Why Space Has Three Dimensions

You put your earbuds in your pocket, neat and untangled. Ten minutes later you pull out a knot.

You didn't tie it. Nobody ties it. The cord does it on its own, in the dark, with no help from you.

Here's the part nobody mentions: knots only exist because space has three dimensions. In a flat world a cord can't cross over itself, so it can't knot. In a world with four directions, every knot slides loose without cutting. Three is the one size of space where two loops can hold on to each other.

Now a puzzle that looks unrelated. Socks, then shoes. Shoes, then socks. Same two items, same two actions, and only one gets you out the door. "Dog bites man." "Man bites dog." Same three words, and only one of them is news.

Order matters. And in a world built from yes/no directions, order can't change what a round trip says until there are three of them. Three again.

So why three? Not two, not four, not eleven? This post gives two reasons that land on three. The first is why space has three directions. The second is what three directions can do that fewer can't. They are separate arguments with separate proofs, and I'll keep them apart.

The short answer

Here's the whole answer in five steps. The rest of the post takes them one at a time and shows how we know each one.

1. A record holds. What's written can't be unwritten by moving things around, however smoothly you push. That's what "written in pen" means for something that sits in space.
2. Two loops can hook. Thread one ring through another and no amount of wiggling pulls them apart. A mark in the record and whatever reads it are loops, and a hooked pair is a mark that holds.
3. Only three dimensions will do. Two loops can hook in exactly three dimensions. In two they would have to cross. In four or more, one steps around the other. So space has three dimensions.
4. Three directions make a cube, and a cube makes eight. Three yes/no's give 2 × 2 × 2 = 8 corners, and a repeating cycle that flips one yes/no per tick needs at least 8 ticks to touch them all. That's the eight in Part 1.
5. Separately, the cube is where order starts. On a cube with fewer than three directions, a round trip can only say how many times round. On the three-direction cube it can say what came before what. A round trip can look like nothing to a simple count of roads and still not be nothing. The shortest one takes 14 steps.

Why this matters: most of physics starts by assuming three dimensions. Here the number has a reason, and the reason is about what a record needs. It also pays off Part 1, which said a moment is eight ticks, 2 × 2 × 2 for three dimensions, and moved on. And the second half is about something you do every day, telling a story. A story is events in an order, and the order is where the news is.

Here's the route. I'll stop after each step to zoom out. The Ledger keeps the map.

Step 1: A record has to hold

A record holds. Nothing you do by moving things around can wipe out what's written.

Part 1 gave us a ledger with no eraser. Think about what that means for something that sits in space. Suppose a mark in the record were a smudge, a bump, a pattern you can flatten by pushing things around gently. Then a stiff breeze would erase last Tuesday. That's not a record. "Written in pen" has to mean that no smooth motion undoes it. (By smooth motion I mean anything that bends, stretches, slides or turns a thing without cutting it or passing it through something solid.)

That's the first of two things this argument starts from. This post doesn't derive it from anything earlier in the series. It's where the argument begins. From the second one, below, to the number three, every step is worked out, given one condition on the shape of space that Step 2 spells out.

So what kind of mark survives every smooth motion?

Take two links of a chain. Each link is a loop, and each loop passes through the other. Wiggle them, stretch them, spin them: they stay hooked. The only way to part them is to cut one open.

That hook has a number. Fill one ring with a soap film, like a bubble wand, then count how many times the other ring pokes through the film, with a plus or minus for the direction. Hooked once: 1. Apart: 0. Call it the hook count. It can't change while the rings move smoothly, because to change it one ring would have to pass through the other, and solid rings can't.

A pair of loops with a nonzero hook count is a mark that holds. "Hooked" and "apart" are two situations that no smooth motion can turn into each other, so "written" and "not written" stay different for good.

The second thing the argument starts from: a mark in the record, and whatever reads it, are loops. Each one is a closed pass, a cycle that comes back to where it started, like the eight-tick cycle in Part 1. In space, a closed pass is a loop. So a mark that holds is a pair of loops hooked through each other.

Zoom out. Step 1 is done. A record holds, a mark and its reader are loops, and a pair of loops hooked through each other is a mark that nothing smooth can undo. Next, and why: whether two loops can hook depends on how much room there is. The number of directions decides it, and that's where the number three comes from.

