← 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 10 · 18 min

A history of the fourth dimension: from Schläfli to Interstellar

Two hundred and seventy years of the fourth dimension: an encyclopedist’s aside, Möbius’s mirror, a treatise that waited half a century for print, séance knots, cardboard models, the cubists, Dalí’s tesseract and the black hole of Interstellar.

The fourth dimension has lived a remarkably colourful life. Philosophers imagined it, geometers built it on paper, spiritualists looked for ghosts in it, and artists, film-makers and game designers turned it into pictures, films and games. The timeline below puts the whole story on one axis: the colour of a dot tells you which world an event belongs to, and a click opens its card.

Try it

Switch off every filter except “art & literature” and see how tightly the events crowd into 1880–1915: it was a genuine craze. The ← and → keys walk along the timeline in order.

Time as a fourth measure

One of the earliest printed guesses about a fourth dimension does not even belong to the man who printed it. In the article “Dimension” in the fourth volume of the Encyclopédie (1754), d’Alembert writes that more than three dimensions cannot be conceived, and adds: “A clever man of my acquaintance believes, however, that one could regard duration as a fourth dimension, and that the product of time by volume would in some way be a product of four dimensions; this idea may be disputed, but it has, it seems to me, some merit, were it only that of novelty.” Who the acquaintance was is unknown. And the subject was not spacetime but algebra: “dimensions” then meant the factors, like length, width and height, in the formula for a volume.

Lagrange, in his Theory of Analytic Functions (1797; quoted here from the revised edition of 1813), put it more firmly: the position of a point is given by the coordinates $x, y, z$, in mechanics all of them are functions of time $t$, and so “one may regard mechanics as a geometry of four dimensions”. The young Kant, in his Thoughts on the True Estimation of Living Forces (written in 1746–47, printed in 1749), argued that space is three-dimensional because of the inverse-square law by which forces act, and that under a different law space would have a different number of dimensions. “A science of all these possible kinds of space,” he wrote, “would undoubtedly be the highest geometry a finite understanding could undertake.”

Twenty years later, in an essay of 1768, Kant gave an example that would become a classic: a right hand is “entirely equal and similar” to a left one, yet the two cannot be made to coincide. He called such bodies incongruent counterparts and used them as an argument for absolute space. The essay says nothing about a fourth dimension — it was Möbius who brought one to mirror images.

Möbius: a mirror through the fourth dimension

In his Barycentric Calculus (1827), August Ferdinand Möbius noticed something odd. Two mirror-image figures on a line coincide if one is turned over through the plane; mirror-image triangles in a plane coincide if one is flipped over through space. For mirror-image solids, Möbius continues, one would by analogy need “to be able to give one system a half-turn in a space of four dimensions. But since such a space cannot be conceived, the coincidence is impossible in this case.” Möbius was not inviting anyone into the fourth dimension — he was explaining why we cannot go there. But his reasoning is exact mathematics.

Theorem

The reflection $(x, y, z) \mapsto (-x, y, z)$ of three-dimensional space is obtained by a continuous rotation of four-dimensional space: the rotation of $\R^4$ by the angle $\pi$ in the $xw$-plane sends every point of the hyperplane $w = 0$ to its mirror image. Within three-dimensional space itself, no continuous motion can do this.

The rotation by $\theta$ in the $xw$-plane is $x' = x\cos\theta - w\sin\theta$, $w' = x\sin\theta + w\cos\theta$, with $y$ and $z$ unchanged. It is a rigid motion: distances are preserved, the determinant of the matrix is $\cos^2\theta + \sin^2\theta = 1$, and the motion is continuous as $\theta$ runs from 0 to $\pi$. At $\theta = \pi$ we get $x' = -x$, $w' = -w$, so the point $(x, y, z, 0)$ goes to $(-x, y, z, 0)$. In three dimensions a rotation has determinant $+1$ and a reflection $-1$, and along a continuous motion the determinant, which can only be $\pm1$, cannot jump from one value to the other. ∎

A right glove becomes a left one if there is somewhere to turn it. What this looks like for flat creatures is shown in the chapter on Flatland.

