← The Fourth Dimension
Contents
  1. 01 What a dimension is
  2. 02 Flatland
  3. 03 Projections
  4. 04 Rotation
  5. 05 Cross-sections
  6. 06 Regular polytopes
  7. 07 The hypersphere
  8. 08 Many dimensions
  9. 09 Where dimensions live
  10. 10 History
  11. 11 How the showcase is drawn
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Chapter 09 · 17 min

Where dimensions live: from spacetime to neural networks

A fourth coordinate is not just a geometer’s fancy: time in physics, a robot’s joints, colour, tables of data, the curled-up dimensions of string theory and the infinite-dimensional spaces of quantum mechanics.

In the previous chapters the fourth dimension was pure geometry: one more axis, perpendicular to our three. But spaces with many coordinates turn up whenever the state of something is fixed by several independent numbers. Sometimes those numbers are time and position, sometimes the angles of a robot’s joints, the brightness of pixels or the coefficients of a Fourier series. This chapter is a tour of such spaces. In some of them the fourth dimension works exactly as it does for the tesseract; in others it is quite different, and telling the two apart matters.

Time as a fourth coordinate

To arrange a meeting you need four numbers: three for the place and one for the time. Physicists call such a quadruple $(t, x, y, z)$ an event, and the set of all events spacetime. The idea itself is old, but relativity gave it physical content. On 21 September 1908, at the meeting of German Natural Scientists and Physicians in Cologne, Hermann Minkowski gave a lecture called “Space and Time” (Raum und Zeit). It contains the famous sentence:

“Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.”

Hermann Minkowski, “Space and Time”, 1908 (translated by W. Perrett and G. B. Jeffery)

In the same lecture Minkowski introduced the term world line (Weltlinie): the history of a point particle is a curve in the four-dimensional world of events. Henri Poincaré had already written the Lorentz transformations in four-dimensional form in 1905, with imaginary time as the fourth coordinate, but it was Minkowski who made it the geometry of spacetime. He did not live to see what it grew into: he died in January 1909. Einstein, according to recollections relayed by his biographer Abraham Pais, at first thought this mathematical dressing was superfluous learnedness. Yet without Minkowski’s geometry there would have been no general relativity.

The minus sign

The main difference between spacetime and the four-dimensional space where the tesseract lives is how “distance” is measured. In Euclidean $\mathbb{R}^4$ the squared distance is $x^2 + y^2 + z^2 + w^2$. Between events, the invariant quantity is the interval $$s^2 = c^2t^2 - x^2 - y^2 - z^2,$$ where $t, x, y, z$ are the differences of the two events’ coordinates and $c$ is the speed of light. All observers, however they move relative to one another, get the same $s^2$ for two events, even though they disagree about distances and durations. And this “square” can be positive (the events can be linked by something slower than light), zero (a light ray links them) or negative (no signal can get from one to the other). The events with $s^2 = 0$ around a given one form its light cone.

Theorem

The transformation of the $(ct, x)$ plane $$\begin{aligned} ct' &= ct\cosh\varphi - x\sinh\varphi,\\ x' &= x\cosh\varphi - ct\sinh\varphi \end{aligned}$$ preserves $c^2t^2 - x^2$. It is the Lorentz transformation to an observer moving at speed $v = c\tanh\varphi$.

Subtract and add the two formulas, using $\cosh\varphi \pm \sinh\varphi = e^{\pm\varphi}$: $$\begin{aligned} ct' - x' &= (ct - x)\,e^{\varphi},\\ ct' + x' &= (ct + x)\,e^{-\varphi}. \end{aligned}$$ Multiplying them gives $$\begin{aligned}(ct')^2 - x'^2 &= (ct - x)(ct + x)\\ &= c^2t^2 - x^2.\end{aligned}$$ The origin of the primed frame, $x' = 0$, moves as $x = ct\tanh\varphi$, that is, at speed $c\tanh\varphi$. Substituting $\cosh\varphi = 1/\sqrt{1 - v^2/c^2}$ gives the familiar Lorentz formulas. ∎

The proof shows what the transformation does: in the “light” coordinates $ct - x$ and $ct + x$ it stretches one axis by a factor $e^{\varphi}$ and squeezes the other by the same factor. Compare this with a rotation of the Euclidean plane: there you have $\cos$ and $\sin$, and $x^2 + y^2$ is preserved. Here you have hyperbolic functions, and a difference of squares is preserved. So as a set of points spacetime is $\mathbb{R}^4$, but its geometry is not Euclidean; it is pseudo-Euclidean. The tesseract, the 24-cell and every other figure in this guide live in Euclidean $\mathbb{R}^4$, where all four axes are on an equal footing. You cannot “rotate a tesseract into time”: spacetime has no rotations that turn the time axis into a space axis.

