What Is Space? Not a Box. How the Record Is Arranged.

Empty a room. Take out the furniture, then the air, then the light. Nothing is left in it.

Is the room still there?

The easy answer is yes. The room is an empty box, waiting to be filled. Stretch that picture to the whole universe and you get a vast empty box, with the stars and us sitting inside it.

Isaac Newton said the box is real. In the Principia (1687) he wrote that absolute space, "in its own nature, without relation to anything external, remains always similar and immovable." Gottfried Leibniz said there is no box. Space, he held, is an order of things existing together, the way time is an order of things following one another. In 1715 and 1716 he argued it out in letters with Samuel Clarke, who spoke for Newton.¹

So what is space?

The short answer

Space isn't a box. Space is how the entries of reality's record are arranged: where each one sits, and which marks can hook through which.

Reality keeps a permanent record of every event, and nothing in it is ever erased: the basic laws of physics never destroy information, and the record only grows. A mark is something written in the record, and a mark and whatever reads it are loops. Because the record holds, the pair has to be one that nothing smooth can undo: two loops hooked through each other, like two links of a chain. Two loops can hook in three dimensions and in no other number. That's why space has three.

Inside one small piece of the record, which I'll call a cell, the arrangement can be counted exactly. A cell has four accounts, which are four kinds of entry, and each account is posted (written) or not yet. That gives 2 × 2 × 2 × 2 = 16 possible records of the cell, and exactly 24 ways to fill it one entry at a time.²

In the cell, time and space are two different jobs. Time is the count: how many entries have been written. Space is every difference between two records with the same count. Time isn't one of the four accounts. Ask that no account be special, that space be measured by squares and that every one of the 24 ways run forward in time, and the cell's geometry has one time and three directions of space: spacetime's shape.

Why this matters: a box is something that can be empty, and an arrangement can't be. With no entries there's nothing arranged, so there's no empty space left waiting for the record to fill. It also changes how to read spacetime. In the cell, time is a count and space is a set of differences, so they're two different kinds of thing, and spacetime's one-time, three-space shape follows from how a record gets written, given three stated starting points.

Here's the whole answer in five steps. The rest of this post takes them one at a time and shows how we know each one. A last section says where this stops.

1. Space is how the record is arranged. Everything space has ever shown of itself is a comparison of one thing with another, and every comparison is an entry in the record. No entry is a box.
2. Three dimensions come from linking. A mark that holds is two loops hooked through each other, and loops hook in three dimensions and in no other number.
3. One cell has 16 records and 24 routes. Four accounts, each posted or not yet. The 24 ways to fill the cell are the 24 pieces of a four-dimensional cube cut the standard way.
4. In the cell, time is the count and space is the differences. Time is how many entries are written. Space is every difference between two records with the same count.
5. Time isn't a fourth account. Treat the accounts alike, measure space by squares and make every route run forward, and the cell has one time and three directions of space.

Here's the route. I'll stop after each step to say where we are and why the next step comes next.

What this post stands on

This post starts from three claims, each with its reason. What You Are builds the record from the ground up, and The Universe Has No Delete Key follows what happens when something tries to erase it.

Reality keeps a permanent record of every event, and nothing in it is ever erased. The past still exists: it's written down. The reason starts in known physics. The basic laws of physics never destroy information. In quantum mechanics this is unitarity: the present fixes the past exactly, and running the laws backward recovers every earlier state.³ Our work gives the record a reason of its own: it only grows. Each new entry is added with every earlier entry kept, and writing a new entry never alters an earlier one.⁴

An entry is a comparison. The smallest thing that can happen is one thing being told apart from another, a single yes or no. You're doing it right now: reading this tells one letter from another. Telling two things apart needs information about both of them, so each entry is a comparison, written against what came before. That says what an entry is. The reasons above are why entries stay.

What happens is the record being written. This is the premise every post in this series stands on. Here's the reason. Every way anything shows up, whether a measurement, a perception or a collision, is an event that tells one thing from another, and an event like that is an entry in the record.

Step 1: Space is how the record is arranged

Space isn't a box that things sit in. It's how the record's entries are arranged: where each one sits, and which marks can hook through which.

Here's how we know. Start with everything you've ever learned about space. All of it arrived as a comparison. A ruler laid against a table. Two events with a clock running between them. A step counted from the door to the window. Each one sets a thing against another and tells them apart, and that is exactly what an entry in the record is. None of them ever returned the box itself. What came back was always how some things stand to others. Everything that shows up is an entry, so everything space has ever shown of itself is in the record.