Geometry without pictures

In the 1840s higher dimensions entered mathematics from several sides at once. In 1843 Arthur Cayley published “Chapters in the analytical geometry of (n) dimensions” — though the geometric language appears only in the title. In 1844 Hermann Grassmann brought out his Lineale Ausdehnungslehre, an algebra of directions which, as he put it, in the pure theory “can increase up to infinity”. Hardly anyone understood the book, and even Möbius declined to review it. And on 16 October 1843, walking with his wife along the Royal Canal in Dublin, William Rowan Hamilton came up with quaternions — numbers of the form $a + bi + cj + dk$ — and carved $i^2 = j^2 = k^2 = ijk = -1$ into the stone of Broome Bridge. A quaternion is a point of four-dimensional space that can be multiplied by other points. Quaternions underpin rotations in 4D, the Hopf fibration and the Julia fractal in the showcase.

On 10 June 1854 Bernhard Riemann gave his trial lecture in Göttingen, “On the hypotheses which lie at the foundations of geometry”. Of the three topics he offered, Gauss, against Riemann’s expectations, chose geometry. Riemann spoke about manifolds of any number of dimensions and their curvature — the language in which general relativity would be written sixty years later. Dedekind published the lecture in 1868, after the author’s death.

Meanwhile the Bernese mathematician Ludwig Schläfli, who had started out as a schoolteacher, wrote his treatise Theorie der vielfachen Kontinuität (1850–1852), a geometry of $n$ dimensions. In it he found all regular polytopes of higher dimensions, which he called “polyschemes”: six in four dimensions and three in every dimension from five up. The Vienna Academy turned the treatise down, and so did Berlin; extracts published in the 1850s went unnoticed. The whole work appeared only in 1901, six years after Schläfli’s death. Coxeter later wrote that perhaps it was just because these ideas were ahead of their time, “like the art of van Gogh”. Why there are exactly six regular polytopes is explained in the chapter about them.

The first pictures: Stringham, Schlegel, Boole Stott

Unaware of Schläfli, people discovered the polytopes all over again. In 1880 Washington Irving Stringham, a student of Sylvester at Johns Hopkins, published “Regular figures in $n$-dimensional space” in the American Journal of Mathematics, with drawings and photographs of models — among the first published pictures of four-dimensional polytopes. The paper was so much more vivid than Schläfli’s treatise that, as Coxeter put it, “many people imagined Stringham to be the discoverer of the regular polytopes”. In 1882 Reinhold Hoppe seems to have coined the German word Polytop, and in the 1880s Victor Schlegel began drawing polytopes in perspective from a point just outside one of the cells, so that the whole figure appears inside that cell. This is the Schlegel diagram; you can switch it on in the showcase with the “perspective” lens.

The most remarkable figure in this story is Alicia Boole, the third of the five daughters of the logician George Boole. She was never taught mathematics; according to Coxeter she never learned analytic geometry. At about eighteen she came across the wooden cubes of Charles Hinton, who married her eldest sister. To her sisters the cubes were “a meaningless bore”, but they led Alicia, in Coxeter’s words, to “an extraordinarily intimate grasp of four-dimensional geometry”. By purely Euclidean constructions she found the three-dimensional sections of all six regular polytopes and built them in cardboard. In 1900 her paper on series of sections appeared in the proceedings of the Amsterdam Academy, and she worked with the Dutch geometer Pieter Schoute until his death in 1913. It was she who brought the word polytope into English, and in 1914 the University of Groningen gave her an honorary doctorate. What sections are and how they look is the subject of the chapter on slices.

Years later “Aunt Alice” worked with the young H. S. M. Coxeter, whose Regular Polytopes (1948) became the book on polytopes in higher dimensions. In its preface Coxeter thanks Mrs. E. L. Voynich for biographical material about her sister: the youngest of the Boole sisters, Ethel Lilian Voynich, was the author of The Gadfly.