The minus sign has observable consequences. A clock carried along with a body measures its proper time: moving at constant speed $v$ for a time $t$, it ticks off $\tau = t\sqrt{1 - v^2/c^2}$. At $v = 0.6\,c$ that is $0.8\,t$. This is why the travelling twin comes home younger than the one who stayed.

World lines and world tubes

We cannot draw four-dimensional spacetime, but we can drop one space dimension and let the world be flat, as in Flatland. Then events are points $(x, y, t)$, and the whole history of the flat world becomes a three-dimensional object. A house that stays put becomes a vertical prism, a creature walking in a straight line a slanted cylinder, a creature circling the room a helix. A point has a world line; a body of finite size has a world tube. “Now” is a horizontal slice $t = \text{const}$ through all the tubes.

Try it

Choose “Circles” and switch on the light cone. The orange cone contains every event a flash of light, sent by the creature at $t = 0$, can reach; in units where $c = 1$ its walls slope at $45°$. The creature’s world line never leaves the cone, because it moves slower than light. Compare the clocks: at $0.6\,c$ the creature’s clock falls behind, showing $0.8$ of the house’s time.

This way of looking at things is called the block universe: the whole history — past, present and future — is a single four-dimensional block, and “now” is no more special than “here” is in space. The phrase goes back to the philosopher William James, who used it about something else. The view that past and future events are as real as present ones is called eternalism; the opposite view, that only the present exists, is presentism. Relativity makes eternalism feel natural, since different observers have different “nows”. But physics does not settle whether time really flows; that is a question for philosophy.

Configuration spaces

Picture a robot arm with two links and two rotary joints. Its position is fixed completely by two angles: $\theta_1$ at the shoulder and $\theta_2$ at the elbow. Each angle is a point on a circle: $0°$ and $360°$ are the same thing. A pair of angles is a point of the product of two circles — the torus $T^2 = S^1 \times S^1$. It is convenient to draw it as the square $[0, 2\pi) \times [0, 2\pi)$ with opposite sides glued: leave through the right edge and you come back on the left, leave through the top and you come back at the bottom. Glue the top of the square to the bottom and you get a cylinder; glue the ends of the cylinder and you get a doughnut.

The set of all positions of a mechanism is called its configuration space. Obstacles cut forbidden regions out of it, and the task “move the arm from one position to another without hitting anything” becomes the task of finding a path for a point through the free part of the torus. That is how robotics has posed the problem since Tomás Lozano-Pérez’s work in the early 1980s.

Try it

Switch on the autopilot: the point moves along a straight line in the square, leaves through an edge and comes back through the opposite one. In the “Torus” view the same thing is a line winding around the doughnut. Then add obstacles and drag them around: the pink regions on the torus are the positions where the arm hits an obstacle. Sometimes they cut the free part so that one position cannot be reached from another.

From here the dimension grows by itself. An arm with $n$ jointed links and no limits on the angles has the $n$-dimensional torus $T^n$ as its configuration space. The orientation of a rigid body in space takes three numbers (Euler angles, for example), and the set of all orientations is a three-dimensional manifold, the rotation group $SO(3)$. Topologically it is not a torus but the three-dimensional projective space $\mathbb{RP}^3$: a ball with opposite points of its boundary glued together. A rigid body flying freely has six degrees of freedom — three for position, three for orientation.

For $N$ particles the configuration space has $3N$ dimensions, and adding the momenta gives $6N$. This is phase space, where the state of the whole system is a single point and its evolution a single curve — the way Boltzmann and Gibbs looked at mechanics. A mole of gas holds about $6 \cdot 10^{23}$ molecules, so its phase space has some $3.6 \cdot 10^{24}$ dimensions. Statistical mechanics works in spaces like that, and it is rescued by the very concentration described in the previous chapter.

Three-dimensional colour

The light entering the eye is described by its spectrum — a function assigning an intensity to every wavelength. That is an infinite-dimensional object. But the human retina has three kinds of cones (S, M and L, for short, medium and long wavelengths), and each produces a single number. The brain receives only those three numbers, so the space of colours we can tell apart is three-dimensional. Thomas Young guessed as much, Hermann von Helmholtz developed the idea, and in 1931 the International Commission on Illumination (CIE) fixed the three-dimensional XYZ colour space.

Vision is a projection of an infinite-dimensional spectrum onto three-dimensional space. Like any projection it has large “blind” directions: different spectra can give the same three numbers, and then we see the same colour. Such spectra are called metamers. Screens depend on this: three kinds of glowing dots, red, green and blue, mix a triple the eye cannot tell from the intended spectrum. Many birds have four kinds of cones, including ones sensitive to ultraviolet, and their colour space is four-dimensional.

