Slice view is the 3D cross-section of the 4D world through your eyes, the way a 3D visitor would see it. Turning toward ana sweeps the slice through the landscape. Lighting and shadows are computed in full 4D, so hills beside you in ana, outside the slice, still cast shadows into it.
Radar. The ground of a 4D world is three-dimensional, so its minimap is a ball, with you at the centre. The disc through the middle is exactly the ground your slice shows (the bright wedge is what is on screen); above the disc is ana, below is kata. It starts with the way's goal (the bright diamond), the things you can carry (accent squares, filled where your slice cuts them; a cache is three small squares), places (the cave's mouth, the ruins, the launcher and the boats) and your trail of the last two minutes; the keys 1–9 add the rest. Each mark stands on a stalk to the disc: up toward ana (orange), down toward kata (blue); the shorter its stalk, the nearer it lies to your slice. Big (Tab), it adds its scale, the compass and the sun's height. With solid mountains on, ground above 24 m floats in the ball, and on the disc orange marks mountains lying toward ana from that spot and blue toward kata; with water on, blue haze is water. The dot in the middle of your view and the ring on the disc are the same point.
The whole planet. Zoom all the way out and the ball holds the entire planet: its whole outer surface is a single point, your antipode, 785 m away by any straight walk. P pins the ball where you stand; walk and you move through it as a dot, and your slice becomes a curved sheet, a great 2-sphere through you and your antipode.
Compass. Unlike Earth, a 3-sphere can carry a compass with no poles: three perpendicular directions (i, j, k, after the quaternion units) defined everywhere. Walking straight keeps your compass heading fixed but slowly rolls the other two around it, once per lap of the planet.
Boulders are 4D balls. Your slice cuts each in an ordinary sphere whose size depends on how far its centre lies toward ana or kata, so as you turn they swell out of nothing, shrink and vanish. The radar's boulder layer (8) shows them all and rings the ones your slice cuts. They are solid, with friction: push into one and you stop; hold Space and you climb its surface (which can carry you toward its centre in ana).
Plants. Bushes, flowering shrubs, cushions of flowers, tufts of grass and young conifers cover the ground near you, grown afresh as you walk and the same every time you come back. They are made of what a 4D slice can cut, balls of leaves and cones of grass, so as you turn through ana they come and go like everything else. You walk through them.
Trees and rocks. A tree is a 4D tree: its trunk and boughs spread in all three directions of the ground, so your slice usually cuts it in floating pieces of crown and branch, and only near its trunk does it join up. A conifer's needles are a 4D cone; through its axis your slice shows a cone, off it a dome that shrinks as you turn. Rocks are 4D solids: slabs and blocks (tesseracts), crystals (16-cells and 24-cells), smooth stones (duocylinders). Turn through ana and watch their slices change shape. They are solid; hold Space against a big one to climb it.
Ruins, cairns, caches. Someone built here for four dimensions: a ring of pillars stands on a sphere round its centre, so your slice cuts it in a circle of pillars, and a different one as you turn; a room's walls are 3D cells. Caches hold things to carry and impulse stones. Lines of cairns lead away from where you arrive.
Things are 4D too: balls (glomes) and tesseracts, which tumble by turning in two planes at once, so a thrown tesseract's slice keeps changing shape: cubes, prisms, hexagons. What you hold turns with you: carry a thing, turn through ana, set it down, turn back, and it is its own mirror image. It keeps its weight in your hands: heavy things come up slowly and swing when you turn, and what you carry knocks into what it meets. Impulse stones: take one, and G puts it in your pouch (it holds nine) or takes one out; the ruins and cairns grow new ones on their pedestals, a few minutes after the last was taken. Throw one and you recoil (momentum is conserved), which is how you steer where there is nothing to push against. Lanterns light a small, sharp pool: in four dimensions light fades as 1/r³.
Space. On the level top of the highest summit stands a launcher: a metal pad, and a barrel on a gimbal beside it. Stand on the pad and the radar becomes a map of the system, with the path the launch would take, and four levers appear beside it: the first turns the barrel round, the second tips it toward ana or kata, the third raises it, the fourth sets the speed (up to 72 m/s, enough to reach the other planet). Free the mouse (Esc) and drag them, or turn the wheel over one; click the view to look around again. Jump (Space) and it throws you the way the barrel points. In flight the map shows your path ahead; throw impulse stones to change it.
The way. One goal at a time: with your pouch empty, the nearest impulse stone; with a stone in it, the launcher. Or choose your own: free the mouse (Esc) and click a mark on the radar (a thing, a creature, a ruin, a boat, a summit or any ground), and the column stands over it; click it again to let it go. A column of light stands over it, set into your slice so that you see it whichever way you face; while the goal itself lies off your slice the column is dashed. A diamond in the view marks it, with its distance; beside it a key and a caret say which way to step to reach its slice and how far (E toward ana, Q toward kata; close to a stone, in tenths of a metre: it is a 4D ball 18 cm across, and your slice cuts it only within 9 cm of its middle). The key under the dot says what to do there: F take the stone, G stow it, Space jump from the pad. What is fixed in place is the builder's dark stone, its edges faintly lit; what you can carry is pale, and the dot opens with an F under it when it rests on one.
Creatures. Herds of crabs live all over the dry land, each about its own home (found where you are, the same each time you come back). Each herd is a kind: six legs along the ground's three directions, eight toward a cube's corners, or twelve toward an icosahedron's; small, medium, large or big; tawny, rust, slate or sand; with young among them. A crab near you turns square to your slice and stands in it on some of its legs; step a little through ana and its legs leave your slice while its shell floats. They are not afraid of you: they graze on, and a curious one now and then comes over to look at you, then goes home. They are solid: walk into one and it steps aside; run into one and it runs. The small ones you can pick up (F); set one down and off it goes. Roll a ball near a herd and the playful ones chase it. Rollers, duocylinders glazed in two colours and rolling by double rotation, roll loops about their homes and chase and knock on a ball you set rolling.
The lighthouse. On a headland stands a tower with a lamp that flashes twice every eight seconds. The planet's copies round the hoop are this planet, so its light flashes on each of them too: look along the hoop with your slice through the lamp and the whole row flashes. (Light reaches you only along your slice: if you can't see it, step through ana.)
The other planet. Planet B is a crystal, a 120-cell you can walk on. Marks show its console and its return pad. The console's screen holds the universe with itself in it; its dials change the laws.
Water. You can't swim. Wade in up to your chest; any deeper and you are back where you stepped in. Things float by the 4-volume they displace; a raft carries you, and you can climb onto it from the water.
The ground is a 3-sphere: walk straight in any direction and you come back to where you started, about 1.6 km later. The planet spins in two planes at once, so there are no poles and the days never repeat.
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Music: J. S. Bach, played by Kimiko Ishizaka (The Open Goldberg Variations, The Open Well-Tempered Clavier), dedicated to the public domain (CC0). Every sound is synthesised in code. Made by bengreff. MIT licence; the source is on GitHub.
A planet in four spatial dimensions that you can walk on.
Growing the terrain of a 3-sphere…