Leibniz made the same point with a thought experiment. Shift the whole universe a mile to the east and keep every distance between things exactly as it was. If space were a box, that would be a different universe. Yet no comparison anywhere would come out differently, so no measurement would ever tell the two apart. Leibniz held that the two are one universe named twice.⁵ The record works the same way. It holds comparisons, and "where in the box" isn't one, so there's no entry for a box to be.

Known physics already treats space as something that acts, not something that sits. In Einstein's general relativity, matter bends the geometry of space and time, and the geometry tells matter how to move. In September 2015 the two LIGO detectors recorded the stretch and squeeze of space between their mirrors as two black holes merged about 1.3 billion light-years away.⁶ A box that ripples isn't a box.

Here is a drawing of the real mechanism. A family tree has no sheet of paper under it holding the people apart. How close two cousins are is nothing but who connects to whom, and with the people taken away no empty tree is left over. Space is like that, and the record's entries are the people.

Next, and why: the arrangement has a size. How many directions does it have? Step 2 asks.

Step 2: Three dimensions come from linking

A mark that holds is two loops hooked through each other, and two loops can hook in three dimensions and in no other number. So space has three dimensions.

Why Space Has Three Dimensions works this out step by step, with drawings. Here is the short form, with its reasons. A record holds: nothing you do by sliding, bending or turning things, short of cutting one or passing it through another, wipes out what's written. If a gentle push wiped out a mark, a stiff breeze would erase last Tuesday, and that isn't a record. A mark is something written in the record, and its reader is what registers it. Each is a cycle that comes back to where it started, and in space a cycle is a loop. Two loops hooked through each other (mathematicians say linked) can't be parted without cutting one, so a mark that holds is a hooked pair.

Whether two loops can hook depends on how much room there is. In two dimensions a loop on a balloon's skin cuts the skin in two, and a second loop can't cross it, so the second shrinks to a dot and was never hooked. In four or more, one loop lifts off along a spare direction, slides clear and sets back down. Only in three is there no spare direction: every motion that frees one loop runs it through the other. Forward and back, left and right, up and down, and nothing else. The argument starts from three stated conditions: a record holds, a mark and its reader are loops, and space is the plain kind, with no edge and no handles.⁷

That is the first fact about the record's arrangement: its number of directions. The rest of this post is about what the arrangement does when the record is written one entry at a time.

Next, and why: to see that, take the smallest piece of the record that has an order to it, and count.

Step 3: One cell has 16 records and 24 routes

A cell of four accounts has 16 possible records and exactly 24 ways to fill it, one entry at a time. Those 24 ways are the 24 pieces of a four-dimensional cube, cut the standard way.

Start with the words. An account is one kind of entry in the record, a tally of the entries of that kind. A posting writes one entry to an account. A cell is a small piece of the record with four accounts, which I'll call A, B, C and D. Each account is posted once, or not yet. Why four? Space has three directions, from Step 2, and there is one time: 3 + 1 = 4. The cell is set up with four accounts to match. That's a setting, not a result, and it doesn't make any account time: Step 5 shows that none is.

A record of the cell says which accounts are posted. Each account is one yes or no, posted or not yet, so the cell has 2 × 2 × 2 × 2 = 16 records. At one end is the blank record, with nothing posted. At the other is the full record, with all four posted. Sort the 16 by how many postings they hold:

• 0 postings: 1 record, the blank one
• 1 posting: 4 records (A, B, C, D)
• 2 postings: 6 records (AB, AC, AD, BC, BD, CD)
• 3 postings: 4 records (ABC, ABD, ACD, BCD)
• 4 postings: 1 record, the full one

1 + 4 + 6 + 4 + 1 = 16.

A route is one way to fill the cell: from blank to full, one posting per step. There are 4 choices for the first posting, 3 for the second, 2 for the third and 1 for the last, so there are 4 × 3 × 2 × 1 = 24 routes. It's the number of orders four friends can line up for a photo. Every route has four steps. None can be longer, because each step uses up one of the four accounts, and none steps back, because the record only grows. The cyan path in the drawing is one route: A, then C, then B, then D.