Digression

The record holder for photoreceptors is the mantis shrimp. For colour vision it has 12 spectral classes of receptors (four of them ultraviolet), and with its polarisation receptors about 16 types in all. The obvious guess is that its colour space is twelve-dimensional. But an experiment by Thoen, How, Chiou and Marshall (2014) showed the opposite: mantis shrimps are poor at telling similar colours apart. They already confuse wavelengths 12–25 nm apart, while humans can tell differences of a few nanometres. The authors suggested that their vision works differently: rather than comparing receptor signals as we do, it quickly recognises a colour by which of the twelve channels fired.

A table is a cloud of points

Any table with $n$ numeric columns is a set of points in $\mathbb{R}^n$: each row is a point. The classic example is Fisher’s irises (1936): 150 flowers of three species, each with four measurements — length and width of sepal and petal. That is 150 points in four-dimensional space, and the species form visible clouds in it. To see them, the cloud is projected onto a plane — exactly the trick from the chapter on projections. Principal component analysis (Pearson, 1901; Hotelling, 1933) picks the directions in which the cloud is most stretched out.

The dimensions quickly get large. A $28 \times 28$ greyscale image of a handwritten digit from the MNIST data set is a point in $\mathbb{R}^{784}$. A one-megapixel colour photo is a point in a space of three million dimensions. A neural network is a chain of maps from one such space to another, and its internal representations — embeddings — are points in spaces of hundreds or thousands of dimensions. All the strange geometry of the previous chapter — concentration, nearly perpendicular random vectors, the curse of dimensionality — is everyday reality here.

The curled-up dimensions of physics

In 1919 Theodor Kaluza sent Einstein a paper with a startling idea: write general relativity in a five-dimensional spacetime. Einstein liked it, and in December 1921 he presented the paper, “On the unification problem of physics”, to the Prussian Academy of Sciences. If the five-dimensional metric does not depend on the fifth coordinate, the five-dimensional Einstein equations split into four-dimensional gravity, Maxwell’s equations for the electromagnetic field, and one more scalar field. Electromagnetism turns out to be part of geometry.

Why do we not notice a fifth dimension? In 1926 Oskar Klein offered an answer: it is curled up into a circle of negligible size. A garden hose looks like that. From a distance it is a line, and an ant’s position on it is given by one number. Up close you see that the hose has a second direction — around — and that it closes on itself.

Klein brought in quantum mechanics. A wave running around a circle must match itself after a full turn, so only a whole number of wavelengths fits, and momentum along the fifth dimension is quantised: $p = k\hbar/R$ with integer $k$. In the four-dimensional picture this momentum looks like electric charge, and its quantisation explains why all charges are multiples of one. Requiring agreement with the electron’s charge, Klein estimated the circumference at about $0.8 \cdot 10^{-30}$ cm. Exciting a wave with $k \ne 0$ on such a circle costs an energy of order $\hbar c/R \sim 10^{17}$ GeV — trillions of times more than the Large Hadron Collider reaches. That is why the fifth dimension stays invisible at the energies we can reach.

Try it

Drag “Zoom in” all the way to the left: the hose becomes a fraction of a pixel thick, and only one coordinate is left — “along”. Then switch to “Wave” and set $k = 1.5$: after a full turn the wave does not match itself, and a break appears at the seam. For whole numbers $k$ there is no seam.

Kaluza–Klein theory in its original form did not describe the real world: charged particles came out unimaginably heavy, and the extra scalar field had no place in experiment. But the idea of curled-up dimensions came back in string theory. Consistent superstring theories need ten spacetime dimensions (the bosonic string needs twenty-six), and the extra six are taken to be curled up into tiny manifolds; a popular choice is Calabi–Yau manifolds (Candelas, Horowitz, Strominger and Witten, 1985). In 1995 Edward Witten argued that the five versions of superstring theory are different limits of a single theory, which also has an eleven-dimensional limit; it came to be called M-theory. String theory has no direct experimental confirmation, and its extra dimensions are most likely so small that they cannot be tested in the foreseeable future.

In 1998 Arkani-Hamed, Dimopoulos and Dvali proposed a bolder version: the extra dimensions could be “large”, up to a fraction of a millimetre, if only gravity spreads into them while we and all matter are confined to a four-dimensional “brane”. That would explain why gravity is so weak compared with the other forces. The model makes testable predictions: Newton’s law should break down at short distances, and at a collider gravitons should carry energy off into the extra dimensions. So far the tests only set limits. The Eöt-Wash torsion balance (2020) checked the law of gravity at separations down to 52 microns and ruled out an extra force as strong as gravity with a range of 39 microns or more; hence the largest extra dimension has a radius under 30 microns. At the Large Hadron Collider, ATLAS and CMS have seen neither “missing energy” beyond expectations nor microscopic black holes; for two extra dimensions the fundamental scale of gravity is pushed above about 11 TeV. Extra dimensions are not ruled out, but so far there is no sign of them.