Step 2: Only three dimensions will do

Two loops can hook in exactly three dimensions. Not two, and not four or more.

Turn the number of directions like a dial.

Two dimensions: too cramped. Picture the skin of a balloon: two directions and no edge. (A balloon, not a donut: no handles, nothing to fall off.) Draw a loop on it. The loop cuts the skin into two patches. Draw a second loop. It can't cross the first, so it sits entirely in one patch, and a loop in a patch shrinks down to a dot and vanishes. It was never hooked to anything. Hook count: 0.

A flat world can still hold things. A loop traps a dot: a dot inside the loop can't get out without crossing the line. But a loop can't hold another loop.

Three dimensions: just right. Thread one ring through another. Every move that frees one runs it through the other, so there's nowhere to go. The hook count stays 1 forever.

Four dimensions: too roomy. Now give the world a fourth direction. Take the hooked pair, lift one ring off into the new direction, slide it clear of the other ring, and set it back down. They're apart, and nothing was cut. In four dimensions every pair of loops comes unhooked. A chain made of closed links is a heap of loose rings.

That's a drawing of the real mechanism, one dimension down. A dot inside a circle on a page is trapped. Lift it off the page, over the line, and put it down outside. The line only blocks the page. The fourth direction does the same for a ring in a hooked pair.

Yeah-But, from his usual seat, has a question. More room has to be better, right? Why settle for three? Because more room means less holding. In four directions your bike chain is a pile of loose links.

There's a rule behind all this, and it fits on a sticky note. Two simple shapes can hold each other exactly when their dimensions add up to one less than the space they sit in. A dot has dimension 0 and a loop has dimension 1: 0 + 1 = 2 − 1, so in two dimensions a loop holds a dot. Two loops: 1 + 1 = 3 − 1, so three dimensions. A loop and a bubble (the skin of a ball, dimension 2): 1 + 2 = 4 − 1, so four. Every size of space has its own kind of hold.¹

A mark and its reader are both loops, and they hold each other: 1 + 1 = 3 − 1. That gives one size of space and no other. Throughout, space is the plainest kind: it closes up on itself like the balloon's skin, with no edge to fall off and no handles to wind around. That shape is a starting condition too, and it matters. On a flat sheet that runs on forever, a ring drawn around another ring can't get free, because there's no back of the balloon to slide round. The closed-up shape is what rules that out.

Space has three dimensions because three is the only size of space where two loops can hold each other, and a record's marks are loops that have to hold.

Sit with that for a second. Forward and back, left and right, up and down, and nothing else. The count isn't arbitrary. It's the one size of space where a mark in the record can hold.

Zoom out. Step 2 is done. Two loops hook in exactly three dimensions: two is too cramped and four is too roomy. So space has three dimensions. Next, and why: the eight in Part 1. With three directions in hand, we can count what one moment has to cover.

Step 3: Three directions make a cube, and a cube makes eight

Three directions give 2 × 2 × 2 = 8 corners of a cube. A cycle that changes one yes/no per tick needs at least 8 ticks to visit them all.

Part 1 said a moment lasts eight ticks, 2 × 2 × 2 for three dimensions, and moved on. Here's the count behind it.

Give each direction of space one yes/no, as Part 1 does. Three directions, three yes/no's. A combination is three answers in a row, written as three digits, 0 for no and 1 for yes: 000, 001, 011 and so on. There are 2 × 2 × 2 = 8 combinations.

Draw each combination as a point, and join two points when they differ in exactly one digit. You get a cube: eight corners and twelve roads. A tick flips one yes/no, so each tick is one step along one road.

A complete moment is one turn of a cycle, and it touches every combination. Skip one and part of the moment is missing. In one turn, each tick lands on one corner, and the last tick lands back home. Eight corners and one landing per tick means at least 8 ticks, because with fewer there aren't enough landings to go round. No complete pass can be shorter.

Is 8 enough? Yes. The cube has a loop that touches every corner once and comes home: 000, 001, 011, 010, 110, 111, 101, 100, and back to 000. Eight ticks.

So eight is the floor, and the cube has a pass that sits right on it. That's the eight in Part 1: the shortest a complete moment can be. Why the clock runs on the floor, with no spare ticks, is still being worked out.

