A space station for four-dimensional people. Our stations spin to make gravity, and spinning has a flaw: there's an axis. A spinning cylinder has gravity on its curved floor and none on its ends; a spinning sphere has full gravity at its equator and none at its poles.

In four dimensions a body can spin in two planes at once, and at equal rates something happens that can't in ours: every point is carried round a circle at the same speed, and every point's spin gravity points straight away from the centre. So a hollow 3-sphere spun this way has the same gravity, straight down, on every square metre of its floor. No axis, no poles, no ends. Its floor is a whole three-dimensional world, finite, without an edge: walk straight in any direction and you come home.

Each point is carried round a great circle, the Hopf circles, and no two of them meet. They're painted here as roads; the gold one is the circle you're standing on, the way the floor is moving. The arrow (top right) shows where spinward lies; its bar, how much of it is in w, out of your slice.

The floor is a three-dimensional world, so the map (top right) is a ball. Each point sits at its walking distance from you, in the direction you'd walk to reach it, on your own axes: forward (the white arrow), right, and ana (the teal tick), so the map turns as you do. The whole habitat fits in the ball; its rim is the point exactly opposite you. Lakes are blue clouds; the track is teal where it runs ana of you and pink where kata, dashed on bridges; the 24 towers are lettered landmarks in their own colours (A and B stand in lakes: lighthouses); home is the gold ring. Drag the map to spin it, ⤢ to make it big. In the view, tags float over the towers ahead with how far, and ▲ ▼ for ana and kata of your slice.

Each track is laid through where you're standing. The bike follows it and leans the way a balancing rider would: so the ground's push runs along its frame. The walk pad (or W S) is the throttle.

With the spin, your speed adds to the floor's and you're heavier; you never lean. Against it, lighter, and at the floor's own speed (about 54 m/s in a 300 m habitat; make it smaller to reach it) you float: from outside you're standing still and the habitat is turning under you.

Across the spin, the road is dead straight and you still have to lean, hard, the whole way round: Coriolis pushes you sideways. It pushes the same way at every point of the road, toward one turn of your heading. You start with that side to your right, so the world rolls. Now hold the twist pad sideways and turn your slice until the lean lies in w: the world stops rolling, you ride upright as far as you can see, and you're leaning 27° into a direction you can't look.

Any straight road keeps the same angle to the spin all the way round, so it rides the same everywhere. The slalom doesn't: it weaves left and right and through w.

Each track is laid on the driest line it can take from where you mount (the spin road has no choice), and crosses what lakes it can't miss on bridges. The road is a tube in the three-dimensional floor: its edge is teal on the side that's ana of your slice and pink on the kata side, white where it's in your slice, so you see the track turning into w before you can see it go.

Seen from outside, anything you let go of flies in a straight line, and the floor turns under it. From inside it curves. Drop a ball and it lands a little against the spin, exactly as in a spinning cylinder of the same size (it's the same problem). Throw one along the spin and it doesn't swerve; it only rises or falls short. Throw it across the spin and it swerves sideways, the same amount whichever way you throw, always turned the same way. In a cylinder, a throw along the axis goes straight; here there's no direction that does.

And the swerve can be in w. Turn so your hidden direction lies across the spin (face spinward does it), throw across, and the ball curves ana or kata, out of your slice. Your slice still cuts it while it's within its radius of you; past that it's a ring, labelled with how far off it is. Step after it with the lever.

Walking, you're heavier going spinward and lighter going against it (your speed adds to the floor's), the same as in a cylinder; walking across the spin you're very slightly heavier, by your own speed squared over the radius.

The hull carries its own spin weight evenly, so a thin 3-sphere shell's stress is σ = pR/3t, a third of a cylinder's pR/t. For the same material and gravity, the Hopf habitat can be three times the radius:

The hull alone, carrying nothing else, no safety factor, and assuming 4D materials are as strong for their weight as ours. Four hasn't worked out its solids' strengths yet.

Drag to look. The walk pad (or W A S D, shift to run) walks you, and on a bike it's the throttle; the twist pad (J L I K) turns your slice into w; the lever (R F) steps you ana and kata. Space or double-tap throws, G drops, Z faces spinward.

Checked by test/habitat.selftest.mjs: every point of the equal double rotation accelerates straight at the centre, so the floor's gravity is the same at 500 random points (and an unequal one isn't); exact flights match a numerical integration in the spinning frame; a drop lands the same distance antispinward everywhere, matching the cylinder worked separately; a throw along the spin doesn't swerve and one across it swerves the same in all directions, always turned the same way; the weights walking; the hull's force balance, σ = pR/3t, by sampling; the Hopf circles never meet and stay the same distance apart; a 6.5 km walk keeps your frame true. On the bike: every great circle keeps the same angle to the spin all the way round; the spin never pushes along any track; with the spin no lean and (ωR + v)²/R; against it at the floor's speed, weightless; across it a constant lean, tan φ = 2ωv/(ω²R + v²/R), toward J of your heading; the slalom closes and its lean swings.