The Paradox of Giants: Strange Consequences in High Dimension—Lectures on Infinity (lecture 3)
The paradox of giants, the paradox of Gabriel's horn, the painter's paradox, and further paradoxes of dimension.
Welcome to the Lectures on Infinity, a series of lectures exploring all my favorite paradoxes and conundrums.
In this third lecture, we shall explore the paradox of giants, showcasing Galileo’s argument that the traditional giants of folklore—taking human form but at much larger scale—are physically impossible. He argued on the basis of an understanding of how size scales differently in different dimensions. Similar ideas lead to the paradox of Gabriel’s horn, the painter’s paradox, paradoxical fractals, and to many further paradoxes of dimension. By the end of the lecture, we shall glimpse some genuinely troubling conundrums in high dimension.
I shall be sharing the individual infinity lectures here on Infinitely More in the coming weeks and months.
Please enjoy!
Find the lectures here on Infinitely More in the lectures-on-infinity tag.
The lectures will appear on YouTube.
The whole lecture course is hosted at Ergo: Lectures on Infinity.
Find other philosophy lecture courses at Ergo.org.
This lecture is based on my essay The Paradox of Giants.
The essay also appears in my new book, The Book of Infinity.
The Paradox of Giants—Strange Consequences in High Dimension
A lightly edited transcript. Timestamps link to the video on the Ergo website.
Giants of Legend and Literature
Welcome to these lectures on infinity. Today, I want to tell you about the paradox of giants. According to legend, giants once roamed the Earth. Everyone knows that Odysseus met the Cyclops, who lived in a great cave and grabbed sheep and men with his hands and ate them whole. And in the time of King Arthur, there was the young boy who earned the title Jack the Giant Killer because he was able to use his sharp wit to outsmart and slay the various giants that plagued the land.
It is the same Jack, I think, as the Jack of Jack and the Beanstalk, who planted the seeds that grew into the beanstalk, climbed to the castle in the sky, and tricked that giant as well. But there is also Jonathan Swift’s character Gulliver, who travels to distant lands and finds the Lilliputians, those tiny human beings to whom Gulliver himself seemed the giant. And yet in those same travels he also encountered the Brobdingnagians, who were giants to whom Gulliver seemed Lilliputian, even though Gulliver had never changed his size at all.
In all of these legends, the giants tend to have an ordinary human form and they do ordinary human things. They walk around, they stomp, they dance, they drink wine from goblets, they carry heavy stones, they climb ladders. They move in a human manner, but simply at scale.
Why Physics Makes Giants Impossible
Galileo wrote in his Dialogues Concerning Two New Sciences a wonderful criticism of this whole manner of thinking about giants, arguing that giants are actually impossible. Physics cannot work like that. The very idea of a giant is, on his account, contradictory. Let me explain his argument.
He asks us to imagine a great oak beam, sturdy enough to hold up a heavy stone or a load of bricks. Now imagine making it ten times bigger, keeping exactly the same dimensions and proportions but scaling everything up: ten times longer, ten times thicker, ten times wider, made of the same material. This larger beam might be the kind of beam you would find in a giant’s house, as opposed to the ordinary beam in our own.
Of course, we expect the bigger beam to be stronger and able to hold a greater load. But how much stronger, exactly? Galileo argued that the strength of a beam is related to its cross-sectional area, because the fibers of the wood run lengthwise, and when the beam breaks, it breaks along the cross-section. It is the strength of the fibers passing through that cross-section that determines the overall strength of the beam. If the beam is ten times bigger in every direction, the cross-sectional area is multiplied by one hundred, since it scales as ten times ten. So the larger beam is one hundred times stronger, which seems quite promising.
But now consider the stone the beam was holding up. If we make the stone ten times bigger in every direction, its volume increases by a factor of one thousand, since it scales as ten times ten times ten across all three dimensions. A stone ten times larger in every direction therefore weighs one thousand times as much, assuming it is made of the same material. Suddenly the situation looks far less favorable: the beam is one hundred times stronger, but the load it must bear is one thousand times heavier.