Now the surprise. Cut a square along its diagonal and you get 2 triangles. Each is the part of the square where one account is ahead of the other, A ahead of B or B ahead of A, so the 2 triangles are the 2 routes for two accounts, and a triangle's three corners are the three records along its route. Cut a cube along the planes where two accounts are equal and you get 6 pieces, each a tetrahedron with four corners, one for each of the 3 × 2 × 1 = 6 routes for three accounts. A four-dimensional cube cut the same standard way, named for the mathematician H. W. Kuhn, comes apart into 24 pieces, one for each route of the cell, and each piece has five corners: the five records along its route. Every piece contains the long diagonal from the blank corner to the full one, the cyan line in the drawing.⁸

That's exact, and it's counting, nothing more. I like this part best: the number of ways to write four entries in order and the number of pieces in a four-dimensional cube are the same number because they're the same list. Reading the cell as a piece of space and time is Step 1's premise at work: a piece of the record's arrangement is a piece of space. Steps 4 and 5 do the reading.

Next, and why: the cell has an order, from blank to full. Which part of it is time, and which part is space? Step 4 answers.

Step 4: In the cell, time is the count and space is the differences

Time is how many entries have been written. Space is every difference between two records with the same count.

Time first. Take a clock that reads 0 on the blank record and goes up by 1 with every posting, one step in the writing of the record. On every record of the cell, that clock reads one thing only: the number of postings written. Every record is reached from the blank by writing its postings one at a time, which takes exactly as many steps as it has postings, and the clock adds 1 per step. So the clock can't read anything else. In the cell, time is the count. It's a tally of postings, not a stopwatch: it says how many entries have been written, and it isn't measured in seconds.

Three earlier posts go further on time. Why Your Coffee Never Un-Stirs shows why the count only goes up. Time, Part 2 of The Universe Writes in Pen, is about its front edge, the present. Did Time Have a Beginning? is about where a count of entries starts.

Two records with the same count read the same time, so they're the same moment. The cell has five moments, counts 0 to 4, with 1, 4, 6, 4 and 1 records in them.

Now space. I'll call a change from one record to another a move. Take two records at one moment, say AB and AC. Both have count 2. They differ by a swap: B out, C in. The count doesn't change, so the swap takes no time. It's a move through space. Every difference between two records with the same count is a move like that, and the cell's space is the set of them.

Picture the middle moment, count 2, with its six records. Join two records when one swap turns one into the other. AB joins AC, AD, BC and BD. The sixth record, CD, shares none of AB's postings, so no single swap reaches it: it's AB's opposite. Every record has four neighbors and one opposite, and the lines make the skeleton of an octahedron.

How many directions of space is that? A move changes each account by +1, −1 or 0. A move through space keeps the count, so its four changes add up to 0, and four numbers that add up to 0 have three free choices. So the cell has three directions of space: four accounts minus the count. The cell has four accounts because it's set up that way, so this three is a setting. It isn't a second proof that space has three dimensions. The proof is the linking argument in Step 2.

So inside the cell, time and space are two different jobs. Time is one number, the count. Space is a set of differences.

Next, and why: the obvious guess is that the fourth account is time and the other three are space. Step 5 tests the guess, and why it fails shows the shape the cell's geometry has to take.

Step 5: Time isn't a fourth account

In the cell, no account is time. Time is the count. And the geometry in which every route runs forward in time has one time and three directions of space.

Here's the natural guess: D is time, and A, B and C are space.

Test it. A step that posts D is a step in time. A step that posts A, B or C is a step through space, and a step through space isn't a step forward in time. Every route posts all four accounts, so every one of the 24 routes has three steps through space, and no route runs forward in time at every step. The same happens whichever account you pick for time, because every route posts every account. So no account is time.

So time is the count, and the count has to go into the way a move is measured. Here's the measure. Call it the interval of a move, its squared length. Add up the squares of the move's changes in the four accounts, then subtract κ (kappa) times the square of the change in the count. A move that adds postings runs forward in time when its interval is zero or negative, and a move goes through space when its interval is positive. That's how special relativity sorts the separations between events, and here the same sorting is applied to the cell's moves.

Three starting points go into this result, each with its reason.

1. No account is special. The 24 routes treat the four accounts alike: each account is posted first on 6 routes, second on 6, third on 6 and last on 6. Nothing in the routes singles one out, so the interval treats every account the same.