A craze: spiritualists, the tesseract and Flatland

In the 1870s the fourth dimension became fashionable, and not among geometers. The Leipzig astrophysicist Johann Karl Friedrich Zöllner held a series of sittings in November–December 1877 and May 1878 with the American medium Henry Slade, who had shortly before been convicted of fraud in London (the conviction was quashed on a technicality). According to Zöllner’s record, on 17 December 1877 at 11 in the morning, four knots appeared within minutes in a cord whose ends were sealed together and held against the table by his thumbs. Zöllner’s explanation: a being able to move in a fourth dimension can tie a knot in a closed cord without cutting it.

Mathematically Zöllner was right; the flaw was in the experiment. He named as witnesses the founder of psychophysics Fechner, the physicist Weber and the mathematician Scheibner; nine years later the secretary of the American Seybert Commission interviewed them and noted that Fechner had an incipient cataract and Scheibner was short-sighted. The decisive tests, the ones no conjuring trick could fake, failed: no knot ever appeared in a band cut from a single bladder, and two rings of different woods, which the “spirits” were supposed to link, turned up on the leg of the table instead.

Theorem

Any two disjoint closed curves in $\R^3$, however they are linked, can be pulled apart continuously in $\R^4$ without ever meeting on the way.

Take our space to be the hyperplane $w = 0$, which contains both curves $A$ and $B$. Shift $B$ along the $w$-axis: the curve $B + t\,e_w$ lies in the hyperplane $w = t$ while $A$ lies in $w = 0$, so for $t > 0$ they are disjoint, and at $t = 0$ they were disjoint by assumption. With $B$ lifted to $w = 1$, move it parallel to our space wherever you like: as long as $w = 1$, it cannot meet $A$. Having taken it far from $A$, lower it back to $w = 0$. At the end both curves are back in our space and no longer linked. ∎

The same goes for knots (this is a sketch of a proof): any knot becomes trivial if some crossings in its diagram are changed, the upper strand made the lower one. And changing a crossing in $\R^4$ is easy: lift a small piece of the strand into the fourth dimension, carry it over the other strand and lower it. Knots in a cord exist only in three dimensions.

The fourth dimension’s chief evangelist was Charles Howard Hinton. His essay “What is the Fourth Dimension?” came out in 1880 and then as a pamphlet subtitled “Ghosts Explained”. In the pamphlets of his Scientific Romances (1884–1886) he coined the words ana and kata — “up” and “down” in the fourth direction. In A New Era of Thought (1888) the word tessaract appears — spelled with an “a”; the familiar tesseract is a later spelling. Readers could order sets of coloured cubes to learn, by memorising them slice by slice, to see the four-dimensional cube. Hinton’s own life was no less strange: in 1886 he pleaded guilty to bigamy and was sentenced to three days in prison; he taught in Japan and then at Princeton, where he built a gunpowder-powered pitching machine for the baseball team.

In 1884 came Edwin Abbott Abbott’s Flatland, signed “A Square”: an inhabitant of a two-dimensional world meets a Sphere, cannot believe in a third dimension and, once convinced, starts talking about a fourth — whereupon the Sphere refuses to listen (see the chapter on Flatland). Three years later the fourth dimension was something to joke about: Oscar Wilde’s Canterville ghost, pelted by the twins, vanishes through the wainscoting, “hastily adopting the Fourth Dimension of Space as a means of escape”. In Wells’s The Time Machine (1895) the Time Traveller explains that “there is no difference between Time and any of the three dimensions of Space except that our consciousness moves along it.” And in “The Plattner Story” (1896) a schoolteacher, Gottfried Plattner, vanishes from our world for nine days after an explosion in the school laboratory and returns mirror-reversed: his heart beats on the right, and he can only write with his left hand, from right to left. That is exactly Möbius’s turn.

In Russia the fourth dimension found its enthusiast in Pyotr Ouspensky: his books The Fourth Dimension (1909, imprint 1910) and Tertium Organum (1911) turned Hinton’s ideas into a mystical philosophy, and the avant-garde artists read them.

Time becomes a coordinate

On 21 September 1908 in Cologne, Hermann Minkowski opened his lecture “Space and Time” with words that are still quoted: “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.” Einstein’s special relativity (1905) acquired a geometry: four-dimensional spacetime, in which the “distance” between events is the interval $c^2t^2 - x^2 - y^2 - z^2$. Minkowski died in January 1909, before the lecture was published. Ever since, for most people “the fourth dimension” means time, although for Hinton and Abbott it was a fourth direction of space. How spacetime works is told in the chapter on where dimensions live.