If the original beam was just barely holding up the original stone, the scaled-up beam will not be strong enough to hold the scaled-up stone. Galileo went further still, arguing that the beam would not even be able to support its own weight, because the mass of the enlarged beam itself grows by that same factor of one thousand, while its strength grows by only one hundred. The structure collapses under itself. This is why, on Galileo’s reasoning, giants are not merely unlikely but physically impossible.
The Square-Cube Law Destroys Giants
Galileo argued that because of this difference in dimension, strength increases with the square of the scaling, but mass increases with the cube of the scaling, which is significantly greater. This mismatch between the two quantities means that the whole concept of a giant becomes incoherent. The beams of the giant’s house would not be able to hold up the roof. The goblet that is ten times bigger would not be able to hold the wine it contains. The giant would not be able to climb a ladder, because the ladder would not even be able to support itself, let alone a giant.
The bones of a giant are essentially beams, and if we take a human being and scale up to make the giant ten times bigger in every direction, ten times taller, ten times thicker, and so on, then the bones become one hundred times stronger, but the mass of the giant increases by one thousand. Therefore the giant will not be able to stand up or walk around. The whole concept of a giant is incoherent.
Galileo writes: “Clearly then, if one wishes to maintain in a great giant the same proportion of limb as that found in an ordinary man, he must either find a harder and stronger material for making the bones, or he must admit a diminution of strength in comparison with men of medium stature, for if his height be increased inordinately, he will fall and be crushed under his own weight.”
Why Elephants Are Stocky and Bugs Are Thin
We can see this as almost obvious if we think about the nature of large animals versus small animals. Consider the typical large animals: elephants, rhinoceroses, and so on. They are characteristically stocky, with very stocky limbs. The reason is precisely the dimensional issue that Galileo pointed out: in order to support greater weight, the limbs need to be not only bigger, but proportionally bigger, and that is what produces a stocky animal.
The same principle works in the other direction. Tiny animals and insects typically have very slender, thin limbs, and yet they support their weight just fine. But if you were to take a housefly and make it ten times bigger, it could no longer crawl along a wall, because the electrostatic forces would simply not be strong enough. Making it ten times bigger makes it one thousand times heavier, and the electrostatic forces do not scale to match that increase.
Similarly, a bug that walks on the surface of water relies on surface tension, and surface tension does not scale in the right way either. The paradox of giants, then, is that you cannot simply take a functioning animal with a given architecture, scale it up, and expect it to work in the same way. It simply will not work that way.
The Paradox of Miniature Humans
A similar issue arises not just with giants but with what we might call the paradox of the miniature human. You may have encountered this theme in Hollywood films such as Downsizing, Ant-Man, or Honey, I Shrunk the Kids, or indeed in the Lilliputians of Gulliver’s Travels. The idea of a miniature human is a recurring cultural fascination, but it runs into the same scaling problems we have been discussing.
If you take an ordinary human and make them ten times smaller, they will be proportionally stronger for their height, for exactly the same kind of reason we considered with giants. This is precisely why grasshoppers can jump many times their own height: a tiny human would likewise be able to jump very high in comparison with their height, not in absolute terms, but relative to its new, smaller stature. Such a creature would not move through the world the way ordinary humans do.
Interacting with water, for instance, would become very complicated, because at that scale water would behave as far stickier than it does at our scale. The nature of physical existence simply does not scale in that way, and this is the core of Galileo’s argument.
Evolution and Body Size Genes
This is related to evolution and body size. It seems to be the case that the body-size architecture for many different kinds of animals must be controlled by relatively few genes, because when you look at the evolutionary history of certain animals, their size varies quite a lot. In prehistoric times, for instance, there were enormous dragonflies, far larger than the ones we see today. Horses, too, were much smaller when they first evolved, and their size went up and down repeatedly over their evolutionary history.
We can see some residual evidence of this in miniature horse breeds, those very tiny horses that still exist, which carry what we might call smallness genes still present in the horse population. One can imagine natural selection acting on those genes to produce changes in the body-size architecture of a species. There might be some advantage to becoming larger, even though greater size makes an animal heavier and proportionately less strong, if that size helps the animal compete more effectively within its ecological niche. So it is easy to imagine evolution acting on those genes to shift body size in response to environmental pressures.
There are closely related effects that arise from differences in dimension, particularly the difference between surface area and volume. I want to turn now to some interesting mathematical examples that illustrate this distinction.
