Nobody can picture a four-dimensional solid: our imagination grew up in three dimensions and stops there. But we can picture a creature that cannot imagine three. We see its troubles from outside, in every detail, and each one hints at what would happen to us if we met a fourth dimension. This trick is called reasoning by analogy: take a fact about the pair “a plane inside space” and carry it over to the pair “space inside four dimensions”. Nobody has shown it better than a Victorian headmaster in a slim book of 1884.
A romance of many dimensions
Edwin Abbott Abbott (1838–1926) was a theologian and a Shakespeare scholar (his Shakespearian Grammar came out in 1870), and from the age of 26 the headmaster of the City of London School. In 1884 the London publisher Seeley & Co. brought out his Flatland: A Romance of Many Dimensions. The title page named the author as “A Square”: the story is told by a square who lives in a plane. A second, revised edition followed the same year.
The first part, “This World”, describes flat society, and it is a sharp satire of Victorian England. Your rank is your number of sides. Women are straight lines, as dangerous as needles. Soldiers and the lowest workmen are isosceles triangles with sharp angles, the middle class are equilateral triangles, professional men and gentlemen are squares and pentagons, the nobility begins with hexagons, and at the top stand the priests, polygons with so many sides that they cannot be told from circles. Regular figures obey a “law of nature”: a son has one side more than his father, so the son of a Square is a Pentagon. The law does not apply to the isosceles: their sons stay isosceles. Irregularity of shape counts as vice and crime.
The second part, “Other Worlds”, is what made the book famous. The Square dreams of Lineland, a one-dimensional world whose king is sure that his line is the whole of space. Then, on the last day of the 1999th year of the Flatland era, a visitor from Space appears: a Sphere. He shows the Square the third dimension and lifts him up, and for the first time the Square sees his world from above. Carried away, the Square immediately proposes the next step: if there are three dimensions, why not a fourth? The Sphere replies that there is no such land and the very idea is inconceivable, and in his anger throws his pupil back down into the plane. (Later, in a dream about Pointland, whose only inhabitant believes it is the whole universe, the Sphere admits he was wrong.) Back home, the Square tries to preach the “Gospel of Three Dimensions” and is sentenced to life imprisonment. He writes the book from prison, where he has already spent seven years.
The Sphere laughs at the King of Lineland for failing to imagine a plane, and becomes just such a king himself the moment a fourth dimension is mentioned. Abbott is warning the reader: don’t be so sure you are smarter than the Sphere.
What a two-dimensional being sees
Before travelling, let’s sort out eyesight. The Square’s eye lies in the same plane as everything else. Light travels within the plane and reaches the eye from the directions of a single fan, so the Square’s “retina” is one-dimensional. He does not see shapes, he sees segments: a circle, a triangle and a square all look the same to him, a piece of a straight line.
So how do Flatlanders recognise one another? Abbott gives three ways: by voice, by touch (the method of women and the lower classes) and, among the educated, by sight, thanks to fog. In fog, distant things look dimmer than near ones. If a triangle turns an angle towards you, the middle of the segment is bright and the ends fade quickly; if it turns a side towards you, the segment is lit almost evenly. From how fast the brightness fades towards the ends, a trained eye judges the angle, and with it the other person’s rank. There are no shadows in Flatland, since there is no sun and no light that could cast them, so depth comes from the fog alone.
It is the same with us, one step up. Our retina holds a two-dimensional picture, and we rebuild the third dimension from indirect cues: perspective, shading, the difference between what our two eyes see. A being from a four-dimensional world would look through a three-dimensional “retina”. It would see our world the way we see a drawing on a sheet of paper: all of it at once, inside and out, with nothing hidden behind anything else.
A sphere passes through the plane
To prove he is not a circle, the Sphere passes through the plane of Flatland. The Square sees a point appear from nowhere and turn into a circle; the circle grows, then shrinks back to a point and vanishes. That is exactly how the Sphere describes himself: many circles, from a point to thirteen inches across, stacked one on top of another.