Zoom out. Step 3 is done. Three yes/no's make a cube with eight corners, and a complete pass takes at least eight ticks. Next, and why: we have the cube, so we can ask a second question about it: what can a trip around it say? That's a separate argument. It's a count on the cube, so it doesn't use the linking proof from Steps 1 and 2, and it doesn't prove that space has three directions. It shows what three directions can do that fewer can't.

Step 4: The cube is where order starts

On a cube of yes/no's, order first carries content at three directions. With fewer, a round trip can only say how many times round.

Socks, then shoes. Dog bites man. Gary plays kick, kick, snare, snare, and it's a beat. Then he plays snare, kick, snare, kick, and it's a different beat. Same four hits, and Gary says it's the same song. Gary is still not getting the gig.

Add up the pieces and the order disappears. So where does order live? The ledger records events in order, and the same events in a different order are a different history. Let's find the smallest cube where order changes what a round trip says.

Round trips. A round trip is a trip that ends where it started. On the cube that happens exactly when every direction was flipped an even number of times: each flip has a matching flip back, the way balanced books have a credit for every debit.

How many different round trips does a cube hold? Endlessly many, because you can go round as often as you like, in any mix. So count the independent ones: the round trips you can't build by gluing others together.

Here's how. Pick a trunk: a set of roads that reaches every corner without closing a loop. A trunk takes one new road per corner after the first, so it has corners minus 1 roads. Add any other road and you close exactly one new round trip. So the number of independent round trips is the roads minus the trunk.

A point has no roads: 0 round trips. A line has 1 road and it's all trunk: 0. A square has 4 roads and a trunk of 3: 1. A cube has 12 roads and a trunk of 7: 12 − 7 = 5. The five leftover roads are the cyan ones, numbered 1 to 5, each with a small arrow.

Now see what that count means. With at most one round trip, order has nothing to change. With five, order starts to matter.

On a square there's a single round trip, the lap. Every round trip on it comes down to some number of laps, one way or the other, once you cancel the steps that undo each other. Three laps and then two is the same trip as two laps and then three. There's nothing to put in a different order. A world with at most one independent round trip can count, but it can't say anything in order.

The cube has room for different laps. Let A be a lap round the floor and B a lap round the front wall.

Meet Tally. Tally keeps score. He has one running number for each of the five leftover roads, the cyan ones: plus 1 each time you go along a road in the direction of its arrow (call that forward), minus 1 each time you go against it (back). That's all Tally does, and he does it very well.

Do A, then B. Then do B, then A. You take exactly the same steps either way, in a different order. Tally's sheet comes out the same for both: road 1 reads −1, road 2 reads +1, road 3 reads −1, and roads 4 and 5 read 0. Anything that adds up a score for each step reads the two trips the same, whatever scores you hand out, because addition doesn't care about order.

But they aren't the same trip. The Ledger doesn't add. It writes each trip down in order, as the list of leftover roads the trip crosses. A then B: road 1 back, road 2 forward, road 3 back. B then A: road 2 forward, road 3 back, road 1 back. Neither list has a road crossed one way and then straight back the other way, so nothing cancels and the lists stay different.

That's "dog bites man" and "man bites dog" on the smallest cube that can say it: same pieces, same count, different news. I'm not saying English is a walk around a cube. The words are a drawing of the real mechanism: when order carries content, adding up the pieces throws it away. The cube is a clean place to watch that happen.

Noted. In order.

Zoom out. Step 4 is done. On a cube of yes/no's, order starts at three directions. With at most one independent round trip, a trip can only count. With five, it can say what came before what. Tally adds, so he can't see order. The Ledger keeps it. Next, and why: the sharpest case. Is there a trip that Tally reads as nothing at all, and that still isn't nothing?

Step 5: The 14-step trip

Do A, then B, then A backwards, then B backwards. Tally reads the whole trip as zero. It isn't nothing. And no shorter round trip does this.

If you've ever held a Rubik's Cube, you've seen this one. Turn the top face, turn the right face, turn the top back, turn the right back. Every move got undone. Count them: top, one turn forward and one back, zero. Right, one forward and one back, zero. And the cube is scrambled. (The Rubik's Cube is a drawing of the mechanism. Ours is a different cube.)

On our cube, A is four steps, B is four, A backwards is four and B backwards is four: 16 steps. But A and B share a road, and where A backwards meets B backwards the trip goes along that road and straight back. Those two steps cancel. Fourteen are left.

