Four is how many directions there are. Ten is how many ways there are to slide, turn and tilt everything without changing any distance that matters. Both numbers are checked on this page, in your browser, against operations that can fail — and two deliberately broken ones are run alongside so you can see the checks catching something.
Every picture here is watched from outside except the last one. Section 11 puts you inside it and does all ten to you.
Robin, 2026-09-08: “3d + t being the four dimensions and 10 more exist as almost matrix operations on the 4 eigen vectors. like l w h and t are just the unit vectors of space.”
That is the Poincaré group, reconstructed from intuition, and “almost matrix operations on the four unit vectors” is close to literally what they are. Four length-width-height-time unit vectors, and ten operations on them: 4 slides (one per direction), 3 turns (one per pair of space directions), 3 tilts (one per space direction, mixing it with time). 4 + 3 + 3 = 10. The six turns and tilts really are 4×4 arrays of numbers acting on those unit vectors; the four slides are the unit vectors themselves, added on.
One correction, and it is the whole lesson. The 4 and the 10 do not count the same kind of thing, so “14 dimensions” adds two different words. Four is how many directions you can move. Ten is how many independent knobs the symmetry has. Section 08 separates the three senses of the word; section 09 asks whether the several tens floating around four-dimensional physics are the same ten, and answers it.
Four arrows from one event. Up is later. Everything on this page is these four arrows and what happens to them.
Your screen has two directions. This has four. Every picture below flattens like this:
It is linear, and it is not faithful: four numbers go in, two come out, so two different events can land on the same pixel and the y and z arrows are drawn shorter than they are. A hidden projection is where a viewer's misunderstanding gets manufactured, so it is written here and the one picture that carries the argument — section 04 — uses no projection at all.
Faint arrows: before. Solid arrows: after. The thin curve is the path each arrowtip took on the way.
A square of numbers. That is all a “generator” is. Rows and columns are labelled by the four directions, plus one extra slot that lets a slide be written the same way as a turn. The amount above is how far along you run it.
Highlighted cells are the non-zero entries. Every one of the ten has at most four of them.
There are two ways to run any one of the ten, and they are the source of nearly every confusion about relativity. Either the object moves and you stand still, or the object stands still and you move — your axes tip underneath it. Physicists call these the active and the passive transformation. They are the same matrix used two ways, and they look nothing alike.
Both are drawn below, always, from the one slider. On the left the object's arrow swings and the grid is nailed down. On the right the arrow never moves a pixel and the grid swings the other way. The numbers underneath are identical in both — that is not a claim, it is checked further down, over every generator and every amount.
The white arrow is the object — something with a direction through spacetime. You are the grid, and you have not moved. Run the amount up and the object leans over, sliding along the hyperbola it is trapped on.
Now the white arrow is the thing that has not moved — it is exactly where it was. The tipped lines are your own axes, leaning under it. Nothing was done to the object at all. Everything that changed, changed about you.
Notice which way the axes lean. To make an object appear to move one way, you have to move the opposite way. The right-hand grid tips against the left-hand arrow. That inversion is the distinction: the passive version is the active one run backwards, and it is why “is time dilating, or am I” has an answer that depends entirely on which of these two pictures you had in your head.
And the reading on the dial does not care. Ask “what are the object's coordinates in my frame” and both pictures answer with the same four numbers, to the last bit. The picture is a choice. The measurement is not.
Left column: the object is multiplied by the matrix. Right column: the object is left alone, the four basis vectors are moved instead, and the coordinates are recovered by solving a linear system against that new basis — a different computation, not the same one reprinted.
Every one of the ten, every amount on the slider, four different object vectors. If the two conventions ever disagreed by more than rounding, this would say so.
On a phone, tilting the device left and right can drive the amount. Tilting your phone to turn the frame is literally you performing the passive transformation — you are the one moving, and the arrows respond. Then switch the generator from a turn to a tilt and do the same wrist movement: it feels wrong. The arrows lean instead of turning, and they never come back round no matter how far you go. That is the minus sign, in your wrist.
The permission prompt only appears when you press the button — never on load. Nothing on this page needs the gyroscope: the slider, the arrow keys and the address bar all reach every state it can reach, and the gyroscope does nothing but set the same amount the slider sets.
Try it against J_z — a turn and then against K_x — a tilt. Same wrist. Completely different feeling.
This is the frame worth the rest of the page. On the left, a turn in the x–y plane. On the right, a tilt in the t–x plane. Same slider, same pixels-per-unit, same grid. Both are genuine flat slices of the four directions, so nothing here is projected and nothing is foreshortened.
The two pictures are the same operation with one sign changed. The left one keeps
x² + y² fixed, so the arrowtip is trapped on a circle. The right one keeps
x² − t² fixed, so the arrowtip is trapped on a hyperbola. Plus versus minus.
That minus sign is the entire difference between space and time.
On the right, the time arrow leans toward the space arrow and the space arrow leans toward the time arrow. Push the slider far and they close on the diagonal together without ever crossing it. That diagonal is where distance travelled equals time elapsed — light. This is “time is just another direction” stopping being a slogan: you are watching an operation turn a space axis into the time axis, one continuous amount at a time.