Gabriel’s Horn Has Finite Volume
One of my favorite examples is the Paradox of Gabriel’s Horn. We begin with the function y = 1/x, looking at the portion of the curve starting at x = 1 and extending out to infinity. To form Gabriel’s Horn, we take that curve and revolve it around the x-axis, producing a kind of symmetric shape. It is, in a sense, like a horn from heaven, perhaps sounding some sonorous, multi-toned note, and that is why it carries the name Gabriel’s Horn.
Now, the paradox centers on a simple question: what is the volume of Gabriel’s Horn? It is an infinite object, since it extends forever, but the function is 1/x, so the horn becomes very thin as we move far out along the x-axis. We can compute the volume using a standard technique from calculus for finding the volume of a solid of revolution. The idea is to slice the solid into thin disks perpendicular to the x-axis.
At a given point x, the radius of such a disk is 1/x, since that is the value of the function being rotated, and the thickness of the disk is dx. The volume of one disk is therefore the area of the disk times its thickness, and since the area is π r squared, the volume of a single disk is π times (1/x)2 dx. To find the total volume, we integrate this expression from 1 to infinity, adding up the contributions of all the disks. The total volume is thus the integral from 1 to infinity of π over x squared dx.
This is an elementary calculus integral. The antiderivative of 1/x squared is minus 1/x, so we evaluate minus π/x from 1 to infinity. As x goes to infinity, minus π/x approaches zero, and by the Fundamental Theorem of Calculus we subtract the value at x = 1, giving zero minus (minus π/1), which equals π. The volume of Gabriel’s Horn is precisely π, a finite number. That is the first paradoxical observation: Gabriel’s Horn is an infinite object, yet it encloses a perfectly finite volume.
Gabriel’s Horn Has Infinite Surface Area
The second part of the paradox is to ask: what is the surface area of Gabriel’s horn? Rather than computing the volume of each disk, we now concentrate on the band around the outside, which is what is called the frustum of a cone. It is slightly angled, and the infinitesimal length along that angled piece is commonly written as ds, equal to the square root of dx squared plus dy squared. Factoring out a dx, this becomes the square root of 1 plus (dy/dx) squared, times dx.
The total surface area is therefore the integral from one to infinity of the circumference of each frustum times that infinitesimal slant length. The circumference is π times the diameter, which gives 2π over x, so the surface area integral becomes the integral from one to infinity of (2π over x) times the square root of 1 plus (dy/dx) squared, dx. Since y equals 1 over x, which is x to the minus one, we get dy/dx equals minus 1 over x squared, so (dy/dx) squared equals 1 over x to the fourth.
The resulting integral is more complicated, but we can sidestep the difficulty with a simple observation. The square root of 1 plus 1 over x to the fourth is always at least 1, so the surface area is greater than or equal to the integral from one to infinity of 2π over x, dx. That integral equals 2π times the natural log of x, evaluated from one to infinity, which diverges to infinity. Therefore the surface area of Gabriel’s horn is infinite.
Can You Paint an Infinite Surface?
The surface area of Gabriel’s Horn is infinite, but its volume is finite. How can that be? Consider this: if we point the horn downward and fill it with paint, we use only a finite amount of paint, and that paint would be touching every part of the interior surface. It seems, then, that with a finite amount of paint we have painted Gabriel’s Horn. This is the heart of the paradox. Gabriel’s Horn is a geometrical object we can understand in a deep way, and yet it has finite volume and infinite surface area.
But does the filling argument really work? Should filling a container with paint count as painting its surface? I would say we are cheating a little, because Gabriel’s Horn grows thinner and thinner as it extends outward. The paint inside the horn is spread more and more thinly the farther out you go. If we require a uniform thickness of paint on the surface, say one millimeter, then eventually that condition is violated, because the horn itself becomes less than one millimeter across. So even though the horn is full of paint, it does not follow that we have painted the surface to any uniform thickness.
This reveals why filling Gabriel’s Horn with paint should not count as painting its surface: the paint is spread so thin in the region far out along the tail. And it is precisely that tail region which accounts for the infinite surface area. If we chop the tail off, what remains has obviously only a finite area. So the infinite area lives out in the part where the paint has been spread vanishingly thin. Filling the volume with paint is therefore entirely the wrong way to think about painting the surface, and the apparent paradox dissolves once we see how the argument was cheating all along.