Tally's verdict: every number on his sheet is 0. Road 1 went −1, then +1. Road 2 went +1, then −1. Road 3 went −1, then +1. Score the corners instead, any scores you like, and you get zero again, because on a round trip every climb is matched by a drop. "All zeros! Nothing happened!"

The Ledger disagrees. Its page for the trip has six lines: road 1 back, road 2 forward, road 3 back, road 1 forward, road 3 forward, road 2 back. Every road shows up once each way, which is why Tally's totals cancel. But no line cancels against its neighbor. Something happened.

How does the Ledger read it? Take roads 1 and 2, and plot Tally's running totals for them as a dot on graph paper: road 1 across, road 2 up. Each time the trip crosses road 1, the dot steps sideways. Each time it crosses road 2, the dot steps up or down.

Follow the 14-step trip. Road 1 back: the dot steps left. Road 2 forward: up. Road 1 forward: right. Road 2 back: down. (Road 3 doesn't move this dot.) The dot walks around a unit square and ends where it began.

Tally looks at where the dot ends: back at the start, so zeros. The Ledger looks at the walk. The square has area 1. Double it: 2. (Why twice? The order reading is a count of what came first. Take every pair made of one road-1 crossing and one road-2 crossing, score +1 if the road-1 crossing came first and −1 if the road-2 crossing did, flip the score once for each crossing in the pair that went back, and add it all up. That total always comes out to exactly twice the area the dot circles.) The dot went clockwise, and clockwise counts as minus. The order reading is −2. Walk the whole trip backwards, which is B, then A, then B back, then A back, and the dot walks the square the other way: +2.

This is my favorite part. The dot comes home, so Tally says nothing happened. But the dot went somewhere on the way, and where it went is the news.

That's what Tally can't see. Not where you went, and not how many times, but which came before which.

Fourteen is the floor. No round trip shorter than 14 steps has all of Tally's numbers at zero and still fails to cancel down to nothing. A trip with a road walked out and straight back can always be shortened, and shortening it changes none of Tally's numbers and nothing the Ledger writes. So only trips with no out-and-straight-back need checking. There are 196,592 walks like that with 13 steps or fewer, counting every starting corner and counting walks that don't come home. A computer checked every one. Not one is a round trip that reads all zeros and still isn't nothing.

That's the exact condition: a round trip, all of Tally's numbers zero, and not nothing. Under it, the shortest trip takes 14 steps, and the one we walked reads −2.

Zoom out. Step 5 is done. Tally's zeros hide a trip that isn't nothing, and the Ledger reads it by its order: −2. No shorter trip does it. Next, and why: both halves are in hand, and they're two separate arguments. Time to put them side by side.

Two reasons, one number

I said I'd keep the two reasons apart, so here they are side by side.

Reason one is about space. A record holds, a mark and its reader are loops, and in a space that closes up on itself, two loops can hold each other in three dimensions only. That fixes how many directions space has.

Reason two is about the cube. Give each direction one yes/no, and three directions make a cube with eight corners. A complete pass takes at least eight ticks. And that cube is the first one with room for order: with fewer directions a round trip can only count, and with three there are five independent round trips and a 14-step trip that Tally reads as nothing.

Neither one proves the other. The first says why space has three directions. The second doesn't say why. It says what three can do that fewer can't. (The cube comes with its own rules: one yes/no per direction, one flip per tick, and a complete moment touches every combination. They go in alongside the inputs of reason one. This post counts with them and doesn't derive them.) Two different arguments, and they land on the same number.

The whole trip, in one breath

A record holds, and a mark and its reader are loops that no smooth motion can unhook. Two loops can hook in exactly three dimensions: in two they would have to cross, and in four or more one steps around the other. So space has three dimensions. Three yes/no's make a cube of 2 × 2 × 2 = 8 corners, and a cycle through all of them takes at least 8 ticks. And that cube is where order starts. Below three directions a round trip can only count. At three there are five independent round trips, and a 14-step trip that Tally reads as nothing has an order reading of −2. On the smallest cube that can say it, "dog bites man" and "man bites dog" are different trips.

Three directions: the one size of space where a loop can hold a loop, and the first size of cube with room for order. Two reasons. One number.

(The Ledger would like you to know that it noted the order in which you read the steps. Step 4 came before Step 5. It says that matters.)