And the pair stays square. On the left the two arrows visibly stay at a right angle. On the right they visibly do not — but the number below the picture says they do, because the right-angle test on the right uses that minus sign too. They look like they are closing; by the only measure this geometry has, they never stopped being perpendicular.
Four slides, three turns, three tilts. Each card is ?generator=<name>&t=<amount>
— the same address always draws the same pixels.
Here is the most surprising thing in the whole list, and it is not an interpretation, it is arithmetic. Tilt along x. Then tilt along y. Then untilt along x. Then untilt along y. You have undone everything you did, in the same amounts, and you are not back where you started — you have turned. Four tilts, no rotation anywhere in the recipe, and the thing left over is a rotation about z.
This is not a curiosity. It is why a gyroscope carried around a curved orbit comes back pointing somewhere else, an effect called Thomas precession, and it falls straight out of the fact that the four tilts do not commute.
The x–y plane, looking down the time direction. Faint arrows: where the two space arrows started. Dashed arrows: where the closed loop leaves them.
Do the two tilts in one order, then the other, and subtract. Fit the difference against all ten. This is computed here, not quoted:
Neither tilt has a single rotation entry in it. The leftover is a pure rotation, and the leftover residual is what tells you nothing else is hiding in there.
Walk the actual four-leg loop at tilt amount a, read the angle it left behind, and
compare with a². If “the leftover is the commutator” is true, the
ratio has to go to 1 as the loop shrinks. It does.
The leftover lean column is how far the closed loop still falls short of being a pure rotation. It dies one power of a faster than the turn does, which is exactly what “the commutator is the second-order term” means.
Run the loop in the other order and the turn goes the other way. Both are in the table. That sign flip is the whole content of the word commutator: the answer depends on the order, the dependence is itself one of the ten, and reversing the order negates it.
Everything above is a claim about numbers, so the numbers are checked here rather than asserted. A check that cannot fail is not a check, so two operations that are deliberately not in the group are run through the same machinery and are expected to blow up. If those two ever read PASS, the checks are broken and nothing on this page should be believed.
A shear tips time into space without the compensating term; a dilation scales all four directions at once. Both are perfectly good matrices. Neither preserves the interval, and neither lies in the span of the ten — which is what makes the ten a closed thing rather than an arbitrary list.
Do two of the ten one after the other, then do them in the opposite order, and compare. The answers differ — a turn then a tilt is not a tilt then a turn. The difference is always another one of the ten. Not something new, not something outside the list. That is the test that the list is exactly ten and not nine or eleven: run all 45 pairs, subtract off whatever combination of the ten fits best, and see whether anything is left over. Nothing is. The dilation control shows what “something left over” looks like: .
Most of the confusion around numbers like 4, 10, 11, 14 and 26 is one word doing three jobs. Every count on this page is one of these three, labelled.
Coordinate axes. How many numbers it takes to say where and when something happened. Spacetime is four: three of place, one of time.
This is the everyday sense. It is the one the word was invented for.
The number of independent dials on the group of motions — how many numbers it takes to specify one whole transformation. 4 slides + 3 turns + 3 tilts.
These are directions in the space of operations, not directions you can walk in. Nothing here moves along a “rotation axis”.
How many independent numbers an object carries at each point. A symmetric 4×4 has 10 of them — which is why Einstein's field equations are ten equations.
These are not directions and not knobs. They are entries in a table that lives at every point of spacetime.
Where the counts come from. A 4×4 table that is unchanged when you swap its two labels has 4·5/2 = 10 independent entries. One that flips sign when you swap them has 4·3/2 = 6 — and that 6 is exactly the turns and tilts. Add the 4 slides and the symmetry has 6 + 4 = 10 knobs. Two tens, arrived at by two different routes. The next section asks whether they are the same ten.
Four-dimensional physics is littered with tens. The symmetry group has 10 knobs. The metric has 10 components. The Ricci curvature has 10. The Weyl curvature has 10. Treating those as one fact is the single most tempting mistake available here, so here is the arithmetic, varied.
Symmetry knobs and metric components: n(n+1)/2. Riemann:
n²(n²−1)/12. Ricci: n(n+1)/2. Weyl:
n(n+1)(n+2)(n−3)/12. Checked for n = 3 … 30 in your browser:
.
Three of the four are the same count, for a reason, and they stay equal in every
dimension. The symmetry's knobs, the metric's components and Ricci's components are all
n(n+1)/2. That is not luck. Asking a motion to leave distances alone is asking a
symmetric 4×4 object to vanish — so the number of conditions a symmetry must satisfy
is the metric's component count. Einstein's equations are ten for exactly that reason:
one per independent metric component.
Weyl's ten is the odd one out, and provably so. It equals ten only in four dimensions. Move to five and the others go to 15 while Weyl goes to 35. Solve Weyl(n) = Ricci(n) and is the only solution above three. That ten is a coincidence of the dimension we happen to live in, not a shared structure.
