Two loops hook in three dimensions, and in no other.

Aristotle closes the question in On the Heavens: "the three dimensions are all that there are", and three is "the first number to which the term 'all' has been appropriated." That is a definition. In 1747 Kant dedicated his first book. It calls Leibniz's proof from three perpendicular lines circular, ties the number to the inverse-square law of force, and is "ready to reject" the idea when a riper judgment finds it weak. By 1783 the perpendicular lines are back in Kant's own argument, now resting "immediately on intuition." A word, a law, a mind: each hands the number to something else. This page hands it to what a record needs.

On 22 January 1833 Gauss wrote that of the geometry of position "we know and possess, after a century and a half, very little more than nothing", and named a main task: count how often two closed lines wind round each other. His integral gives 4π times a whole number, the same from either line, and a whole number can't change smoothly. It is the dial's hook count. On 10 June 1854, in the same town, Riemann called three dimensions "an assumption" and let space be finite without an edge. A balloon's skin is its two-dimensional picture. The argument starts from that shape: space closes up on itself, with no edge and no handles.

Early in 1867 Peter Tait showed William Thomson, later Lord Kelvin, a box with a hole in one side and a towel across the other. One blow on the towel fired a ring of smoke. Kelvin saw atoms: closed swirls in a frictionless fluid, with "an absolutely unalterable quality". Two rings linked together can't be undone however the fluid moves, he wrote, and he thought a sodium atom was "two approximately equal vortex rings passing through one another like two links in a chain". The atoms were wrong. Tait set out to list every knot by its crossings, and with the Reverend Thomas Kirkman he carried the tables to ten crossings. Kelvin wanted something that holds, made of loops that hook. A record's mark is exactly that.

In 1877 Tait wrote that Felix Klein had shown there are no knots in four dimensions. That December in Leipzig the astrophysicist Friedrich Zöllner tied a cord's ends together, sealed the join with his own seal, and wrote that a being with four directions must be able to knot it without loosening the seal. With the American medium Henry Slade present, he wrote, the cord had four knots in minutes. His geometry was right and Slade was a fraud: a University of Pennsylvania commission found what it saw of him "fraudulent throughout" (1887). In Flatland (1884) a Sphere takes a tablet out of a locked cupboard: "the doors, you see, remain unmoved." Zeeman's 1960 unknotting result covers it: a loop can't stay knotted in four or more dimensions.

On 20 June 1877 Georg Cantor sent Richard Dedekind a proof that the points of a square match the points of a line, one to one. On the 29th, still waiting for the verdict, he wrote: "je le vois, mais je ne le crois pas." What does "two dimensions" count, then? Dedekind replied that he still held dimension to be "the first and most important invariant" of a continuous space, and Brouwer proved it in 1911. In 1912, the year he died, Poincaré asked why space has three dimensions. Apart from our instruments, he wrote, space is "amorphous": it keeps only what survives continuous deformation, and "its fundamental property, I was about to say its only property, is to be a continuum of three dimensions." A record's mark holds the same way: no smooth deformation undoes it.

Paul Ehrenfest opened his 1917 paper, communicated by Lorentz to the Dutch Academy, by doubting the question: does space "exist"? Is it three-dimensional? "And then the question 'why'!" He answered with physics. Let gravity fall off as Gauss's law requires in each dimension, and follow a planet. In four or more dimensions it "falls on the attracting centre or flies away", in spirals. In two, orbits never close. Only in three do bound orbits close into ellipses while a body can still escape. So do the fields: the electric field has a number for each direction, the magnetic field one for each pair, and the counts match only at three. His reason is about motion. This page's is about records.

Give each direction one yes/no, the smallest distinction, and three directions make 2 × 2 × 2 = 8 combinations. Join combinations that differ by one answer and you have a cube: eight corners, twelve roads. In Flatland the Sphere walks the Square through the count: one terminal point for a point, two for a line, four for a square. The Sphere asks, "What is the next number?" and the Square answers, "Eight." In 1888 Charles Howard Hinton printed a cube with every point, line and surface named. A trip that flips one yes/no per step, touches every corner and comes home needs at least eight steps, and the cube has one of exactly eight. Why exactly eight, with none to spare, is not yet derived.

On 16 October 1843, beside the Royal Canal, Hamilton found how to multiply triplets. Next day he wrote to his friend John Graves that "we must admit, in some sense, a fourth dimension of space for the purpose of calculating with triplets", and that the order of multiplication "is not indifferent." With fewer than three directions a round trip can only say how many times round; with three it can say what came before what. A floor lap then a wall lap takes the same steps as wall then floor. A tally that adds can't tell them apart; the record, which writes each trip in order, can. Floor, wall, floor backwards, wall backwards: the tally reads 0 everywhere, yet the trip isn't nothing. It takes 14 steps. A computer checked all 196,592 shorter walks that don't double back: none does this. Its order reading is −2, and +2 backwards.

The atoms failed and the knots stayed. In 1969 Keith Moffatt proved that in a frictionless fluid the linkage of vortex lines is conserved as they move with the fluid. In 1980 Liu, Liu and Alberts showed how enzymes unknot DNA: they break a double strand, pass a second strand through the break and seal it, the cut that parts hooked loops in three dimensions. Vaughan Jones found a new knot polynomial in 1985, and in 1989 Edward Witten explained it in three-dimensional terms, with quantum fields; both won Fields Medals in 1990. In 2013 Dustin Kleckner and William Irvine tied vortex knots in water and watched them untie by reconnection: a cut again.