Mooring a 4D boat

Knots don't hold in four dimensions.

September 27, 2026

Ahoy there, sailor!

You’ve been away on a long voyage across the four-dimensional sea. The horizon all around you is a sphere rather than a circle. The sea’s roiling surface isn’t a sheet of water, but a whole volume of it, slamming into your hull from directions no 3D sailor has a name for. As well as fore and aft, port and starboard, and up and down, there’s a fourth axis, and your boat can steer along it. Charles Hinton, who spent the 1880s trying to teach people to picture four dimensions, named its two directions ana and kata, and they’ve served 4D sailors ever since.

Now the harbour wall is coming up off your ana bow, and it’s time to moor.

1. The bowline slips out ana

You throw a line around the bollard, which is infinitely tall at this particular port, tie a bowline and step ashore. Behind you, the knot slides open. One strand steps ana, slips past the other, and the bowline falls apart.

A knot holds in three dimensions because rope can’t pass through rope. In four dimensions, there’s a way around. Every point in the sea, the sky and the space around you has four coordinates, \((x, y, z, w)\). The first three are the usual ones, but \(w\) measures how far a point is ana (\(w > 0\)) or kata (\(w < 0\)). Mathematicians call this space \(\mathbb{R}^4\).

Exploring your rope, you cross it over itself then lift the upper strand ana. It’s now at a different \(w\) from the lower strand, meaning the two can share an \((x, y, z)\) position without touching. You move it down along \(z\), bring it back kata, and the crossing has changed without the strands ever meeting.

In the diagram, we express \(w\) as colour. Blue is kata, cream is \(w = 0\) and orange is ana. Two sections of rope only touch if they are in the same place and share the same colour.

Changing one crossing turns a trefoil into a plain loop.

Freely changing rope crossings is enough to untie any knot. Pick a starting point on any knot diagram and walk once around the entire geometry. Whenever you reach a crossing for the first time, make your strand the upper one, leaving it be if it already is. The new diagram is always equivalent to the unknot.

So every knot turns into the unknot after some crossing changes, and in \(\mathbb{R}^4\) every crossing change is free. Even friction doesn’t hold your knot taut as the cascading four-dimensional waves batter your vessel, bullying your rope until it comes undone.

The problem is codimension, the dimension of the space minus the dimension of the object in it. A rope is 1D, contorted in three dimensions, giving it a codimension of 2. That’s enough room to go around another strand but not enough to get past it. In \(\mathbb{R}^4\) it has codimension 3, and the spare dimension lets every crossing undo itself.

2. Something to tie to

No knots, then. But even the unknot doesn’t work. Passing a closed loop of rope around a bollard just results in it slipping past the bollard ana once again. The harbourmaster smiles. He’s seen plenty of visiting sailors struggling with this, and kindly points you down the quay to a different kind of bollard.

The trouble is the linking number between the surface spanned by your rope (a 2-dimensional disc) and the bollard (a 1-dimensional line with a healthy bit of body fat in the other dimensions, including the fourth). Only the line matters, since the bollard can slim down to it without ever touching the rope. Call it the core. The linking number counts the points where the core passes through the disc, \(+1\) for each pass from front to back and \(-1\) for each pass from back to front. So a loop around a bollard has linking number \(\pm 1\), and a loop lying on the quayside has 0.

In three dimensions, the count can’t change while the loop remains intact. Crossings can only be destroyed if they pass outside the disc’s edge, and the edge is the rope. In other words, in three dimensions you cannot turn a configuration with linking number \(\pm 1\) to one with \(0\) without cutting the rope.

This only works when the core and disc cross at isolated points. In \(\mathbb{R}^n\), an \(a\)-dimensional object and a \(b\)-dimensional one cross at isolated points when \(a + b = n\), and can be nudged apart entirely when \(a + b < n\). On a 3D quay, \(2 + 1 = 3\), so the disc and the core cross at a point and the bollard holds. In \(\mathbb{R}^4\), \(2 + 1 < 4\). Nudge the core ana and it misses the disc altogether and your boat floats off. To cross the disc at a point again, the core needs one more dimension, \(2 + 2 = 4\). It has to be a plane.

The same count works for closed sheets too. Just as our loop of 1-dimensional string creates a 2-dimensional disc, looping a 2-dimensional surface makes a 3-dimensional ball. A closed \(p\)-dimensional loop spans a \((p+1)\)-dimensional disc, so a core of dimension \(q\) can hold it in \(\mathbb{R}^n\) when

\[p + q = n - 1.\]

This is also satisfied by a 0-dimensional point in two dimensions.

Each bollard slims down to its core. The 4D bollard is drawn as three slices, at w = −1, 0 and 1.

