kata

four dimensions, ridden

A bicycle squeezes the world down a dimension. It goes forward in one direction, turns in one plane, and the two are tied together: you steer by leaning, and you lean by steering. kata is about piloting one through a world with four dimensions of space, where that squeeze doesn't quite work.

Two wheels still touch the ground at two points, and the bike can still fall about the line between them. But in four dimensions a line has two directions across it, not one. So a lean has a direction as well as a size: some of it can tip into the side you can see, and some of it into w, kata or ana. Nothing in gravity chooses between them.

Demos

LEAN walk it

The track. A road on our ground is a curve made wide in one direction; here it's a curve made wide in both, a tube of tarmac marked out on the flat floor of a closed four-dimensional basin. Its bends turn partly into w, and the plane of each bend keeps turning as you go round. That's torsion, which no road on our ground has. A bicycle rides it, balanced by its own dynamics (the benchmark Whipple bicycle, lifted to 4D) with either steering mechanism: walk beside it, ride along, or ride it yourself: tilting your phone pushes the handlebars, and nothing balances it but the bicycle itself. Walk it with IRIS⁴'s controls, and watch the bend split between your slice and the direction you can't see, with a map of the whole ground in x, z and w that shows the track as the tube it is.

TAILSPIN ten levels

The Cheshire cat rides a bicycle through four dimensions, and you teach yourself to lean into w. The stick is the lean: right, left, up for ana, down for kata. Kerbs turn teal or rose where the track leaves your slice. Ten levels, each harder by an oracle's measure, from a first lean to a corkscrew that twists the other way, and made courses for any seed.

FALL LINE eight hills

Downhill. A hill in four dimensions is a height over a ground with three directions in it, and marked trails run down it, benched in as real ones are. The slope sets your speed and the stick still sets your lean. From the fourth hill on there's more than one way to the bottom: a long gentle way, or a steep one that falls away ana. An oracle rides every way down, and par is its quickest. Paint the ground with Kelvin's foam in 4D, or the tesseract grid, and see which shows you the axes.

WAKE eight courses

A boat on a four-dimensional lake. The stick is the rudder, and a turn can go ana as easily as right. A boat here heels two ways, so which way it falls is set by its shape: three hulls in a triangle are steady every way, a catamaran is tender in w, and an ordinary one-hulled boat, flat in w, can't float upright at all. Behind you Kelvin's 19.47° V becomes a cone, and your slice cuts it.

WAVELAB solved exactly

Water waves on a surface with three directions in it. Drop a pebble: in your slice a ring, on the whole surface a sphere, fading faster than ours. Drop one out of your slice and its ring appears from nowhere. In shallow water the pulse is a clean shell with flat water behind it, Huygens' principle, which our water doesn't have. Swell from ana looks too fast for its length. And the standing waves of a round pond are the orbitals. Twist and move your slice through w, and put corks on a rough sea: pushed down the water's slope in every direction, they bob out of your slice and back.

THE W-CYCLE next

A tricycle whose three wheels sit in a triangle in the plane you can see. It's stable in that plane, but a triangle is flat, and on this ground a flat base can still tip into w. So you balance in w alone: one unfamiliar direction, like learning a bicycle again.

THE BICYCLE then

Two hyperwheels, a lean that is a 2-vector, and a handlebar that steers in two planes. Lean sets how hard you turn, the lean's direction sets which way, and turning the lean's direction is the track's torsion. Ridden by balance alone, like a unicycle on our ground. And a wheel here can spin in two planes at once, which may let the lean precess into w by itself.

What we think we know

The lean is a direction on a circle. Gravity picks out up and nothing else. For a perfectly symmetric bike, y against w is just labelling, and the first nudge sets the direction.
Near upright it's two of our bicycles. Linearized, the lean and steer in y and the lean and steer in w don't talk to each other, and the 3D rule (steer into the fall) works in each. The spinning wheel's gyroscopic coupling keeps the same direction too.
The machine breaks the symmetry. Which planes the fork can steer in, the tyre's cross-section, and how the frame and rider spread their mass in y against w (an ordinary frame is flat in w, so it falls fastest kata).
The calibration is our world's, as vectors. In a steady turn tan φ = v²κ/g, as on our ground, and the lean points along the bend's principal normal. So the lean's size is set by how hard the path curves, and its direction by which way, in your slice or into w. Checked in Lean's selftest, which also recovers the textbook tan φ = v²/gR wherever the bend lies in our plane.
Two ways to carry the lean. With two-axis steering the frame never twists, and round Lean's track it comes home turned 13°, the track's holonomy. With one-axis steering the frame has to roll about its direction of travel at τv to keep its one steering direction along the bend.
Tracks have torsion. A bike's path on our ground is a flat curve. Here the lean's size is the path's curvature, and the rate its direction turns is the path's torsion.
Near upright it's two of our bicycles, now computed. The benchmark Whipple bicycle (Meijaard et al. 2007), reproduced to its published numbers, twice: once leaning into y, once into w, with each body's spread in y swapped for its spread in w. A bike built like ours is a little lighter to roll into w, and balances itself into w only above 4.30 m/s instead of 4.29.
One-axis steering has to twist. With its steer locked, a fall into the frame's w grows as e^(3.1t) at any speed. The frame must twist to bring its one steering axis round to the fall, with a torque nothing on the ground can supply. It rides the track twisting up to 0.11 rad/s at 5 m/s, and falls when asked to swap its lean faster than it can twist.
A 4D cat rides the one-axis bike. The Cheshire cat's front paws stand across z and its hind paws across w: front paws on the bar steer in z, hind paws pedal at ±w. To twist the frame it whirls its tail round z and w, the falling cat's trick. The tail is 1/182 of its twist inertia, so it spins 182 times as far as the frame, about 2 turns a second round Lean's track. Pick it under ⚙ when riding.
Boats heel two ways. A boat's righting is its metacentric height, and here that's a 2×2 matrix over the plane across its bow. One hull flat in w (ours) has negative GM in w and falls over at once. Two side by side are steady across but only barely in w. Three in a triangle are steady every way. And a round hull floating at its axis is neutral in any dimension: its metacentre is its axis.
Kelvin's wake is a cone. Deep-water waves carry their energy at half their speed in any dimension, so what a boat stirs spreads over spheres that each subtend asin(1/3) from it: on our water a 19.47° wedge, here a cone of the same angle about the track.
Waves: the same law, a different geometry. A plane wave on 4D water obeys our ω² = g k tanh(k h) exactly. What changes is the spreading. A pebble's ring is a sphere and fades as 1/r, not 1/√r. In shallow water its front leaves flat water behind it (Huygens' principle, odd dimensions), where ours leaves a tail. Swell from out of your slice looks too fast for its length. Measured in WAVELAB's selftest.
Open: a hyperwheel spinning in its axle plane as well. The algebra says that spin turns lean in y into lean in w. Whether it helps or hurts is for the simulator to say.