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. In the time of King Arthur, there was a young boy who earned the title Jack the Giant Killer because he used his sharp wit to outsmart and slay the various giants that plagued the land. This is the same Jack, I believe, as in 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.
Then there is Jonathan Swift's character Gulliver, who travels to distant lands and encounters the Lilliputians, tiny human beings to whom Gulliver himself seems a giant. Yet in those same travels he also meets the Brobdingnagians, who are giants to whom Gulliver seems Lilliputian, even though Gulliver's size never changes at all.
In all of these legends, giants tend to have an ordinary human form and to do ordinary human things. They walk and stomp, they dance, they drink wine from goblets, they carry heavy stones, they climb ladders, they move in a thoroughly human manner, only at scale.
Galileo, in his Dialogues Concerning Two New Sciences, offers a wonderful criticism of the whole idea of giants, arguing that they are actually impossible, that physics simply cannot work that way. The very concept of a giant, he contends, is contradictory. His argument runs as follows.
Imagine a sturdy oak beam, the kind that might hold up a heavy stone or a load of bricks in an ordinary house. Now imagine making that beam ten times bigger in every direction: ten times longer, ten times thicker, ten times wider, constructed from the same material. This scaled-up beam would be the sort of thing a giant might use in his house. The question Galileo poses is: how much stronger will it be?
Galileo argued that the strength of a beam depends on its cross-sectional area, because the fibers of the wood run lengthwise, and when a beam breaks, it breaks along the cross-section. It is therefore the strength of the fibers passing through that cross-section that determines the beam's overall load-bearing capacity. If the beam is ten times larger in every direction, its cross-sectional area is multiplied by one hundred, ten times ten, meaning the larger beam is one hundred times stronger. That seems promising: it should be able to bear one hundred times the load.
But now consider the stone the beam was holding up. If that stone is also made ten times larger in every direction, its volume grows by a factor of one thousand: ten times ten times ten, one for each of the three dimensions. Since the material has the same density, the scaled-up stone weighs one thousand times as much. The beam is only one hundred times stronger, yet the load it must bear is one thousand times heavier. If the original beam was just barely supporting the original stone, the scaled-up beam will not be strong enough to support the scaled-up stone.
Galileo pressed the argument even further: the enlarged beam would not even be able to support its own weight. The mass of the beam itself grows by that same factor of one thousand, while its strength grows by only one hundred, so the beam collapses under itself. This is why Galileo concluded that giants are not merely impractical but physically impossible, the very laws governing the strength of materials and the scaling of volume forbid them.
Galileo argued that because of this difference in dimension, strength increases with the square of the scaling, while mass increases with the cube of the scaling, a significantly greater quantity. This mismatch between the two means that the very concept of a giant becomes incoherent. The beams of a giant's house would not be able to hold up the roof; the roof would not be able to support itself. A goblet ten times larger would not be able to hold the wine it contains. A giant could not climb a ladder, because the ladder would collapse under its own weight before the giant ever set foot on it.
The giant's own bones illustrate the problem most vividly, since bones function essentially as structural beams. If we take a human being and scale up every dimension by a factor of ten, ten times taller, ten times thicker in every direction, the bones become one hundred times stronger, but the mass of the giant increases by one thousand. The giant would therefore be unable to stand or walk. 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."
This can be seen as almost obvious when we consider the nature of large animals versus small ones. Large animals, elephants, rhinoceroses, and the like, are typically very stocky, with thick, heavy limbs. The reason is precisely the dimensional relationship that Galileo identified: in order to support a greater weight, limbs must be not merely larger, but proportionally larger, and that is what produces a stocky animal.
The principle works in the other direction as well. Tiny animals, insects in particular, have extremely slender limbs, yet those limbs support their weight with ease. But if you were to take a housefly and make it ten times larger, it could no longer crawl along a wall, because the electrostatic forces that allow it to cling to surfaces would be nowhere near sufficient. The fly would be ten times bigger in each dimension, making it a thousand times heavier, while the adhesive forces would not scale to match.