The geometry is simple. If the centre of a sphere of radius $R$ is at height $h$ above the plane, the plane cuts the sphere in a circle of radius $$r = \sqrt{R^2 - h^2}, \qquad |h| \le R.$$ That is Pythagoras: the sphere’s radius drawn to a point of the cut is the hypotenuse, while $h$ and $r$ are the legs. While $|h| > R$, there is nothing in Flatland; at $h = \pm R$ a point appears; at $h = 0$ the circle is largest.
A cube is more interesting: everything depends on how it is turned. Let its edge be 1.
- Face first. A unit square appears all at once, stays unchanged while the cube passes, and vanishes all at once.
- Edge first. First a segment, then a rectangle that widens to $1 \times \sqrt2$ (the diagonal cross-section of the cube) and narrows back to a segment.
- Vertex first. A point, a growing equilateral triangle, then a hexagon, a triangle again (upside down) and a point.
The last case is easiest in coordinates. The cube is the set of points with $0 \le x, y, z \le 1$, and a plane perpendicular to its long diagonal is $x + y + z = s$, with $s$ running from 0 to 3. For $s \le 1$ the cut is an equilateral triangle with vertices $(s,0,0)$, $(0,s,0)$, $(0,0,s)$ and side $s\sqrt2$. For $1 < s < 2$ the plane cuts off the corners of that triangle where some coordinate exceeds 1, leaving a hexagon. At $s = 1.5$ it is regular, with side $\sqrt2/2 \approx 0.707$ and area $3\sqrt3/4 \approx 1.299$. For $s \ge 2$ everything runs backwards: the substitution $x \mapsto 1 - x$ (and likewise for $y$, $z$) takes the cut at $s$ to the cut at $3 - s$, turned through 180°.
Pick the cube “vertex first” and set the cut to the middle: you get a regular hexagon. Look at the strip at the bottom right: that is all the Square sees. When a corner faces him, the strip is bright in the middle and darkens towards the ends; when a side faces him, it glows almost evenly. From strips like this, Flatlanders have to guess shapes.
One floor up: a visitor from the fourth dimension
Now let’s go up a floor. Let our space be the “plane” $w = 0$ inside four-dimensional space with coordinates $(x, y, z, w)$. Such a three-dimensional subspace is called a hyperplane. All a four-dimensional solid can show us is its cross-section by our hyperplane: the three-dimensional piece of it that happens to be “here” at the moment.
A hypersphere. A four-dimensional ball of radius $R$ is the set of points with $x^2 + y^2 + z^2 + w^2 \le R^2$. If its centre is at distance $h$ from our space, the section is $$x^2 + y^2 + z^2 \le R^2 - h^2,$$ an ordinary ball of radius $\sqrt{R^2 - h^2}$. Pythagoras again, with one extra term. What would such a visit look like? In the middle of the room — not coming in through the door, just in mid-air — a point appears, swells into a ball, the ball grows, then shrinks and vanishes. It could happen inside a locked safe, too: the walls of a safe block only our three directions.
A tesseract, cell first. Take the four-dimensional cube $[0,1]^4$ and move it along $w$. The section $w = c$ with $0 < c < 1$ contains all points with $0 \le x, y, z \le 1$, which is a cube. It appears all at once, stays put while the tesseract passes through us, and vanishes all at once, exactly like the square when a cube passes through Flatland face first.
A tesseract, vertex first. Here the analogy with the cube says more than it seems to. For the cube, a vertex gave a triangle and a hexagon; for the tesseract we get a tetrahedron, a truncated tetrahedron and an octahedron.
The section of the tesseract $[0,1]^4$ by the hyperplane $x + y + z + w = s$, perpendicular to its long diagonal, is:
- for $0 < s \le 1$, a regular tetrahedron with edge $s\sqrt2$;
- for $1 < s < 2$, a tetrahedron with edge $s\sqrt2$ with its four corners cut off: four triangles with side $(s-1)\sqrt2$ and four hexagons (a truncated tetrahedron);
- for $s = 2$, a regular octahedron with edge $\sqrt2$;
- for $2 < s < 4$, the same in reverse order.