In 1919 Theodor Kaluza sent Einstein a paper that added a fifth dimension to the four: five-dimensional general relativity produced the equations of electromagnetism along with gravity. The paper appeared in 1921, and in 1926 Oskar Klein suggested that the fifth dimension is curled up into a tiny circle and therefore goes unnoticed. The “compact dimensions” of modern physics descend from this.

Artists: cubism, Duchamp, Malevich, Dalí

In 1903 the former artillery officer and actuary Esprit Jouffret published an Elementary Treatise on the Geometry of Four Dimensions full of projections of polytopes. Picasso’s circle in Montmartre included another actuary, Maurice Princet, known as “the mathematician of cubism”. In 1910 Metzinger wrote that from Picasso’s “free, mobile perspective” the shrewd mathematician Princet “has deduced an entire geometry”; Gleizes and Metzinger’s Du “Cubisme” (1912) invoked Riemann; and Apollinaire, in The Cubist Painters (1913), wrote that the new measures of extension were known in the studios by a single term, “the fourth dimension” — and promptly called the expression utopian.

How much all this shaped painting itself is disputed. The art historian Linda Dalrymple Henderson, in The Fourth Dimension and Non-Euclidean Geometry in Modern Art (1983; expanded edition 2013), shows that the idea of a higher, fourth dimension of space was central to the development of modern art. Arthur I. Miller (2001) went further, suggesting that in 1907 Princet showed Picasso Jouffret’s book while he was working on Les Demoiselles d’Avignon. There is no document showing that Picasso saw it; this is a hypothesis built on the resemblance of drawings.

Marcel Duchamp’s interest is documented: in his notes for the Large Glass (The Bride Stripped Bare by Her Bachelors, Even, 1915–1923) he wrote that the shadow cast by a four-dimensional figure on our space is three-dimensional. At the “0,10” exhibition of 1915 Kazimir Malevich showed Painterly Realism of a Boy with a Knapsack – Colour Masses in the Fourth Dimension: a black square and a red one. Henderson links Suprematism to the books of Ouspensky and Hinton, but that is her reading; Malevich himself noted in the catalogue that one should not look for real forms in his pictures by their titles.

The best-known painting of the fourth dimension is Salvador Dalí’s Crucifixion (Corpus Hypercubus) (1954, Metropolitan Museum of Art). Its cross is made of eight cubes: an unfolded tesseract, just as a cross of six squares is an unfolded cube (you can fold it up in the chapter on polytopes). In 1975 Dalí saw a Washington Post photograph of the mathematician Thomas Banchoff with an unfolded hypercube next to his painting and invited him over; they kept meeting for about ten years.

Fiction: a tesseract house and a fold in space

In Robert Heinlein’s story “—And He Built a Crooked House—” (Astounding, February 1941) an architect builds a Los Angeles house shaped like an unfolded tesseract: eight cubic rooms in a cross. An earthquake folds it into a real tesseract, and the next shock drops it “into another section of space”. In 1947 George Gamow gave the four-dimensional world a whole chapter in One Two Three… Infinity. Madeleine L’Engle’s A Wrinkle in Time (1962) was, by her count, turned down by at least twenty-six publishers, and in 1963 it won the Newbery Medal; its tesseract is the fifth dimension, a fold that lets you cut across space. In 1980 Carl Sagan talked about Flatland and the four-dimensional hypercube in the tenth episode of Cosmos, and the mathematician and science-fiction writer Rudy Rucker wrote several books about the fourth dimension and collected Hinton’s essays into an anthology.

The computer sees the fourth dimension

In 1965 at Bell Labs, A. Michael Noll programmed an IBM 7094 to make some of the first computer films of four-dimensional objects: a rotating hypercube in stereo pairs, one frame for each eye. In his 1967 paper he admitted frankly that the films gave no profound “feeling” for the fourth dimension: “we are still as puzzled as the inhabitants of Flatland”. In 1978 Thomas Banchoff and Charles Strauss of Brown University made the film The Hypercube: Projections and Slicing, which won a prize at the International Congress of Scientific Films in Brussels.