The same logic applies to a bug that walks on the surface of water: surface tension does not scale in the right way to support a greatly enlarged version of the creature. This is the paradox of giants. You cannot simply take a functioning animal with a given architecture, scale it up, and expect it to work in the same way, because the physics simply will not allow it.
A similar paradox arises not just with giants, but with the miniature human. Perhaps you have seen some of these Hollywood films, Downsizing, Ant-Man, Honey, I Shrunk the Kids, the idea of a miniature human is a common theme in popular cinema, echoed as well in the Lilliputians of Gulliver's Travels. If you take an ordinary human and make them ten times smaller, they will be proportionally stronger for their height, for exactly the same reason we have been discussing.
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 its new, smaller stature, not in absolute terms, but relative to that reduced height. Such a creature would not move through the world the way ordinary humans do. Interacting with water, for instance, would become enormously complicated, because at that scale water would behave far more viscously and adhesively than it does at ours.
The nature of physical existence simply does not scale in that straightforward way, and this is the core of Galileo's argument.
This phenomenon appears to be related to evolution and body size. The body size architecture for many different kinds of animals seems to be controlled by relatively few genes, because when we look at the evolutionary history of certain animals, their size varies quite dramatically. 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 fluctuated repeatedly, growing large, then small, then large again.
We can observe some residual evidence of this in miniature horse breeds, where smallness genes have persisted in the population. Natural selection acting on those genes could drive significant changes in body size architecture. There might be circumstances in which becoming larger confers a competitive advantage within a given ecological niche, even at the cost of being heavier and proportionately less strong, and so we can easily imagine evolution acting on those genes to shift body size in response to environmental pressures.
There are similarly interesting effects that arise from differences in dimension, the relationship between surface area and volume, for example. This brings us to some compelling mathematical illustrations of that principle.
A classic example is the Paradox of Gabriel's Horn. We begin with the function y = 1/x, considering only the part of the curve starting at x = 1 and extending out to infinity. To form Gabriel's horn, we revolve that curve around the x-axis, producing a symmetric shape that tapers as it stretches infinitely far, a horn descending from heaven, which is precisely why it bears the name.
The paradox concerns the volume of this object. It is an infinite object, since it extends forever, yet because the function is 1/x, the horn becomes very thin as x grows large. To compute the volume of the solid enclosed by the horn, we use the standard calculus technique for volumes of revolution: we slice the solid into thin disks perpendicular to the x-axis. At position x, the radius of each disk is 1/x, and its thickness is the infinitesimal dx.
The volume of a single disk is its area times its thickness. Since the area of a disk is πr², the volume of one disk is π(1/x)² dx = π/x² dx. To find the total volume, we integrate this expression from 1 to infinity, adding up the contributions of all the disks. This gives the integral from 1 to ∞ of π/x² dx.
This is an elementary calculus integral. Since the antiderivative of 1/x² is −1/x, we evaluate −π/x from 1 to infinity. As x → ∞, the term −π/x tends to zero, and by the Fundamental Theorem of Calculus we subtract the value at x = 1, which is −π/1. The two minus signs cancel, and the result is exactly π. The volume of Gabriel's horn is precisely π, a finite number. That is the first paradoxical feature of Gabriel's horn: it is an infinite object, yet it encloses a perfectly finite volume.
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 outer band, which is the frustum of a cone, a slightly angled strip. The infinitesimal length of that angled piece is commonly written ds, equal to the square root of dx² + dy², which factors as the square root of 1 + (dy/dx)² times dx.
The surface area of one frustum is that infinitesimal length multiplied by the circumference of the band. The circumference is π times the diameter, giving 2π/x. So the total surface area is the integral from one to infinity of 2π/x times the square root of 1 + (dy/dx)² dx. Since y = 1/x = x⁻¹, we have dy/dx = −1/x², and so (dy/dx)² = 1/x⁴.