If $s \le 1$, the constraints $x_i \le 1$ hold automatically: each coordinate is non-negative and no bigger than the sum of all four, which is $s$. What is left is the set $\{x_i \ge 0,\ \sum x_i = s\}$, a tetrahedron with vertices $s\mathbf e_1, \dots, s\mathbf e_4$, where $\mathbf e_i$ are the unit vectors of the axes. Any two vertices are $|s\mathbf e_i - s\mathbf e_j| = s\sqrt2$ apart, so the tetrahedron is regular. If $1 < s < 2$, we must remove from this tetrahedron the points where some coordinate exceeds 1. Near the vertex $s\mathbf e_1$ this is the set $\{x_1 > 1,\ x_2, x_3, x_4 \ge 0,\ \sum x_i = s\}$, a small tetrahedron similar to the big one, with edge $(s-1)\sqrt2$. Two such caps never meet: a common point would have $x_i > 1$ and $x_j > 1$, hence $s > 2$. Cutting four disjoint corners off a tetrahedron adds four triangular faces and turns each old triangular face into a hexagon. At $s = 2$ the cuts reach the midpoints of the edges, and what remains are the points with two coordinates equal to 1 and two equal to 0. There are six of them, such as $(1,1,0,0)$, and each is $\sqrt2$ away from four others: the vertices of a regular octahedron. Finally, the substitution $x_i \mapsto 1 - x_i$ maps the tesseract to itself and the hyperplane $s$ to the hyperplane $4 - s$. ∎
A few numbers. At $s = 1.5$ all edges of the truncated tetrahedron equal $\sqrt2/2$: it is the Archimedean solid, with volume $23/24 \approx 0.958$. The regular tetrahedron at $s = 1$ has volume $1/3$, the octahedron in the middle $4/3$. The middle of the tesseract is “fatter” than its cell: the octahedron’s volume exceeds the unit volume of a cubic cell. And passing vertex first takes twice as long as passing cell first: the long diagonal of the unit tesseract is $\sqrt{1+1+1+1} = 2$.
Pause the animation and slide along the strip at the bottom: it is the $w$-axis, with the volume of the section plotted above it. The faces of the section are coloured by the cell of the tesseract they were cut from: each face is a slice of one of the eight cubes, and the colour tells which axis that cube is perpendicular to. In the octahedron at $w = 0$, neighbouring faces have different colours.
Notice what is missing: no section ever shows a tesseract. We see only tetrahedra, octahedra and cubes, three-dimensional solids that replace one another. The Square, likewise, saw only circles and polygons. There is more on sections, including those of the regular polytopes, in the chapter on slices.
The view from above: everything laid open
A house in Flatland is a pentagon with walls made of segments. For a Flatlander that is real protection: to get in, you have to go through the door. But the Sphere, looking from above, sees every room at once. He tells the Square which members of the household are where, sees what is inside a locked cupboard (boxes of money and tablets of accounts) and takes one tablet out without opening the door. Then, to be convincing, he gives the Square a light touch right in the stomach, from the inside. The insides of a flat creature are open to a view from the third dimension, just as a drawing on paper is open to us.
Here is what it means for us. A closed curve is a prison only for someone who cannot leave the plane. A closed surface, the walls, floor and ceiling of a cell, is a prison only for someone who cannot leave space. A being that can move along $w$ sees the whole content of our safe, our organs, bones and blood, all at once. It could take a coin out of a locked safe: lift it a little along $w$, move it sideways and put it back down. To us the coin would vanish inside the safe and reappear outside without ever crossing a wall. It could take out an appendix without an incision.
Only a barrier one dimension lower can split space into “inside” and “outside”: a curve in the plane, a surface in space. In a space with one more dimension, the same barrier separates nothing.
Left and right
Draw the letter L on a plane. Its mirror image is an L with its foot pointing the other way. No motion within the plane, no sliding or turning, will turn one into the other. But lift the L off the plane, flip it like a pancake and put it back, and it lands mirrored. A Flatlander would see a miracle: the shape vanishes (apart from the points on the line it was flipped about) and comes back reversed.
Shapes that cannot be moved onto their own mirror image are called chiral, from the Greek word for “hand”. The most familiar example is a pair of gloves. In his 1768 essay “Concerning the ultimate ground of the differentiation of directions in space”, Immanuel Kant called such a pair incongruent counterparts (incongruentes Gegenstück): the right hand is similar and equal to the left in every respect, yet the two cannot be enclosed in the same boundaries. Kant came back to the example in the Prolegomena (1783): what could be more like my hand than its image in a mirror, and yet the image cannot be put in the place of the original.
