes every other row. Pick up to five walls, how deep the lace goes, how tall a bit stands and how far apart the walls run; the link carries it all, so it can be sent.

Build it ↗

Presets: three Sierpinski walls · five walls of the edge, tall · combs about the edge, four blocks a bit

◈ The Seam Ranges

Lars's address space has two mirrors in it: one folds on a lattice of powers of two, the other folds on the squares. Put them out of true and they agree almost nowhere — and the rare places where both still cut are the seams. A seam's position is settled by arithmetic alone, so there is no table, no seed and no spawn rate, and the same seam stands in the same place for everyone. The gaps between them grow by exactly 2·step² every time, which is what both games are built on. Next seam ◈ on the travel bar flies you to the next one by the law rather than by searching.

◈ The Seam Ranges pageThe law, its table at five widths, all three ranges, all three craft, and both games playable on the page itself. Call the Seam has a right answer that needs no simulation at all. ◆ The Close SeamsWidth four, the mirrors nearly true: sixty-three seams, the first at g = 15, then 63, 143, 255. Gaps of 48, 80, 112, 144 — close enough to walk one to the next and watch the spacing grow. The best range for seeing what the law does. ◈ The Seam RangeWidth six: thirty-one seams, the first at g = 63, every gap 128 cells longer than the last. Hills, rivers, prizes on the summits and a warp gate on the biggest. ◇ The Wide SeamsWidth eight: fifteen seams, the first at g = 255, every gap 512 cells longer than the last. Fewer marks, longer country, every one provable before you arrive. Surveyor · reach 24A long-range survey truck. Unhurried in a bend, but the next seam's distance is on the dash from twenty-four cells out. Plumbline · reach 10Built for the straight run between two seams and nothing else. Quickest on the range in a line, nervous near a slope. Take it when you already know where you are going. Gnomon · reach 40A slow surveying helicopter that hangs where it is put. The widest reach of the three, and it sits over a summit while you read the arithmetic off the mark. ▶ Call the SeamA stretch of ground, and one question: does a seam fall inside it? The law settles it exactly, so guessing scores about half and working it out scores all of it. Plays on the page, no world needed. ▶ The Seam RunReach six seams in order. Each leg is longer than the last by exactly 2·step², so the whole route can be worked out before you move. The planner gives you the route; the running is done in the range.

Explore

Cities grown by the law, ranges that go on for ever, and the classic valleys. Any world takes any craft.