Although the resulting integral looks complicated, we can obtain the result by a simple estimate. The square root of 1 + 1/x⁴ is always at least 1, so the surface area is greater than or equal to the integral from one to infinity of 2π/x dx. That integral equals 2π log x evaluated from one to infinity, which diverges to infinity. Therefore, Gabriel's horn has infinite surface area.
The surface area of Gabriel's Horn is infinite, but its volume is finite. How can that be? If we point the horn downward and fill it with paint, we need only a finite amount of paint to do so, and that paint would be touching every part of the interior surface. It seems, then, that a finite amount of paint suffices to paint Gabriel's Horn.
This is the heart of the paradox. Gabriel's Horn is a geometrical object we can understand in considerable depth, and yet it possesses finite volume and infinite surface area simultaneously. The filling-with-paint argument appears to resolve the tension, but does it really work?
In fact, the argument is subtly dishonest. Gabriel's Horn grows thinner and thinner as it extends outward, so the paint filling its interior is spread correspondingly thinner and thinner along that tail. If we require that painting a surface means applying a uniform thickness of paint, say, one millimeter, then the argument fails, because the horn itself eventually becomes narrower than one millimeter. The horn may be full of paint, but that does not mean the surface has been painted to any uniform thickness.
This is precisely where the cheat lies. Most of the horn's infinite surface area resides in the distant tail: if we truncate the horn at any finite point, the remaining piece has only a finite surface area. It is therefore illegitimate to claim we have painted the surface by filling the volume, because in the region that contributes the bulk of the infinite area, the paint has been spread infinitely thin.
There is one thing worth pausing on here. I have been writing the infinity symbol on the board, and of course this entire lecture series is about the infinite, so I want to address this particular use of infinity, often the first instance that students encounter, in a calculus class or similar setting. What does this symbol mean? Is infinity 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 them by order, and identify them with points on the number line. From this foundation, we construct what are called the extended real numbers. This is a number system obtained by taking the real numbers and adjoining two idealized objects, positive infinity and negative infinity, and then specifying how arithmetic works with these new symbols.
For example, in the extended real numbers, infinity plus any finite number is still infinity, and infinity plus infinity is infinity. Likewise, negative infinity plus any finite number remains negative infinity. As for multiplication, infinity times a positive number is positive infinity, while infinity times a negative number is negative infinity, exactly as one would expect. There are, however, certain combinations that are simply left undefined: "infinity minus infinity" has no meaning in this system, and neither does "infinity times zero."
Subject to those exceptions, one can work with these infinity symbols in a remarkably intuitive way. What strikes me as philosophically interesting is how ontologically light this approach is. We need not assign any deep or heavy meaning to infinity; we simply introduce it as a symbol and specify rules for calculating with it, and this turns out to go a very long way. It is quite remarkable that such an apparently modest attitude toward such a weighty concept can be so productive.
For many mathematicians, this extended-real-number sense of infinity is the primary one they work with, and we already drew on it when examining the nature of Gabriel's Horn.
We discussed the idea of a paint-based theory of surface area. If we have a geometric object, a container, what does it mean to say that it has finite area? One proposal is that a surface has finite area if and only if it can be covered with paint to a uniform thickness using a finite volume of paint. That might sound like a reasonable criterion for determining when a surface has finite area.
More precisely, the paint-based proposal holds that a surface has finite area just in case a finite volume of paint suffices to coat it to a uniform thickness, say, one millimeter, or whatever scale one chooses.
Let me criticize this proposal, because it does not quite work. Suppose we have an infinite line, such as the x-axis. That line has zero area, yet we could not cover it to a uniform thickness with a finite volume of paint, because any uniform coating around the line would form an infinite cylinder, and since the line is unending, that cylinder would have infinite volume. This is a counterexample to the paint-based account: here is a geometric object with finite area, namely zero area, that nevertheless cannot be painted to uniform thickness with a finite volume of paint.