Extended Real Numbers and Infinity
There is one thing I want to mention. I have been writing the infinity symbol ∞ on the board, and since this whole lecture series is about the infinite, I want to discuss this particular use of infinity, which is often the first instance of infinity that many students encounter in a mathematics class, in a calculus class. So what does this mean? What is that number? Is it a number? How should we think about it?
We begin with the real number system, the set of all real numbers. It is an ordered field: we can add and multiply its elements, compare their order, and identify them with points on the number line. From there, we have what is called the extended real numbers. This is a number system obtained by starting with the real numbers and simply adding infinity and minus infinity as idealized objects. We adjoin these two extra elements to the set and then define how arithmetic works with them.
For example, in the extended real numbers, infinity plus two equals infinity; indeed, adding any finite number to infinity leaves it infinite. Infinity plus infinity is infinity. Similarly, minus infinity plus any finite number remains minus infinity. As for multiplication, infinity times a equals infinity if a is positive, but equals minus infinity if a is negative, so infinity times minus five is minus infinity, and so on.
One has to keep in mind, however, that certain combinations are simply not defined in the extended real numbers. Infinity minus infinity has no meaning, and neither does infinity times zero. Within those constraints, you can work quite intuitively with these symbols according to these rules, and it is remarkable how far this way of treating infinity actually goes.
I think of it, philosophically, as ontologically very light, even deflationary in a way. It says: we do not need to give a robust or heavy meaning to infinity; we can simply add it as a symbol, define how to calculate with it, and things work out beautifully. It is rather remarkable that such a light attitude toward an apparently heavy concept can be so productive. For many mathematicians, this is precisely the use of infinity they encounter most often, and it is the one we already relied on when examining the nature of Gabriel’s horn.
Testing the Paint-Based Theory of Area
We discussed the idea of a paint-based theory of surface area. The proposal is this: a surface has finite area if and only if one can coat it to a uniform finite thickness using a finite volume of paint. If every point on the surface is covered to some fixed depth, say one millimeter, and the total volume of paint required is finite, then perhaps that is a reasonable criterion for saying the surface has finite area.
But let me criticize this proposal, because it does not quite work. Suppose we have an infinite line, such as the x-axis. A line has zero area, and yet one cannot cover it to a uniform thickness with a finite volume of paint, because any uniform coating around an infinite line would form an infinite cylinder, which has infinite volume. So here we have something with finite area, namely zero area, that nevertheless cannot be painted to uniform thickness with a finite volume of paint. This is a counterexample to the paint-based account.
One might object that a line is not a surface at all. It is a one-dimensional object, not a surface, and so it should not count as a test case for a theory about surfaces. Fair enough. So let me offer a different kind of counterexample, a variant of Gabriel’s horn.
A Modified Horn with Finite Area
In this modified version of Gabriel’s horn, I am using the function one over x squared instead of one over x. The two functions look roughly similar, but one over x squared decreases to zero far more rapidly. When x is 100, one over x squared equals one ten-thousandth, which is 100 times smaller than one one-hundredth. When x is a million, one over x squared is a million times smaller than one over x, since it equals one over a million times a million, and so on.
Because one over x squared goes to zero faster, the resulting horn tapers toward the x-axis much more quickly, though it never actually touches it. It is extremely thin in the tail, but we can construct a Gabriel’s horn-type surface from it in exactly the same way. The key difference is that for this version, both the surface area and the volume are finite. Recall that the paradox of the original Gabriel’s horn was the contrast between a finite volume and an infinite surface area; this modified horn has neither of those infinities.
Now consider what happens when we apply the paint-based criterion for finite surface area. The proposal was that a surface has finite area if and only if it can be painted to a uniform thickness using a finite volume of paint. If we try to apply a uniform coat of paint to this tighter, more rapidly tapering horn, we still run into trouble. Even though the horn is extremely close to the x-axis out in the tail, there remains a thin cylindrical shell of paint of, say, one millimeter radius running along that entire infinite tail, and covering it to a uniform thickness requires an infinite volume of paint.