In 2008 came the film Dimensions by Jos Leys, Étienne Ghys and Aurélien Alvarez: nine chapters, two hours, a free licence and voice-overs in several languages, Russian among them; two chapters are about the fourth dimension and Schläfli’s polytopes, two more about the Hopf fibration. In 2014 came Christopher Nolan’s Interstellar, with the physicist Kip Thorne as science adviser and executive producer. At the end the hero finds himself in a “tesseract” — a three-dimensional space which, according to the robot TARS, unknown beings built inside their five-dimensional reality. The film won the Oscar for visual effects, and Thorne in 2017 the Nobel Prize in Physics, for the LIGO gravitational-wave detector.

And finally, games. Marc ten Bosch first showed his four-dimensional puzzle game Miegakure in 2009 (it is still unreleased), and in 2017 released 4D Toys, a sandbox of four-dimensional objects that fall and collide according to four-dimensional mechanics; he described the physics in a paper for SIGGRAPH 2020. This guide’s showcase belongs to the same family: everything is drawn right in the browser, and how is told in the last chapter.

Four is a special number

Meanwhile mathematics kept finding surprises in four dimensions. In 1931 Heinz Hopf discovered that the three-dimensional sphere — the “surface” of a four-dimensional ball — can be split into great circles, any two of which are linked (chapter on the hypersphere). In 1904 Henri Poincaré ended a paper with a question that became a famous conjecture: is every closed three-dimensional manifold on which every loop can be shrunk to a point the three-sphere itself? The analogue for dimensions 5 and up was proved by Smale (1961), for dimension 4 by Michael Freedman (1982), and the original conjecture by Grigori Perelman in three preprints of 2002–2003. He declined both the Fields Medal (2006) and the Clay Institute prize (2010).

Four dimensions allow things that happen nowhere else. The space $\R^n$ with $n \ne 4$ has exactly one smooth structure, while $\R^4$ has infinitely — even uncountably — many: one and the same topological space can be “smoothed” in ways that cannot be carried into one another smoothly. This follows from the work of Freedman and Simon Donaldson (1982–1983) and of Clifford Taubes (1987). The smooth Poincaré conjecture in dimension 4 is still open.

And in 2016 Maryna Viazovska proved that the densest packing of equal balls in eight-dimensional space is given by the $E_8$ lattice: the balls fill $\pi^4/384 \approx 25.4\%$ of the space. A week later, with Cohn, Kumar, Miller and Radchenko, she settled the same problem in dimension 24, and in 2022 she received the Fields Medal. For comparison: in three dimensions this problem (the Kepler conjecture) was solved only at the end of the twentieth century, and in four it is still open. Why high dimensions are so unlike ours is the subject of a chapter of its own.

Summary

  • Time as a fourth dimension is first mentioned in the Encyclopédie (1754), “mechanics as a geometry of four dimensions” in Lagrange, and spacetime as geometry in Minkowski (1908).
  • Möbius (1827) saw that mirror-image solids could be made to coincide by a rotation in four dimensions — and held such a space to be inconceivable. It is a theorem: a reflection is the trace of a 180° rotation in the $xw$-plane.
  • Schläfli found all six regular polytopes (1852), but his treatise appeared only in 1901; the first pictures came from Stringham, Schlegel and Alicia Boole Stott.
  • From the 1880s to the 1910s the fourth dimension was a craze: spiritualists, Hinton and his tesseract, Flatland, Wilde, Wells, Ouspensky, the cubists, Malevich, Duchamp; its influence on painting is debated by art historians.
  • In four dimensions any linked rings and any knots come apart.
  • Since the 1960s computers have “seen” the fourth dimension: from the films of Noll and Banchoff to Interstellar and 4D games.
  • Dimension 4 is special for modern mathematics as well: exotic $\R^4$, the Poincaré conjecture, sphere packings.

Next: how the showcase is drawn — how these ideas turn into a picture in the browser.

Sources