One might object that a line is not a surface at all: it is a one-dimensional object, not a surface in the relevant sense. Fair enough. So let us consider a different version of Gabriel's Horn that addresses this concern directly.
This is a modified Gabriel's horn using a different function: 1/x² instead of 1/x. Although they look roughly similar, 1/x² decreases to zero far more rapidly than 1/x. When x is 100, for instance, 1/x² equals one ten-thousandth, which is 100 times smaller than one one-hundredth. When x is a million, 1/x² is a million times smaller than 1/x, since it equals one over a million squared. So 1/x² goes to zero much faster, and the resulting horn tapers toward the x-axis far more quickly, though it never actually reaches it.
We can still construct a Gabriel's horn-type surface using this function in exactly the same way as before. The key difference is that for this version, both the surface area and the volume are finite. Recall that with Gabriel's horn the paradox arose precisely because the volume was finite while the surface area was infinite. 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, it still requires an infinite volume of paint. Even though the horn narrows dramatically, there remains an effectively cylindrical tail, a tiny, say one-millimeter-radius cylinder of paint extending along the entire infinite length, and covering that tail to any fixed uniform thickness demands infinitely much paint.
This modified Gabriel's horn therefore gives us 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, both examples fall on the same side: each is a finite-area surface that fails the painting criterion. What remains is to produce a counterexample on the other side, a surface that can be painted with a finite volume of paint and yet has infinite surface area.
There exists a surface with infinite area that can nonetheless be painted with a finite volume of paint. This shows that the paint-based criterion is wrong in both directions: it is neither necessary nor sufficient for finite surface area. To demonstrate this, I want to draw on an example we will return to more fully in a later lecture on the infinite coastline paradox, which leads to the concept of fractals.
Consider the Koch snowflake curve. You begin with a line segment of a certain length, divide it into thirds, and replace the middle third with two sides of an equilateral triangle, a small outward kink. You now have four segments, each of length one-third. You then repeat the process: each of those four segments receives its own kink in the middle. Repeating this indefinitely, at every scale, produces a curve that is ever more wiggly at ever finer scales. If you carry this construction around a full triangle rather than a single segment, the resulting shape resembles a snowflake.
The Koch snowflake curve has infinite length. You can see why: each iteration of the process multiplies the total length by a factor of four-thirds, because three segments of length one-third are replaced by four segments of the same length. Since this is done infinitely many times, the length grows without bound. For the length to be finite, it would have to equal four-thirds times itself, which is impossible for any finite value. The curve is also self-similar, the whole figure being a scaled-up version of each of its constituent parts.
Now consider building a three-dimensional surface from this curve by simply extruding it in the perpendicular direction, producing a corrugated shape whose cross-section is exactly the Koch snowflake curve. Enclose this in a rectangular box, so that the lid of the box is the snowflake curve stretched into a surface. Because the snowflake curve has infinite length, this lid has infinite area. Its surface is so intricately wiggly, with so many nooks and crannies at arbitrarily fine scales, that no finite quantity bounds its area.
And yet the entire box is bounded. If you submerge it in a vat of paint, a finite volume of paint covers every part of the surface to within one millimeter, thereby satisfying the paint-based criterion, even though the surface area is genuinely infinite. The paint-based test therefore fails to detect infinite area in this case.
The key difference between this example and Gabriel's Horn is the direction of the geometry. Here, the surface folds back on itself so densely that a single small blob of paint simultaneously covers many different parts of the surface at once, with enormous overlap. In the Gabriel's Horn case, the situation is reversed: to cover even a tiny patch of surface area, a large volume of paint is required, because the paint must wrap all the way around an extremely thin tube. One small surface area demands a great deal of paint. These two examples are, in a sense, mirror-image failures of the paint-based account, and together they illustrate why that account does not constitute a reliable criterion for finite surface area. This is what I call the painter's paradox.