August Ferdinand Möbius, in his book Der barycentrische Calcul (1827), worked through what we have just done with the letter L, and took the next step. Mirror-image triangles in a plane can be brought together by turning one of them half a turn about a line, that is, through the third dimension. By analogy, Möbius wrote, mirror-image solids would coincide if one of them could be given half a turn in a space of four dimensions. “But since such a space cannot be conceived, coincidence is impossible in this case.” The Sphere from Flatland would have said the same.
The formulas show how it works. A rotation through angle $\theta$ in the $xw$-plane changes two coordinates and leaves $y$ and $z$ alone: $$\begin{aligned} x' &= x\cos\theta - w\sin\theta, \\ w' &= x\sin\theta + w\cos\theta. \end{aligned}$$ For a solid lying in our space ($w = 0$) this gives $x' = x\cos\theta$, $w' = x\sin\theta$. At $\theta = 90°$ every point has $x' = 0$: in our three coordinates the solid is squashed flat, although in four it is intact, merely stretched out along $w$. At $\theta = 180°$ we get $x' = -x$, $w' = 0$: the solid is back in our space, reflected in the plane $x = 0$.
Press “Turn over” and watch both panels. On the left, the shadow of the L on the plane shrinks to a segment; on the right, our three-dimensional “shadow” of the four-dimensional rotation squashes the four-cube shape flat. Colour shows the coordinate the inhabitants cannot see: the height above the plane on the left, $w$ on the right. The dashed outline is the mirror copy; at 180° the shape lands on it exactly.
Why this cannot be done in three dimensions can be proved rigorously.
If a rigid body moves continuously in $\R^3$, then at every moment $t$ its position is obtained from the initial one by a map $\mathbf x \mapsto A(t)\,\mathbf x + \mathbf b(t)$, where $A(t)$ is an orthogonal matrix with $\det A(t) = +1$. A reflection in a plane has determinant $-1$, so no motion in $\R^3$ can turn a chiral body into its mirror twin.
A rigid motion preserves distances, and every distance-preserving map of space has the form $\mathbf x \mapsto A\mathbf x + \mathbf b$ with an orthogonal matrix $A$, meaning $A^{\mathsf T}A = I$. Hence $(\det A)^2 = \det(A^{\mathsf T}A) = 1$, so $\det A(t)$ can only be $+1$ or $-1$. The determinant is a polynomial in the entries of the matrix, and they change continuously during a continuous motion, so $\det A(t)$ is a continuous function of time. A continuous function cannot jump from $+1$ to $-1$ without passing through the values in between, and those it cannot take, so it is constant. At the start $A = I$ and $\det A = 1$. ∎
A rotation in the $xw$-plane is a motion too, with determinant $+1$ in four dimensions. There is no contradiction: the theorem is about motions inside $\R^3$, and on the way the body leaves our space. If you look only at the start and the end, the 180° turn acts on our space itself as the reflection $x \mapsto -x$, with determinant $-1$.
Fiction got there before physics. In H. G. Wells’s “The Plattner Story” (The New Review, April 1896), a schoolmaster named Gottfried Plattner sets fire to a greenish powder in the school laboratory, disappears in the explosion and comes back nine days later mirror-reversed: his heart beats on the right, his liver and lungs have swapped sides, and he can only write with his left hand, from right to left. The narrator offers a single explanation: Plattner has been in the fourth dimension. A real person would pay dearly for such a trip. The molecules of life are chiral too (proteins are built from “left-handed” amino acids), and a mirror-image body would most likely be unable to digest ordinary food properly.
Knots come undone
Another consequence of the same trick concerns knots. A knot is a closed loop of rope in space that cannot be turned into a plain ring without cutting. All its stubbornness lives at the crossings: where one stretch of rope passes over another, the two cannot swap places without the rope passing through itself.