So this modified Gabriel’s horn is a surface with finite surface area that nevertheless cannot be painted to uniform thickness with a finite volume of paint. Together with the original Gabriel’s horn, we now have two examples on the same side of the ledger: surfaces of finite area that fail the painting criterion. What I want to do next is produce a counterexample on the other side, namely a surface that satisfies the painting criterion but does not have finite area in the standard sense.
Koch Snowflake Breaks the Paint Rule
The paint-based criterion for finite surface area turns out to be wrong in both directions: it is neither necessary nor sufficient. To see why, I want to introduce an example we will return to more fully in a later lecture on the infinite coastline paradox and the concept of fractals. The example is the Koch snowflake curve.
The construction works as follows. You start with a line segment of a certain length, chop it into thirds, and replace the middle third with two sides of an equilateral triangle, producing a shape with a small outward kink. Where you had one segment, you now have four segments, each of length one-third. You then repeat the process: each of those four segments gets its own kink in the middle. You do this again, and again, adding smaller and smaller triangular bumps at every scale, producing a curve that is ever more wiggly at ever finer scales. If you carry this process all the way around a triangle rather than along a single segment, the resulting shape looks like a snowflake, which is why it bears that name.
The Koch snowflake curve has infinite length, and you can see why directly from the construction. Each iteration of the process replaces three segments of length one-third with four segments of length one-third, so the total length is multiplied by four-thirds at every step. Since this is done infinitely many times, and since these curves converge in a way that makes the infinite iteration well-defined, the length cannot be any finite value. A finite length would have to equal four-thirds times itself in order to satisfy the generation rule, which is impossible. So the length of the curve is infinite.
Now I want to build a surface out of this curve by extending it into a third dimension, producing something like a corrugated roof whose cross-section is exactly the snowflake curve. The surface is extremely wiggly in one direction but consists of straight lines in the other. I then enclose this corrugated lid in a rectangular box to make a solid. Because the snowflake curve has infinite length, the lid of this box has infinite area: there are so many nooks and crannies, at such fine scales, that the cross-sectional length is infinite, and therefore the area of the roof is infinite, larger than any finite quantity.
And yet the whole object is bounded. If you dunk it in a vat of paint, a finite amount of paint will cover every part of the surface to within one millimeter, thereby satisfying the paint-based criterion, even though the surface area is genuinely infinite. This is the opposite situation from Gabriel’s Horn. In the Koch box, one small region of paint simultaneously covers many different parts of the surface, because the surface folds back on itself so tightly that a one-millimeter thickening of the surface produces enormous overlaps. In the Gabriel’s Horn case, the geometry runs the other way: to cover even a tiny patch of surface area, you need a large volume of paint wrapping all the way around an extremely thin tube. One bit of paint covers very little surface there, whereas here one bit of paint covers a great deal.
This pair of examples together constitutes what I call the painter’s paradox. The paint-based account of finite surface area simply does not work, and these two constructions show exactly why: one gives infinite surface area that can be painted with finite paint, and the other gives finite surface area that cannot be painted with finitely much paint. With that, we can move on to some other paradoxes of higher dimension, beginning with curves in the plane.
Beautiful Spirals in the Plane
There are some beautiful curves that can be drawn in the plane. If you are familiar with polar coordinates, where a point is specified not by its x and y coordinates but by its radial and angular coordinates, then consider the curve r equals e to the minus theta, where theta is the angle and r is the radius. This specifies the radius as a function of the angle, and the resulting curve spirals inward, because as theta increases the value of e to the minus theta becomes very small, so the curve spirals very rapidly into the origin. This is called a logarithmic spiral, and one can prove that even though the curve winds around the origin infinitely many times, it still has finite length.
There is another spiral given by r equals theta, called the Archimedean spiral. A characteristic feature of the Archimedean spiral is that the spacing between successive arms is quite regular: each time you go around, the distance between turns is the same. If we traverse it inward toward the origin, we go around only finitely many times, and the curve has finite length.