⛰ The Sieve RangeSeven ranges written by nothing but Lars's range, stride, stripe and sieve law: no noise anywhere, and the Rule panel will tell you why any hill is the height it is. Every hamlet's doorway opens twenty-five courtyards compiled from its own column. ▦ The Indexin TerracesStride seven, mask five, modulo four, dropping residue zero. Fifty-six columns and the skyline is identical, exactly. ◫ The Dropped TwoThe Terraces with one field changed: residue two thrown away instead of zero. No part of the skyline survives it. ✝ The Long CountStride nine, stripe twenty-five, sieve forty-nine, mask 4,095: 45,158,400 columns before the pattern comes round. 🏜 The Indexin CanyonsThe same law with its sign flipped, subtracted from a plateau, so every address it keeps carves down instead of up. 🏝 The Sieve IslesThe same counting with the ground dropped under the water line: the tiers that would have been hills are islands. ☖ Silo LineA tower game. Five waves come up a winding road from the far gate to your keep, each worse than the last. Build a mortar silo, a rail turret, a scatter gun or a pulse mast by driving to the spot, and upgrade between waves. The road is laid by the law, so it is the same road every run. ◢ The Crestline: Hull DownA tank game about reading the ground. Keep your hull behind a crest and only the turret over it: shells that find you there do half, ones that catch you in the open do double, and only hits from cover score. ◢ The Sieve RidgesA tank duel on forty-three blocks of law-built relief. The Longbow reaches a hundred and seventy blocks; the crests decide the rest. ☸ The Ember SiegeClose work in the rift valleys, in fog, with lava along the floors. ▓ The Step GauntletWave after wave across the Dropped Two ground. Hold the high step. 🏁 The Step CircuitThree laps of the Thousand Steps law at a fifth the height: it banks where the ground steps and jumps where it drops. ⛽ The Ember RunSixteen gates through the Ember Steps valleys, fissures still burning along the floors. ⥣ The Thousand StepsThree heights, a hundred and eighty blocks apart, and a cliff between each. Built to be flown at. ☲ The Ember StepsThe selection carving down through rock, the rift palette on the walls, lava along the floors. ⌵ The Deep VaneGorges cut most of the way through five hundred blocks of tableland. Flown through, not over. 🏎 The Sieve RallySixteen gates around the benches, in a car, a buggy or a tank. Every terrace edge is a jump put there by an XOR. 👑 The Vast CitiesYou start on a street of a megalopolis 560 cells across. Capitals, Crowns and an Empire further on. 🏙️ The Skyline EmpireTowers two hundred blocks tall in glass and steel, on a great range. 🌄 The Valley of TowersA rolling wooded valley with mega structures among the hills. 🌆 The Ruler CitiesSixteen settlements a region, hamlet to metropolis, placed by the ruler. 🏜 The Hanging CitiesCities on terraces cut into canyon country. 🔧 The Law ForgeA range of your own from seven knobs, lots among them; the link carries them. ⛰ The Ruler HillsNew: the land itself from the ruler sequence, a hill per column by its rank, a prize on every prize summit, a warp gate on the biggest hill. Wild. 🪜 The Ladder RangeNew: terraces at ladder heights with ramps between; towns on the plateaus. 🌄 The Dyadic DownsNew: a lattice of hills, big where both coordinates are highly divisible. 🪨 The Light ConeNew: rule 30's light-cone edge read as stone, three walls the length of the range, overhangs and pockets from the exact bits. Fly the face. 🟫 The Walsh MesasNew: exact square waves as ground, mesas aligned to powers of two. 🏝 The Gaussian IslesNew: an island on every Gaussian prime, no hash. 🏜 The Ruler DunesNew: the hills drawn with a fluid's steepened wave, gentle west, scarp east. 🔢 The Sudoku MesasNew: solved grids as ground, no two mesas in a line level. 🏯 The Ziggurat RangeNew: nested plateaus from the ruler, wider with rank, ramps at every edge; towns on the flats, rivers in the cuts. 🏔 The Ruler AlpsNew: the hill law at mountain scale, massifs three hundred blocks high; the Tour flies the six biggest. 🏝 The Dyadic IslesNew: the downs with the sea let in; only the big lattice hills are islands. 🏜 The Ruler CanyonsNew: the hill law turned downward; lakes in the deepest cuts, prizes on the canyon floors. ▲≋▲ The Long Range at nightA village by your own lantern; the range goes on for ever. 🏙️ The SprawlCities on broad lots; at every eighth block one estate with a single great building. ⚓ The ArchipelagoHarbours, a volcano, a hovercraft at sunset. 🚤 The Lantern CoastA drowned range; the speedboat's home water. ✦ Your Own ValleyType a name, get a valley. Six true things are its lanterns.

Races

Three laps, a garage, and a catch-up clutch. Enter races again.

Games

Driving and tank games, each its own world. Any arena game takes builder knobs on its link: &cars=3 for the rivals, &clock=30 for the seconds, &ring=28 for the ring; the Arena Cup button in an arena garage runs five of them back to back.

Fly

Plane, gunship, Wraith, UFO: every world takes them.