The bollard you attempted to moor to was the obvious 4D version of a post, round in all three horizontal directions \(x\), \(y\) and \(w\), and running up in \(z\). Its core was a line. The harbourmaster’s bollard is round only in \(x\) and \(y\). It runs up in \(z\) and on forever ana and kata in \(w\), so its core is the \(zw\)-plane.

You can see the difference by looking at the harbour one slice at a time. The slice at a fixed \(w\) is an ordinary 3D space.

Drag the slice through w, or let the rope slip.

The round bollard thins out as you move ana, like the slices of a ball, and then stops. The rope steps ana past its edge, slides sideways and comes back kata beside it. The long bollard is in every slice, so wherever the rope goes, the bollard is still inside it.

The harbourmaster runs a coil of rope around his long bollard and a cleat of the same shape on your deck, and splices the ends together to form the loop. Your boat is at last moored, and without a single knot.

3. Missing knots? Take a tarpaulin

The boat is safe, but every knot you know is useless. You can’t lash a crate, hitch a fender or tie off a sail. The harbourmaster sees you turning a length of rope over in your hands with lament and passes you a tarpaulin. “If it’s knots ye want, give up on rope.”

A tarpaulin is 2D, so in \(\mathbb{R}^4\) its codimension is 2, the same as rope in 3D. That’s the room knots need. Lifting one patch of sheet past another would need a spare dimension, and a sheet in \(\mathbb{R}^4\) has none. That alone doesn’t prove any sheet is knotted, though. We’d best define exactly what we’re after.

Close the tarpaulin up into a bag with no opening, a sphere. The plainest sphere is the skin of a ball sitting in a single 3D slice. A sphere in \(\mathbb{R}^4\) is knotted if no amount of moving it around, without passing it through itself, turns it into that plain one. It’s the same idea as rope, where a knot is a loop you can’t move to a plain circle.

Emil Artin built the first one in 1925, by taking a knotted rope and spinning it.

Spinning in 3D is nice and easy. Spin a semicircle about the line through its two ends and it sweeps out a sphere. In 4D you spin about a plane instead of a line. A rotation that mixes \(x\) and \(w\) leaves the \(yz\)-plane where it is and carries every other point round a circle: out ana, back through ordinary space on the far side of the plane after half a turn, and home through kata.

So let’s do what Artin did and spin a knot in four dimensions. Make a simple trefoil knot and cut it once. Take the two fresh ends of the rope, rest them on a plane, and spin it. The ends stay put, the rest of the arc sweeps round, and out comes a knotted sphere with a copy of the knotted arc at every angle of the turn.

Left: a semicircle spun about a line. Right: a knotted arc turned through w about a plane.

To make it from the tarpaulin, lay tarpaulin everywhere the arc passes as it spins. From the figure, it might look like the tarpaulin passes through itself, but in your four dimensions you can see they’re at different \(w\).

You can look at the knotted sphere the way you looked at the bollards, one slice at a time.

Slices of the spun trefoil.

At \(w = 0\) the slice is the arc joined to its mirror image, a right-handed trefoil joined to a left-handed one. Move ana or kata and the parts nearest the plane drop out of the slice, so the knot breaks into loops that shrink and vanish at the sphere’s edge.

Artin proved that no motion in four dimensions turns this sphere into the plain one, just as no motion in three turns a trefoil into a plain loop.

The tarpaulin rescues the round bollard too. A sheet has \(p = 2\), so a line core holds it. Pick any \(z\) position and look at the \((x, y, w)\) 3D slice there, where the round bollard is a ball. Wrap the tarpaulin all the way round that ball and fuse its edges together, the way the harbourmaster spliced the rope, so the ball is sealed inside a closed bag.

4. Casting off

By morning the four-dimensional swell has calmed, and it’s time to go. Your boat has spent the night moored twice over, because you were having too much fun to stop at one. There’s the harbourmaster’s rope, spliced round his long bollard with its plane core, and your tarpaulin, sealed round the round bollard with its line core and a round cleat on your deck. You reach for them, but there’s nothing to untie. Both are closed up, and the linking number that kept your boat safe all night works both ways. Neither can come off while it stays closed.

The harbourmaster is already walking down the quay with a knife. “Same as mooring,” he says, “only backwards.” He cuts the rope, and the loop falls open. He slits the tarpaulin, and with an edge again it slides off the bollard. He bundles it up and tosses it aboard. “Ye’ll want that where you’re going.”

You push off and wave as the quay drops away off your kata stern. You’re setting out to explore the Bermuda Tetrahedron, where ships slip away ana and are never seen again. You’d better brush up on your four-dimensional fluid mechanics first (coming soon).

Fair winds, sailor, and keep a tarpaulin aboard.

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