2. Space is measured by squares. Squares are how Pythagoras measures a straight-line distance, and they're also the shape the cost of a comparison takes near balance. A comparison has a cost that is zero when its two sides match, grows as they differ and comes out the same whichever side is put first. Those three requirements, the rule for how the costs of combined comparisons add, and one calibration (how sharply the cost rises near balance) leave exactly one formula, J(x) = (x + 1/x) / 2 − 1, where x is the ratio of the two sides.⁹ Take a small mismatch e, so that x = 1 + e. The cost is about e² ÷ 2. A 1% mismatch costs about 0.00005, and a 2% mismatch costs about 0.0002, four times as much: double the mismatch, quadruple the cost. That's why this result takes squares as its measure of space. Taking space to be measured by the cost's own shape is a starting point of the result, not something derived here.

3. Every route runs forward in time. A route is the order the record is written in, and each entry is written after the ones before it. So the interval has to call every step of every route a step forward in time, zero or negative.

Now work three moves through.

• Post one account, blank to A. The changes are 1, 0, 0 and 0, so the squares add up to 1. The count changes by 1. The interval is 1 − κ × 1² = 1 − κ.
• Swap, AB to AC. The changes are 0, −1, +1 and 0, so the squares add up to 2. The count changes by 0. The interval is 2 − κ × 0² = 2.
• Post everything, blank to ABCD. The changes are 1, 1, 1 and 1, so the squares add up to 4. The count changes by 4. The interval is 4 − κ × 4² = 4 − 16κ.

Every swap has interval 2, so every swap is a move through space. A single posting has interval 1 − κ, and that is zero or negative, a step forward in time, exactly when κ is at least 1. So "every route runs forward" says κ is at least 1. That's all it says about κ. Set κ below 1 and a single posting goes through space, so the routes fail. Take away the time term altogether, plain Pythagorean distance with no minus sign, and every posting goes through space and no route runs forward.

That's the result. Among the measures that treat the four accounts alike, take space by squares and use the count as time, the ones where every route runs forward in time are exactly those with κ at least 1, up to a positive overall factor that sets the unit.¹⁰

Here's the geometry this gives the cell. One direction has a negative interval: post everything, from blank to full, the long diagonal of the cube, the line every one of the 24 pieces shares. That's time. The directions at right angles to it, the swaps, have positive intervals, and three of them are independent. That's space. One time and three directions of space is the shape of spacetime, and here it follows from the three starting points above.

Check it on the moments. Every swap has interval 2, and at count 2 a record and its opposite, which share no posting, differ by interval 4. So the six records at count 2 sit at the corners of an octahedron: each has four neighbors at interval 2 and an opposite at interval 4. The four records at count 1 are the corners of a tetrahedron, every pair at interval 2, and so are the four at count 3.

Next, and why: that's the whole cell, with time and space told apart. What about the space we live in?

Where this stops

Steps 3 to 5 are about one cell: 16 records, 24 routes, one time and three directions of space. The space we live in has distances measured in meters, and nothing in one cell ties its lengths to those.

One step is missing, and it isn't derived yet: how the record grows from one cell into the space we live in. Nothing derived says which cells neighbor which, how far apart they are, how big a cell is in meters, how long a posting lasts in seconds, or whether the whole goes on forever. Our models of space use many cells at once, and which cells neighbor which is put in by hand. Is the Universe Infinite? says why no survey of space settles the last question.

Next, and why: that's the edge of what's known. Here's the list of what's open, then the whole trip in one breath.

What's still open

• From one cell to the space we live in. How the record grows from one cell into the space we live in: which cells neighbor which, how far apart they are, how big a cell is and how long a posting lasts, and whether the whole goes on forever. Closing it takes a derivation that starts from the blank record, goes through its first posting and arrives at a space of many cells, with distances and a size.

The whole trip, in one breath

Space isn't a box. It's how the record's entries are arranged. The record holds comparisons, and a bare position in an empty box isn't one, so there's no entry for a box to be. Take a family tree away from its people and no empty tree is left. A mark that holds is two loops hooked through each other, and two loops can hook only in three dimensions, so the arrangement has three directions. Write the record one entry at a time and look at one cell: four accounts, 16 records, 24 routes from blank to full, the 24 pieces of a four-dimensional cube cut the standard way. In the cell, time is the count of entries written, and space is every difference between two records with the same count. Time isn't one of the four accounts, because every route posts all four. Treat the accounts alike, measure space by squares and make every route run forward, and the cell has one time and three directions of space.

That's how reality is built. Space isn't the box the record sits in. It's how the record is laid out, and in one cell, from three stated starting points, the shape of spacetime comes out of the order in which four entries can be written.