The fleet

Twenty-three more ways to travel, in any world: G cycles through them, the Vehicles button lists them. A hypersonic jet, an Apache, two drones, a hoverboard that climbs cliff faces, a jetpack, a hang glider, a snowmobile, a quad bike, a balloon, a mech walker, a rocket bike, a sailboat, a submarine and a maglev sled; then a second fleet for the epic aerial adventures: a tiltrotor, a stealth bomber, a gyrocopter and a rocket, four repulsorlift craft that defy gravity (a landspeeder, a speeder bike, a skiff, a pod racer), and hellfires on the Apache, the Reaper and the tiltrotor: a lock-on, a homing rocket, a splash of nine that kills every hostile inside it, a crater five wide. Four a sortie, rearmed on landing; &hellfire=off on a link leaves the rails empty.

🛩 The hypersonic jet over the Long RangeThree times a plane’s speed, a contrail, afterburners on Shift. 🚁 An Apache over the MarshRocket pods and a chain gun; it lands anywhere. 📷 A quadcopter in the SkylineSeen through its own camera, forty metres up at most. 🦅 A Reaper over the Cinder RangeLaunched high; two missiles that crater the ground. 🛹 A hoverboard on the Walsh MesasWhere a mesa wall stops it, it lifts straight up the face. 🧑‍🚀 A jetpack in the Skyline EmpireFly between towers two hundred blocks tall; land to refill. 🪂 A hang glider off the Ruler AlpsNo engine: dive for speed, flare to stretch the glide. 🛷 A snowmobile in the Frost RangeQuick on snow, slow off it. 🛞 A quad bike on the Ruler DunesClimbs what stops a buggy; the scarp side of every dune. 🎈 A balloon over the Valley of TowersThe wind takes it; Space burns, C vents. 🤖 A mech walker in the SprawlSix blocks tall, every stride shakes the ground. 🚀 A rocket bike through the Vast CitiesThe quickest thing on the ground; a long burn. ⛵ A sailboat in the ArchipelagoFast before the wind, slow into it; the almanac says how hard it blows. 🔱 A submarine in the ArchipelagoC dives to periscope depth, quicker there; torpedoes. 🚄 A maglev sled in the Ruler CitiesNinety a second on the streets and the railway, a crawl off them. 🛩 A tiltrotor over the Lantern CoastA gunship under twelve a second, a plane above; the nacelles swing. Hellfires that lock on. 🦇 A stealth bomber over the Cinder RangeLaunched high; B drops one of six gravity bombs, each a crater. 🪁 A gyrocopter over the Ruler HillsIt never stalls; close the throttle and it sinks gently. A rifle. 🚀 A rocket off the Sixfold AlpsStraight up to six hundred, fireworks at the apogee, a glide home. 🛸 A landspeeder on the Ruler DunesA repulsorlift 1.2 m over any ground or water, cliff faces too; a blaster. 🏍️ A speeder bike through the Ruler CanyonsVery fast, a metre over anything; a wall above forty a second spins it out. ⛴ A skiff over the MesasSlow and wide, three metres up, heavy freight and a rail to stand on. 🏎 A pod racer through the CanyonsA hundred and twenty a second on two engines and cables; a huge nitro. 🚁 Apache hellfires over the Ruler HillsRaiders in the air: a bracket, LOCKED, E, a fireball and a crater.

Missions

Six against the clock in any range: reach, overfly, rooftop, courier, drones, beacons; spires and inner courts in the law cities.

Deliveries

A delivery service: six parcels in a chain, each picked up where the last was dropped, the first next door and the last across the region. Two are rush jobs on a short clock for half again the pay; one is heavy freight that only a buggy, tank, hovercraft or boat can carry, for double. Deliver with time in hand for a tip, keep it up for a streak. Amber beacon for the pickup, green for the drop. Boats and hovercraft on the coast.

Prizes and warps

On the ruler land a prize stands on the summit of every column of rank five or more with an even square remainder. Touch it to take it; the Warp panel (Z) lists the region's prizes, its gate and your bookmarks, and takes any pasted warp address, in this world or another.

Puzzles and lanterns

Craft

G or the Vehicles button in any world; every one is armed, in any colour you like.

Controls

Wander · by Lars Rocha · back to the Range Arcade home page