Another example is the hyperbolic spiral, given by r equals one over theta. This curve also winds around the origin infinitely many times, but it approaches the origin more slowly, and it has infinite length. So we have a contrast: the logarithmic spiral winds around infinitely many times and has finite length, the hyperbolic spiral winds around infinitely many times and has infinite length, and the Archimedean spiral, traversed inward, winds around only finitely many times and has finite length. These examples illustrate some of the range of possible behavior for these one-dimensional curves in the plane.
With that, let us move to higher dimensions and ask: what is the volume of a sphere in higher dimensions?
Hypersphere Volume Peaks at Dimension 5
Let me begin with something familiar. The unit circle in dimension two has radius one, and its area is π r-squared, which at r equals one gives us simply π. Moving up to dimension three, the unit sphere has volume four-thirds π r-cubed, and again with r equal to one, that is four-thirds π. So from dimension two to dimension three, the hyper volume has increased by a factor of one-third. The natural question is what happens as we continue into higher dimensions.
Before going up, it is worth asking what happens when we go down. What is the one-dimensional sphere? A circle is the set of all points at distance one from a given center, and we can apply exactly that definition in one dimension. The result is just two points, one on each side of the center, forming a line segment of length two between them. The relevant notion of size in dimension one is length, in dimension two it is area, in dimension three it is volume, and in higher dimensions we call it hyper volume. All of these are instances of the same concept, and we can use the term hyper volume to cover all cases uniformly.
So the sequence begins: in dimension one, the hyper volume is two; in dimension two, it is π; in dimension three, it is four-thirds π. The question is whether this keeps increasing forever. It turns out there is a recurrence relation one can derive, which I will state without proof. If v sub n denotes the hyper volume of the unit sphere in dimension n, then v sub n equals two π over n, times v sub n minus two. In other words, if you know the hyper volume of the unit hypersphere two dimensions below, you multiply by two π over n to obtain the hyper volume in dimension n.
Applying this formula, we can build a table. v4 equals two π over 4, times v2, which is two π over 4 times π, giving π-squared over 2. v5 works out to eight π-squared over 15. v6 then comes to π-cubed over 6. In approximate decimal terms, π-squared over 2 is about 4.9, eight π-squared over 15 is approximately 5.264, and π-cubed over 6 is approximately 5.168. So the hyper volume increases up through dimension five and then begins to fall in dimension six.
We can see directly from the recurrence why this must happen. The factor two π over n is less than one whenever n is greater than two π, and two π is approximately 6.28. So for n equal to seven and beyond, each step multiplies the previous hyper volume by something less than one, and the values decrease. In fact, comparing dimension six with dimension four already shows a decrease relative to dimension five, which is why the maximum is achieved at dimension five rather than at six or seven. In all dimensions greater than five, the hyper volume of the unit hypersphere is strictly smaller, and it continues to shrink toward zero as the dimension grows. The upshot is that the hyper volume of the unit hypersphere is maximized in dimension five, which is, on reflection, a rather surprising fact.
Why Hypercubes Are All Corners
I want to talk about this in connection with the paradox of giants, because Galileo’s argument was fundamentally about understanding the nature of giants by understanding how scaling works in different dimensions. That is exactly what we are doing here. He was mainly concerned with dimensions up to three, but I see no reason to be limited to three dimensions only.
I want to understand hyperspheres and how they sit inside the cubes that naturally bound them. We have the unit circle sitting inside a square, the unit sphere sitting inside its bounding cube, and similarly, in higher dimensions, we have a hypercube bounding the corresponding hypersphere. In the one-dimensional case, the unit sphere and the unit cube are the same object. As the dimension increases, the sphere begins to fill less and less of the cube.
The question is: what fraction of the cube’s volume does the sphere fill? In two dimensions, we are asking what fraction of the area of the square lies inside the circle. The unit circle has radius one, so its area is π times one squared, which is π. The bounding square is two by two, since the diameter is two, so the fraction of the area inside the circle is π fourths, a little more than three quarters.
In three dimensions, the volume of the unit sphere is four thirds π, and the bounding cube is two by two by two, giving a volume of eight. The fraction is therefore π over six, which is already noticeably smaller. This makes intuitive sense: in the square there are only four small extra corner regions not covered by the circle, whereas in the cube there are eight corners, accommodating more of the volume outside the sphere.