There are some beautiful curves that can be drawn in the plane using polar coordinates, where a point is specified not by its x and y coordinates but by its radial and angular coordinates. Consider the curve r = e−θ, where θ is the angle and r is the radius. As θ increases, the radius becomes very small very quickly, so the curve spirals rapidly inward toward the origin. This is called the logarithmic spiral, and one can prove that even though it winds around the origin infinitely many times, it still has finite length.
Another classical example is r = θ, known as the Archimedean spiral. A distinctive feature of this spiral is that the spacing between successive turns is perfectly regular. If we traverse it inward from some starting angle, we wind around only finitely many times and the total length is finite.
A third example is the hyperbolic spiral, given by r = 1/θ. This curve also winds around the origin infinitely many times, but unlike the logarithmic spiral it has infinite length. These three examples together illustrate the range of possible behaviors: a spiral can wind around infinitely many times with finite length, wind around infinitely many times with infinite length, or wind around only finitely many times with finite length.
Having surveyed some of this one-dimensional behavior in the plane, we now turn to higher dimensions, where the natural question to ask is: what is the volume of a sphere in higher dimensions?
Consider the unit circle: its radius is one, and its area is πr², which gives simply π. That is the two-dimensional case. In three dimensions, the unit sphere is a globe-shaped object, and its volume is ⁴⁄₃π, that is, ⁴⁄₃πr³ with r = 1. So we have gone from π in dimension two to ⁴⁄₃π in dimension three, which is larger by a third.
We can also go down to dimension one. A circle is the set of all points at distance one from a given center, and we can apply the same idea in one dimension: the points at distance one from a center on a line form a line segment of length two. The relevant notion of size in one dimension is length, just as it is area in two dimensions and volume in three. We can call all of these hyper-volume: length is one-dimensional hyper-volume, area is two-dimensional hyper-volume, volume is three-dimensional, and so on.
The natural question is whether this hyper-volume keeps increasing as the dimension grows. It turns out there is a recurrence relation one can derive for Vₙ, the hyper-volume of the unit n-dimensional hypersphere:
Vₙ = (2π / n) · Vₙ₋₂
In other words, if you know the hyper-volume of the (n−2)-dimensional unit hypersphere, you multiply it by 2π/n to obtain the hyper-volume in dimension n. Applying this formula repeatedly, we can build a table: V₁ = 2, V₂ = π ≈ 3.14, V₃ = ⁴⁄₃π ≈ 4.19, V₄ = π²/2 ≈ 4.93, V₅ = 8π²/15 ≈ 5.26, and V₆ = π³/6 ≈ 5.17.
The hyper-volume rises through dimensions one to five, reaches its maximum at dimension five, and then begins to decrease. We can see why directly from the recurrence: once 2π/n is less than one, that is, once n exceeds 2π ≈ 6.28, each successive hyper-volume is smaller than the one before it. Even comparing dimension six with dimension four, the value has already dropped below the peak at dimension five, and for all dimensions greater than seven the hyper-volume continues to fall. The conclusion is that the hyper-volume of the unit hypersphere is maximized in dimension five, which is a rather surprising result.
This topic connects directly to the paradox of giants, because Galileo's argument was fundamentally about understanding how scaling works across different dimensions. That is exactly what we are doing here. He was mainly concerned with dimensions up to three, but there is no reason to stop there. We want to understand hyperspheres and how they sit inside the cubes that naturally bound them.
Consider the unit circle sitting inside a square, or the unit sphere, think of the Earth, sitting inside its bounding cube. The same relationship extends to higher dimensions, even if it becomes harder to draw. In the one-dimensional case, the unit sphere and the unit cube coincide. As the dimension increases, the sphere begins to fill a smaller and smaller fraction of the cube.
The central question is: what proportion of the cube's volume does the sphere occupy? In two dimensions, the unit circle has area πr² = π, while the bounding square is two by two, giving area four. So the circle fills π/4 of the square, a little more than three quarters. In three dimensions, the sphere has volume 4π/3, while the bounding cube is two by two by two, giving volume eight. The fraction is therefore π/6, which is already noticeably smaller.