Infinitely many dimensions

One can go further and take infinitely many coordinates. A periodic function can be expanded in a Fourier series: $$f(x) = \frac{a_0}{2} + \sum_{k=1}^{\infty} \bigl(a_k \cos kx + b_k \sin kx\bigr).$$ The coefficients $a_0, a_1, b_1, a_2, b_2, \dots$ are the function’s coordinates in an infinite-dimensional space whose axes are sines and cosines. The inner product of functions is defined by an integral, $\langle f, g\rangle = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x)g(x)\,dx$, and in this sense $\cos x$ and $\sin 2x$ are perpendicular: the integral of their product is zero.

Theorem

Let the vectors $e_1, \dots, e_m$ be pairwise perpendicular and of unit length, and let $f = c_1e_1 + \dots + c_me_m$. Then $$\begin{gathered} c_k = \langle f, e_k\rangle,\\ |f|^2 = c_1^2 + \dots + c_m^2 . \end{gathered}$$

The inner product is linear: $\langle f, e_k\rangle = \sum_j c_j\langle e_j, e_k\rangle$. All the terms with $j \ne k$ vanish and $\langle e_k, e_k\rangle = 1$, which leaves $c_k$. In the same way $|f|^2 = \langle f, f\rangle$ expands into the double sum $\sum_{j,k} c_jc_k\langle e_j, e_k\rangle$, in which only the terms with $j = k$ survive, giving $\sum_k c_k^2$. ∎

This is Pythagoras’ theorem in $m$ dimensions. For Fourier series it holds for $m = \infty$ too — that is Parseval’s identity — and the formula for $c_k$ is the familiar formula for Fourier coefficients. The space of functions with a finite integral of the square is complete: every sequence of coefficients with a finite sum of squares defines a function (the Riesz–Fischer theorem, 1907). Complete spaces with an inner product are called Hilbert spaces, after David Hilbert, whose work on integral equations led to them. John von Neumann gave the abstract definition in 1927–1929, building the mathematics of quantum mechanics on it.

In quantum mechanics the state of a system is a vector in a Hilbert space. For a particle moving along a line it is a wave function, and the space is infinite-dimensional. Some are finite-dimensional: the state of a qubit is a vector in $\mathbb{C}^2$. But the state space of $N$ qubits has dimension $2^N$. For 300 qubits that is $2^{300} \approx 2 \cdot 10^{90}$ — more than the number of atoms in the observable universe. This is part of why quantum systems are so hard to simulate on ordinary computers.

Key idea

Dimension is the number of independent numbers that fix a state. Time adds a fourth coordinate, but with a minus sign in the interval, so spacetime is not the Euclidean $\mathbb{R}^4$ of the tesseract. Robot joints give tori, a rigid body gives $SO(3)$, a gas gives a phase space of enormous dimension, data give clouds of points in $\mathbb{R}^n$, functions and quantum states are vectors in infinite-dimensional spaces. The curled-up dimensions of physics remain a hypothesis that experiments have so far only constrained.

Summary

  • An event is given by four numbers $(t, x, y, z)$; in 1908 Minkowski turned this into the geometry of spacetime and introduced world lines.
  • The interval $s^2 = c^2t^2 - x^2 - y^2 - z^2$ has a minus sign; Lorentz transformations are “hyperbolic rotations”, and spacetime is not Euclidean. The tesseract lives in Euclidean $\mathbb{R}^4$.
  • The positions of a mechanism form its configuration space: a torus $T^2$ for a two-link arm, $T^n$ for $n$ links, $SO(3)$ for the orientations of a rigid body, $3N$ dimensions for $N$ particles, $6N$ for their phase space.
  • Colour is three-dimensional because we have three kinds of cones; the mantis shrimp, with twelve receptor classes, tells colours apart worse than we do.
  • A table with $n$ columns is a cloud of points in $\mathbb{R}^n$; images and embeddings are points in spaces of hundreds or thousands of dimensions.
  • Kaluza (1921) and Klein (1926): a fifth dimension curled into a circle gives electromagnetism and the quantisation of charge. String theory needs 10 dimensions, M-theory 11. Large extra dimensions are being searched for, so far yielding only limits.
  • A Fourier series gives a function’s coordinates in an infinite-dimensional Hilbert space; quantum states are vectors in such spaces.

Next: history — how mathematicians, writers and artists discovered the fourth dimension.

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