In four dimensions this is no obstacle. At each crossing, nudge the upper stretch a little along $w$ (it is now “beside” rather than “over”), drag it to the other side and bring it back to $w = 0$. The two stretches have passed each other in the fourth direction without touching. By switching crossings like this, any knot can be undone. So every (smooth) knot in $\R^3$ comes undone in $\R^4$: in four-dimensional space any closed curve without self-intersections can be deformed into a circle without ever crossing itself.
The analogy one floor down: two inhabitants of Lineland can never swap places, because one would have to pass through the other. In a plane, they simply walk round each other. An extra dimension gives room to get past.
Knots do not disappear in four dimensions altogether; they move up a step. What you can tie in a knot there is not a rope but a two-dimensional sphere. The first knotted spheres were built by Emil Artin in 1925: he took a knotted arc and spun it in four-dimensional space around a plane, the way a potter’s wheel turns a profile into a vase. Knotted circles live in $\R^3$, knotted 2-spheres in $\R^4$.
A dictionary of analogies
Let’s put it all together. On the left, what we watch from outside; on the right, the same thing for us, if a fourth dimension were next door.
| Flatland inside space | Our space inside 4D |
|---|---|
| the “retina” is one-dimensional; the world is seen as segments | our retina is two-dimensional; a 4D being’s would be three-dimensional |
| a sphere: a point, a growing and shrinking circle | a hypersphere: a point, a growing and shrinking ball |
| cube face first: a square appears and vanishes whole | tesseract cell first: a cube appears and vanishes whole |
| cube vertex first: triangle, hexagon, triangle | tesseract vertex first: tetrahedron, octahedron, tetrahedron |
| a closed curve locks you in if you cannot leave the plane | a closed surface locks you in if you cannot move in $w$ |
| from above, a Flatlander’s insides are all on view | from 4D, all our insides are on view |
| the letter L flips into its mirror image through 3D | a left glove becomes a right one through 4D |
| Linelanders get past each other by stepping into the plane | knots in a rope come undone by moving in $w$ |
The method does not prove that a fourth dimension exists physically; that is a different question, with a chapter of its own. It does something else: it turns the unimaginable into something you can compute. Every row of the table is a statement you can check with a formula, and we have checked them above.
Summary
- Edwin Abbott Abbott’s Flatland (1884) is a satire of Victorian society and a textbook of reasoning by analogy: we look at the two-dimensional Square the way a four-dimensional being would look at us.
- A two-dimensional being sees a one-dimensional picture and judges shape by brightness in fog; we see a two-dimensional one and rebuild the third dimension.
- A solid from a higher dimension shows us only its sections. A sphere of radius $R$ gives circles of radius $\sqrt{R^2-h^2}$, a hypersphere gives balls of the same radius.
- Sections depend on orientation: a cube gives a square, a rectangle or the sequence “triangle, hexagon, triangle”; a tesseract gives a cube or the sequence “tetrahedron, truncated tetrahedron, octahedron, …”.
- From $n+1$ dimensions you see everything inside $n$-dimensional bodies, closed barriers stop locking anything in, mirror twins can be turned into each other, and knots in a rope come undone.
A section is one way to see a four-dimensional solid. The other is a shadow, and it gives the famous “cube inside a cube”: that is the next chapter.
Sources
- Edwin A. Abbott. Flatland: A Romance of Many Dimensions. London: Seeley & Co., 1884. Text: Project Gutenberg #201.
- Edwin Abbott Abbott and Flatland on Wikipedia.
- I. Kant. Von dem ersten Grunde des Unterschiedes der Gegenden im Raume (1768); English: “Concerning the ultimate ground of the differentiation of directions in space”, in Theoretical Philosophy 1755–1770, Cambridge, 1992; Prolegomena, § 13 (1783).
- A. F. Möbius. Der barycentrische Calcul. Leipzig, 1827, § 140, note (pp. 184–185).
- H. G. Wells. The Plattner Story, The New Review, April 1896; Wikipedia article.
- Knot theory, section on higher dimensions, Wikipedia.
- E. Artin. Zur Isotopie zweidimensionaler Flächen im R₄, Abh. Math. Sem. Univ. Hamburg 4 (1925), 174–177.
- H. S. M. Coxeter. Regular Polytopes. 3rd ed. New York: Dover, 1973.