This decrease makes geometric sense. In the square there are only four small corner regions outside the circle, but in the cube there are eight corners not covered by the sphere, so more of the total volume is pushed into those corners. The general pattern follows from the recurrence relation: each time the dimension increases by one, the hypersphere's volume is multiplied by 2π/n, while the bounding hypercube's volume is multiplied by two. Since 2π/n grows smaller as n grows large, this ratio, hypersphere volume to hypercube volume, tends to zero.
The picture that emerges is striking. In high dimensions, 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 their proportion relative to all points in the hypercube vanishes. Almost all points, when the dimension is large, are not near the center; they are pushed into the corners. This is the phenomenon people describe by saying that a high-dimensional hypercube is very corner-y: almost all the hypervolume comes from the corners, and very little from the central region.
This represents a fundamentally different geometric character from the dimensions we are familiar with. Our ordinary spatial intuition is built on dimensions one, two, and three, perhaps extended to four if we think of time. But in dimensions five, six, and beyond, direct visualization fails us, even though we can still calculate and observe. The nature of existence inside a hypercube changes: the center becomes, in a precise measure-theoretic sense, negligible. If you are running a numerical simulation that involves sampling points at random from a high-dimensional hypercube, almost all of those points will be lodged in some remote corner, and very few will be near the origin. Points near the center are not typical; they are, in fact, exceedingly rare as a proportion of all points in high dimension.
Let me show you some more examples of this phenomenon. Take four unit spheres and stack them as shown, placing them inside a square. These are unit spheres, so each has diameter two, making this a four-by-four square. Now I want to place a small blue ball in the middle of them and ask: how big is that ball? We will then repeat the same construction in higher dimensions.
We can calculate the size of the blue circle directly. Placing the origin at the center, the four surrounding circles each have radius one, and their centers sit at coordinates (1,1), (1,−1), (−1,−1), and (−1,1). The distance from the origin to any one of those centers is therefore √(1² + 1²) = √2. If the blue circle has radius r, then r + 1 = √2, giving r = √2 − 1 ≈ 0.414.
Now consider three dimensions. Take eight unit spheres, like billiard balls, arranged in a perfectly orthogonal stack inside a four-by-four-by-four cube, and fit a blue sphere into the central gap. The same analysis applies: the centers of the surrounding spheres now sit at coordinates such as (1,1,1), and the distance from the origin to any such center is √(1² + 1² + 1²) = √3. Therefore r₃ + 1 = √3, giving r₃ = √3 − 1.
The identical reasoning extends to arbitrary dimensions, yielding the general formula: in dimension n, the blue hypersphere that fits exactly in the central gap among the surrounding unit hyperspheres has radius r = √n − 1. This is a remarkable result, and it is worth pausing to appreciate its consequences.
Because √n − 1 grows without bound, the blue sphere becomes arbitrarily large as the dimension increases. In dimension four, √4 − 1 = 1, so the blue hypersphere is exactly the same size as the surrounding ones — you can fit another unit sphere in the middle, even though in two and three dimensions it was considerably smaller. In dimension nine, √9 − 1 = 2, meaning the blue sphere is twice as large as the surrounding spheres and, crucially, its radius of two reaches exactly from the center to the wall of the bounding hypercube. For the first time, the blue sphere touches the walls of the box.
In dimensions greater than nine, √n − 1 > 2, and the blue hypersphere actually protrudes outside the hypercube that was supposed to contain it. This is completely unlike anything we see in two or three dimensions, and it forces us to stretch our geometric intuition considerably. Nine is not even a large number, but imagine dimension one million, where the blue sphere is so enormous that it dwarfs everything else in the construction. That is precisely what the mathematics tells us, and it is one of the genuinely surprising features of high-dimensional geometry.
I hope you have enjoyed this account of the paradox of giants, which led us from Galileo's analysis of how volume and structural strength scale with dimension, through to these further paradoxes of high-dimensional space. I look forward to seeing you next time.