And the same count is still not the same ten. The decisive test is not the total but how each ten breaks into pieces that the turns and tilts cannot mix. The metric's ten splits as 9 + 1 — nine shape-carrying components and one overall scale. The symmetry's ten splits as 6 + 4 — six turns-and-tilts and four slides. Ten equals ten; 9 + 1 does not equal 6 + 4. They are different objects that happen to be counted by the same formula, and one of the two derivations of that formula is literally about the metric's components. Ricci's ten and the metric's ten are the same kind of object — both symmetric 4×4 tables, both 9 + 1 — so those two really are the same ten slots holding different contents.
Stated as a claim that could have come out the other way: if the metric's ten had split 6 + 4, the identification would be on the table. It splits 9 + 1. It is off the table.
Those are counts of sense 1 — coordinate axes — and they are a claim from a different theory, motivated by making a specific quantum calculation come out finite. They propose extra places to move. The ten on this page proposes nothing: it is a count of knobs on the symmetry of the four axes we already have, and it is as settled as arithmetic.
Nothing here is evidence for or against extra spatial dimensions. If the number 10 sent you looking for strings, that is the word “dimension” doing two jobs, which is what section 08 is for.
The exceptional group G₂ has dimension 14. That is a real fourteen and it has nothing to do with this one; no connection is claimed and none should be inferred.
The honest neighbour is 15. Allow one more operation — scale everything up together — and then the four that undo it, and the group grows from 10 to 15: the conformal group. It is a real extension and it is left out here because it does not preserve the interval, only the light cone. That is exactly the sort of thing the checks in section 07 exist to notice: our dilation control is the first of those five, and it fails.
And “14 dimensions” is itself a phrase to be suspicious of, including as used in the title of this page. 4 + 10 is a correct inventory of two different kinds of thing. It is not a fourteen-dimensional anything.
Everything above is watched from outside: arrows on a grid, and you standing off to one side of them. This is the same ten operations performed on you. You are at the origin of a field of fifteen hundred stars. Pick one of the ten, and turn the knob.
The field is seeded, not random: fifteen hundred stars laid down by one fixed number, with
a fixed distance, temperature and brightness each. So an address here is a picture and not a
description of one — ?view=first&generator=K_x&u=2.400177 draws the same
sky on any machine that loads it, and reloading it never shuffles anything.
Three of them you already know in your body. Turning is your inner ear. Walking is your legs. The fourth translation is the one nobody counts: waiting. Sitting perfectly still is a translation in t, and it is a real motion through spacetime. Watch what it does to the sky. And then there are the three you have never done: a boost does something to the view that no human eye has seen, and it is drawn here exactly, not artistically.
Out of the window. A hundred and ten degrees wide, looking along +x.
The whole sky at once, equal-area, centre = straight ahead, rim = straight behind. Area on this disc is solid angle, so the crowding you see is the crowding there is. The marked ring holds everything that was in front of you when you were standing still.
Home means every star is back on the pixel it started on. The number is the largest angle any star has moved, measured as a chord so that “no movement at all” reads as a true zero rather than as arc-cosine noise.
A Lorentz boost is a change of constant velocity — one steady state of motion swapped for another. It is not an acceleration. The shove in your back, the weight in the seat, the blood leaving your head: all of that comes from accelerating, from the changing of the velocity, and none of it is modelled here. Nothing on this page pushes you.
The picture is exact. The seat is not. If you want the felt push as well you need a worldline that curves, which is a different and much longer page.
A picture this striking has to be checked harder, not less. Every star above is placed by the
page's own 5×5 matrix exponential acting on the photon's four-momentum — the same
expm that draws the arrows in section 02. Below, that route is run against closed
forms derived independently, and against a numerical integral that never touches them.
The conventions, stated. Photon four-momentum p = E(1, −n̂) for a star
in direction n̂; components in your frame are p′ = exp(−u·G)·p,
so D = p′⁰/E and n̂′ = −p⃗′/D. Wavelength goes as
λ′ = λ/D, so a blackbody at T is seen as a blackbody at
D·T — the colour of every star here is Planck's law resampled at 610, 550 and 465 nm,
not a lookup table. The beaming exponent is derived, not guessed: specific intensity
Iν/ν³ is invariant and dΩ′ = dΩ/D²,
so bolometric intensity per solid angle goes as D⁴ while the flux of a single
point star goes as D². Both are true at once: each star ahead is D² brighter and
there are D² more of them per unit sky. The dots are drawn at D², and the D² crowding is left to
happen on its own — the D⁴ is what your eye then adds up. The measured exponent is in the table.
On a phone, press this and tilt. Under J_z — a turn the sky swings past you and comes back round: your wrist already knows this one. Switch to K_x — a tilt and make the exact same wrist movement: the sky does not swing, it gathers, piling up ahead of your thumb and draining away behind. Same wrist, same range, and the sky answers in a way it has no business answering. That contrast, in your hand, is the whole thing.
Same rule as everywhere else on this page: the permission prompt only appears when you press the button, never on load, and the gyroscope is an input and nothing more — it sets the same amount the slider sets, and reaches no state the slider, the [ ] keys or a typed address cannot reach.
Try J_z at a quarter turn, then K_x at the same number, then K_x at β = 0.995, and then wait a while.