What happens in higher dimensions? Recall the formula where the volume of the hypersphere is multiplied by two π over n when passing from dimension n to the next. The volume of the bounding hypercube is two to the n, since it is a product of n factors of two. So the ratio we care about is v sub n divided by two to the n. Each time the dimension increases, the denominator doubles, while the numerator is multiplied by two π over n. That fraction becomes tinier and tinier as n grows large.
The picture that emerges is striking. As the dimension increases, more and more of the points in the hypercube lie outside the sphere. The points near the center are precisely those inside the sphere, but the proportion of such points, compared to all points in the hypercube, goes to zero. Almost all the points in a high-dimensional hypercube are not near the center. Instead, they are concentrated in the corners. This is the phrase people use: the hypercube is very “endy.” Almost all the hypervolume comes from the corners, and very little of it comes from the center.
This represents a fundamentally different geometric character from the dimensions we are familiar with. Our ordinary intuition is built on dimensions one, two, and three, perhaps with dimension four imagined as time. But in dimensions five, six, and beyond, while visualization becomes difficult, we can still calculate, and what we observe is that existence inside the hypercube has the property that almost all points are far from the center. If you are running a numerical simulation that involves picking points at random from a high-dimensional hypercube, almost all of those points will be stuck in some corner. Points near the origin are not typical; they are, in fact, extremely rare as a proportion of all points in high dimension.
The Blue Sphere That Escapes Its Box
Let me show you some more examples of this kind of phenomenon. Take four unit spheres and stack them inside a square. Since each sphere has diameter two, the containing square is four by four. Now place a small blue ball in the middle of the four spheres, and ask: how big is that ball? We can calculate this exactly. If we place the origin at the center of the square, the centers of the four unit circles sit at coordinates (1, 1), (1, −1), (−1, 1), and (−1, −1). The distance from the origin to any one of those centers is the square root of one squared plus one squared, which is the square root of two. Since that distance equals the radius of the blue circle plus the radius of one of the unit circles, we get r plus one equals the square root of two, and therefore r equals the square root of two minus one, which is approximately 0.414.
Now let us do the same thing in three dimensions. Take eight unit spheres, like billiard balls, arranged in a perfectly orthogonal stack inside a cube, and fit a blue sphere in the center. The containing cube is four by four by four, and the centers of the eight unit spheres sit at coordinates such as (1, 1, 1), (1, 1, −1), and so on. The distance from the origin to any one of those centers is the square root of one squared plus one squared plus one squared, which is the square root of three. By the same reasoning as before, r plus one equals the square root of three, so the radius of the blue sphere in three dimensions is the square root of three minus one.
Exactly the same analysis applies in any number of dimensions, and in general the radius of the blue hypersphere that fits snugly in the center of the arrangement of unit hyperspheres in dimension n is the square root of n minus one. Let us think about what this means as n grows. When n equals four, the square root of four is two, so the radius of the blue hypersphere is two minus one, which equals one. In dimension four, the blue hypersphere in the middle is exactly the same size as each of the surrounding unit hyperspheres.
When n equals nine, the square root of nine is three, so the radius of the blue hypersphere is three minus one, which equals two. In dimension nine, the blue hypersphere is twice as large as each of the surrounding unit hyperspheres. More strikingly, its radius of two carries it all the way from the center of the hypercube to the wall, so in dimension nine the blue hypersphere is actually touching the walls of the hypercube that contains all the others.
For dimensions greater than nine, the square root of n minus one exceeds two, and the blue hypersphere is so large that it actually protrudes outside the hypercube that bounds the surrounding hyperspheres. This is very hard to imagine if you think only in two or three dimensions, but it is exactly what the mathematics tells us. Nine is not even a particularly large number, yet the geometry has already become radically different from anything our low-dimensional intuition would suggest. In a million dimensions, the blue hypersphere is so enormous that it is difficult to grasp, and yet that is precisely what follows from the formula.
I hope you enjoyed this account of the paradox of giants, which led us from Galileo’s analysis of how volume and structural strength scale with dimension into these further paradoxes of higher-dimensional geometry. I hope to see you next time.
The Lectures on Infinity will appear in the lectures-on-infinity tag. The full collection of essays is available on Infinitely More at The Book of Infinity. And the book is also now available in printed form: