# The Paradox of Giants: Strange Consequences in High Dimension

Speaker: Joel David Hamkins
Course: Lectures on Infinity
Watch: https://ergo.org/videos/joel-david-hamkins-the-paradox-of-giants-strange-consequences-in-high-dimension
Transcript: https://ergo.org/videos/joel-david-hamkins-the-paradox-of-giants-strange-consequences-in-high-dimension/transcript/
YouTube: https://www.youtube.com/watch?v=yiRx6dvC9jk

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Why could giants never actually roam the Earth? Joel David Hamkins begins with a question from Galileo: how do volume, surface area, and strength change as creatures scale up or down? The square-cube law reveals that giants would collapse under their own weight, while tiny humans would be bizarrely strong for their size. This insight about dimensional scaling opens the door to a series of mathematical surprises. Gabriel's Horn holds finite volume but has infinite surface area, raising the question of whether you can paint a surface that stretches on forever. The Koch Snowflake challenges our intuitions about area and perimeter. Hypersphere volumes peak unexpectedly at dimension five and then shrink toward zero. Hypercubes become "all corners" in high dimensions, and a blue sphere somehow escapes its bounding box. Each paradox deepens our understanding of how geometry behaves in ways that defy everyday experience, revealing the strange and beautiful consequences of infinity and dimension.

I'm Joel David Hamkins. Welcome to these lectures on infinity, and today, I want to tell you about the paradox of giants. According to legend, giants once roamed the Earth. Everyone knows Odysseus met the Cyclops who lived in the big cave, and grabbed sheep and men with his hands and ate them whole. And also, in the times of King Arthur, there was the young boy who got 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.

## Giants of Legend and Literature

It's the same Jack, I think, as Jack and the Beanstalk, who planted the seeds that grew the beanstalk and climbed to the castle in the sky and tricked that giant as well. But then again, there's also Jonathan Swift's character, Gulliver, who travels to distant lands and finds the Lilliputians, these tiny human beings to whom Gulliver seemed the giant. But he also met, in those same travels, the Brobdingnagians, who were giants to whom Gulliver seemed Lilliputian, even though Gulliver had never changed his size at all.

And in all of these legends, the giants often have kind of ordinary human form and they do human things. They walk around. They stomp. They maybe dance. They drink wine from goblets. They carry heavy stones and so on. They do human things. They climb ladders. They move in a human manner, but at scale.

## Why Physics Makes Giants Impossible

And Galileo wrote in his *Dialogues Concerning Two New Sciences* a wonderful criticism of this whole manner of thinking about giants, and Galileo argued that giants are actually impossible. Physics cannot work like that. The whole idea of a giant is actually contradictory. And so I want to tell you about Galileo's argument.

He asks us to imagine having a great oak beam. So here, I've drawn it here. Maybe it's holding up a heavy stone or a load of bricks. A sturdy oak beam. But now, let's imagine making it 10 times bigger. So we make a bigger one. I'm not gonna draw it actually 10 times bigger, but let's imagine. A much bigger one. Exactly the same dimensions and proportions, but 10 times bigger, 10 times longer, 10 times thicker, 10 times wider, and so on, but made out of the same material. So maybe this beam is a beam in our house, and this would be the kind of beam in the giant's house. This is the idea.

And Galileo asks us to imagine, well, of course, that bigger beam is gonna be stronger and sturdier and able to hold a bigger load. So let's think about it, though, in detail. How much stronger will it be? Well, Galileo argued that the strength of a beam is related to the cross-sectional area because the fibers of the wood are going lengthwise in the beam, and when the beam breaks, it breaks along the cross-section. And so it's the strength of the fibers going through those cross-sections that really matter for how strong the beam is overall. And so if the beam is 10 times bigger in every direction, then the cross-sectional area would be multiplied by 100, because it's 10 times 10. So therefore, the larger beam, it's 10 times bigger, but it's 100 times stronger, which seems really good, so it's gonna hold a lot more weight, 100 times as much. So that seems pretty good when you think about it.

But then let's think about the stone that the beam was holding up, or the bricks, and how much do they weigh? Well, the stone, the material of the stone has a certain density, and if we make it 10 times bigger, 10 times wider, longer, taller, then the volume of the stone is actually 1,000 times bigger because it's 10 times 10 times 10 in each of the three dimensions. And so a 10 times bigger stone weighs 1,000 times as much if it's made out of the same material.

So now it doesn't look so good anymore because the 10 times bigger beam was 100 times stronger, but the load that we're gonna put on it is 1,000 times heavier, and so it's not gonna be strong enough. If currently the beam was just barely holding up the load, then when you make it bigger, it won't be strong enough. And so Galileo actually argued that not only will it not hold up the new stone, but it won't even hold up itself because the mass of the beam itself will be so great that its strength won't be enough to hold up that weight.

## The Square-Cube Law Destroys Giants

So he argued, Galileo argued that because of this difference in dimension, the strength increases with the square of the scaling, but the mass increases with the cube of the scaling, which is significantly greater. And this mismatch between those 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 roof wouldn't be able to hold up itself. The goblet that's 10 times bigger wouldn't be able to hold the wine that it contains.

The giant wouldn't be able to climb the ladder and step on the steps of the ladder, because the ladder wouldn't even be able to hold up the ladder alone without even a giant on it, or even the giant itself, his bones, they're basically like beams. And if we scale up a giant, if we take a human being and scale up to make the giant 10 times bigger, 10 times taller, 10 times thicker, and so on, in every direction, then the bones become 100 times stronger, but the weight of the giant, his mass has increased by 1,000, and therefore, the giant will not be able to stand up, will not be able to walk around. The whole concept of a giant is incoherent.

So 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." Now,

## Why Elephants Are Stocky and Bugs Are Thin

We can see this as obvious, if you think about the nature of large animals and small animals. If you think of the typical large animals, elephant, rhinoceros, and so on, then they're typically very stocky. They have very stocky limbs. And the reason is precisely because of this dimension issue that Galileo pointed out. In order to hold up the larger weight, the limbs need to be not only bigger, but proportionally bigger, and that makes a stocky animal.

And it works in the other direction too. The tiny animals that you see are typically very slender and have very thin limbs, or insects and so on, have these very thin limbs, and they can hold up their weight just fine. If you take a housefly and you make it 10 times bigger, then there's no way it could be crawling around on the wall and so on, because the electrostatic forces are simply not strong enough anymore. You made it 10 times bigger, so it's 1,000 times heavier, and now the electrostatic forces aren't strong enough for it to stick to the wall and so on.

A bug that crawls on the surface of the water, the surface tension effects simply aren't, don't scale in the right way. And so the paradox of these giants is that you can't just take an animal, a functioning being, with a certain animal architecture, and scale it up and expect it to still work just the same. It just won't work that way.

## The Paradox of Miniature Humans

There's a similar thing with the, not just giants, but the paradox of the miniature human. Maybe you've seen some of these Hollywood films, *Downsizing* or *Ant-Man* or *Honey, I Shrunk the Kids*. It's a common theme in Hollywood films, the idea of having a miniature human, or the Lilliputians of *Gulliver's Travels*. And of course, if you take an ordinary human and make them 10 times smaller, then they are going to be proportionally stronger for their height for exactly the same kind of reason.

And this is exactly why grasshoppers can jump many times their own height and a tiny human would be able to jump very high in comparison with its height, not in absolute terms, but in comparison with their new smaller height. And so they wouldn't walk around like ordinary humans. Interacting with water would become very complicated, because at that scale, it would be much stickier than it is at our scale. So the nature of physical existence doesn't scale in that way, and this is the core of Galileo's argument.

## Evolution and Body Size Genes

I think it's related to evolution and body size. It turns out, 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 some animals, their size varies quite a lot. In prehistoric times, the very old days, dragonfly, there were these huge, enormous dragonflies, and they're much smaller now, of course, and also horses. Initially, when they first evolved, were much, much smaller than they are now, and they got big and small and big and small again. And we can see some of that in the miniature horse breeds, if you're familiar with these kinds of horses that are very tiny. So there's some residual smallness genes still in the horse population. So we can see that.

But one can imagine that natural selection acting on those genes would cause changes in the body size architecture for the animals. Maybe there would be some reason why it would be to your advantage to become larger, a little larger, even though that would make you heavier and less proportionately strong. It might enable you to compete in your niche, as an animal in that niche, in a way that was favorable, and so we can imagine easily evolution acting on those genes to change the body size depending on the environment.

So there's very similar other effects about differences in dimension, the difference between say surface area and volume, and I want to talk about some interesting mathematical examples now, and one of those

## Gabriel's Horn Has Finite Volume

Examples is the paradox of Gabriel's horn. Gabriel's horn, we look at the function of 1 over x. This is y equal 1 over x. And we start at the point one. We're just going to look at this part of the curve here. So it goes all the way out to infinity. And to form Gabriel's horn, we take that line and we revolve it around the x-axis. So we make a symmetric shape like this. And it's like a horn from heaven. Maybe we can imagine it with a sonorous tone, maybe this multi-tone coming from heaven, and this is why it's called Gabriel's horn.

Gabriel's horn is obtained by revolving this function around the x-axis and making this kind of surface. So I could put my arm in it like a horn. Now, the paradox of Gabriel's horn is the following observation. I want to ask, what is the volume of Gabriel's horn? It's an infinite object because it keeps going forever, but it's 1 over x, and so it's very thin out there when you go very far. And so we can compute the volume of this.

And in order to compute the volume of this, it's a standard problem in calculus, to compute the volume of a surface of revolution. And the way that you do it is, in the typical calculus manner, you imagine slicing it into cross-sections. So we're going to slice this horn into these disks. I'm thinking of the solid, the volume of the solid, the part inside the horn as well. That's what I'm talking about. And if I slice it like this, then, if I'm at point x here, the radius of that disk is 1 over x, because that's the function that we're rotating. And maybe the thickness, if you know how to do this, is dx, the infinitesimal dx.

So the volume of one disk is the area of that disk times the thickness, but the area is pi r squared, so therefore, the volume of one disk is pi times 1 over x squared dx. So that's the area times the thickness. So the total volume will be the integral from one to infinity of the volume of each disk. So we took the whole thing and we slice it into these disks. We calculate the volume of each disk. That's the integrand, and now we're going to add them up. That's exactly what integration does. So this is equal to the integral from one to infinity of pi over x squared dx.

But I can compute this integral. This is an elementary calculus integral. The integral of 1 over x squared is minus 1 over x, so we have here minus pi over x from one to infinity, and when I put in infinity in for x, the limit will be zero, so it's zero minus, using the fundamental theorem, the value when I put one in, but there's a minus sign here, so it's minus minus pi over 1, and so I get pi. So the volume of Gabriel's horn is precisely pi, which is a finite number. That's the first paradoxical part about Gabriel's horn. It's an infinite object, but it has a finite volume, pi. So the volume of Gabriel's horn is finite.

## Gabriel's Horn Has Infinite Surface Area

That's nice, but the second part of the paradox is to ask, what is the surface area of Gabriel's horn? So now instead of asking the volume, I want to ask the surface area. For the surface area, we don't want the volume of this disk, but rather, we want to concentrate on this band on the outside, which if you think about it, is what's called the frustum of a cone. It's slightly angled. And the angular piece here is commonly called ds, which is the square root of dx squared plus dy squared, and I can think of that as, if I factor out a dx, 1 plus dy/dx squared dx here.

So the infinitesimal length of the frustum part is this square root times dx, so the total surface area will be the integral from one to infinity of the surface of one frustum, but that's the width of that frustum times the circumference, and the circumference is pi times the diameter, so that would be 2pi over x. So we have 2pi over x times the square root being 1 plus dy/dx squared dx. And dy/dx squared, well, y was 1 over x, which is like x to the minus 1, and so dy/dx is minus 1 over x squared, and so this turns into 1 over x to the fourth.

So this is more complicated, but one can calculate. In fact, we can just ignore this square root. It's always at least 1, so this is bigger than or equal to the integral from one to infinity of 2pi over x dx. If I just observe, this is at least 1, so this whole thing is at least as big as this, and this is 2pi log x from one to infinity, which is infinity.

## Can You Paint an Infinite Surface?

The point I'm trying to make here is that the surface area of Gabriel's Horn is infinite, but the volume of Gabriel's Horn is finite. The surface area is infinite and the volume is finite, but how can that be? Can't we just fill it with paint? Suppose I point it down and I just fill it with paint. It has a finite volume. So with finitely many buckets of paint, I can fill it up, and that paint would be touching every part of the surface. And so, with a finite amount of paint, I can paint Gabriel's Horn.

So that's the puzzle of Gabriel's Horn. The paradox of Gabriel's Horn is that it's a geometrical object that we can understand in a deep way, and yet it has finite volume and infinite surface area.

And so, what about this filling with paint idea? Does it really work? Does it convince you if you have an object, a container, and you fill it with paint, then would it ever take more paint than that to paint that surface? And I would say, well, actually we're cheating a little bit with that argument, because Gabriel's Horn is getting thinner and thinner as you go out here. And so the paint that's inside Gabriel's Horn was spread very thinly when you go very far out.

If you say, "Well, to paint a surface, there should be a uniform one millimeter thickness of paint on it," then eventually we weren't obeying that one millimeter thickness, because the horn itself was less than one millimeter across. And so even though the horn was full of paint, it doesn't mean that we've painted to a uniform thickness. And so that's a way of seeing, well, look, just filling Gabriel's Horn with paint shouldn't count as painting the surface, because you've spread the paint so thin in the part that's way out there.

But the part that's way out there is adding to the infinite surface area. Most of the area is out on the tail, because if I chop it off, then what remains here is obviously just a finite area. So it's totally cheating to try to paint the surface by filling the volume with paint, because the paint will, in effect, be spread so thinly. So let's try to understand what's going on with this.

## Extended Real Numbers and Infinity

Actually, there's one thing I wanna mention. I've used this symbol infinity here. And of course, this whole lecture series is about the infinite, and so I wanna talk about this particular kind of use of infinity, which is often the first instance of infinity that many students find in a math class, in a calculus class, and so on. One will be writing infinity, this infinity symbol on the board in exactly the situation that I just did. And so what does this mean? What is that number? Is it a number? How should we think about that infinity?

And so what one can say is that we have the real number system. This is the set of all real numbers. It's an ordered field. We can add them and multiply them and so on, and compare their order. And then these are the points that correspond to the points on the number line. Then we have what's called the extended reals. The extended real numbers. And this is a number system that's obtained by starting with the real numbers and just adding infinity and minus infinity as idealized objects.

So we just add these two extra things to the set, and we defined a little bit about how to do arithmetic with these objects. So for example, when you're working in the extended real numbers, then infinity plus two is equal to infinity, or if you add anything to infinity, it's still just infinite. And infinity plus infinity is infinity. And similarly, minus infinity plus something is still minus infinity. In the extended real number system, this is how we define how these symbols work in this number system.

Also, infinity times A is equal to infinity if A is positive, but if A is negative, then it turns into minus infinity. So infinity times minus five is minus infinity and so on. You add an infinity symbol and a minus infinity symbol to the ordinary real numbers, and that system together makes the extended real numbers, and we define how to work with those symbols. If you add a finite number to infinity, it's still infinity, or even if you add two infinities together, you get infinity. But if you add it to minus infinity, it's still minus infinity, and if you multiply infinity by a positive number, it's positive infinity. But if you multiply it by a negative number, it becomes minus infinity, just like you would expect.

Except one has to keep in mind there's certain combinations that simply aren't defined in the extended real numbers. So infinity minus infinity doesn't have a meaning. It's not defined. And also infinity times zero is not defined. So in the extended real number systems, you can work basically very intuitively with these symbols for infinity according to these rules, but certain combinations don't have a meaning, and it's quite amazing how far this way of treating infinity goes.

I think of it as, philosophically, it's very ontologically light, because it's deflationary in a way, because it's saying, "Look, we don't have to give a robust, heavy meaning to infinity, we can just add it as a symbol, and then we can calculate with it according to these rules, and things work great." And that goes a very long way. It's remarkable how such a light attitude towards such an apparently heavy concept could be quite productive, but it is. And in fact, for many mathematicians, this is the use of infinity that they encounter, and we used it already when understanding the nature of Gabriel's horn.

## Testing the Paint-Based Theory of Area

We discussed the idea, look, maybe it's a paint-based theory of surface area. I want to say, look, if I have a geometric object, a container, then what does it mean to say that it has finite area? And maybe what you might propose is, look, a surface has finite area if I can cover it with, if and only if I can cover it with paint to a certain thickness, using a finite volume of paint. Maybe that sounds like a good criteria for when are we going to say that a surface has finite area?

The paint-based proposal is that a surface has finite area just in case, with a finite volume of paint, I can coat it to a uniform finite thickness. So that every point on the surface was one millimeter, or whatever the scale, whatever the thickness of paint you want is. But let me criticize this proposal. It doesn't quite work. Namely, suppose I have a line, an infinite line. Like the x-axis.

That has zero area, but I couldn't cover it to a uniform thickness with a finite volume of paint, because any volume of paint to uniform thickness would basically be an infinite cylinder surrounding that line. But that would have an infinite volume because the line was unending. So that would be a counterexample to the paint-based account of what does it mean to say that a surface area has finite area, because this would be a surface that does have finite area. A line has zero area, and yet you can't paint it to uniform thickness.

But actually I want to give another counterexample also. That was an example of something that has finite area, and yet can't be painted with a finite volume of paint. Let me give another one. Maybe you say, "Well, look, a line isn't a surface at all. It's a one-dimensional thing. It's not a surface." So let's do another, a different version of Gabriel's horn. This is a modified Gabriel's horn.

## A Modified Horn with Finite Area

where I'm using a different function. So now, I'm going to use the function one over x squared instead of one over x, and the difference between one over x squared and one over x is, they look roughly the same. Oops. But one over x squared, when x gets big, one over x squared is much tinier than one over x, because, for example, when x is 100, then one over x squared is one ten-thousandth, which is 100 times smaller than one one-hundredth, yeah? Or when x is a million, then this is a million times smaller than one over x because it's one over a million times a million, and so on.

So, one over x squared goes to zero faster than one over x, but I can still make a Gabriel's horn kind of thing. It's a slightly different shape. It tapers to the line much more quickly. It's never actually on the line. It's very thin out here, but I can use it to make a Gabriel's horn-type surface in exactly the same way that I did with Gabriel's horn, and now you'll see that the surface area and the volume are both finite for this version. In Gabriel's horn, the paradox was that they were different because Gabriel's horn had finite volume but infinite surface area, but this one has finite volume and finite surface area.

But now if I think about the paint-based criteria for finite surface area, remember we said the proposal was a surface has finite area just in case you can paint it with a finite volume of paint to uniform thickness. But if I put uniform thickness of paint on this version of Gabriel's horn, this tighter version, the tapered version, then it's still going to take an infinite volume because when the horn is so close to the x-axis and basically almost like a line and there's still going to be basically a cylinder, a tiny one millimeter radius cylinder of paint, that it's going to take infinitely much paint to cover this tail part to finite thickness. And so therefore, this modified Gabriel's horn is a surface that has finite surface area and yet you cannot paint it to uniform thickness with a finite volume of paint.

So therefore, we have counterexamples maybe on both sides. No, I'm sorry. We still, both of those are still on the one side, a finite area which can't be painted. So, what I want to do now is give you a counterexample on the other side. Namely, I want to give you a surface.

## Koch Snowflake Breaks the Paint Rule

That has infinite surface area, but you can paint it with a finite volume of paint. So, let's do that. I'm saying the painter, the paint-based criteria is wrong in both directions. It's neither necessary nor sufficient. And for this, I want to talk about something that we're going to talk again, more fully about in another lecture on the coastline, the infinite coastline paradox, which leads to the concept of fractals.

So, let's just give an example of a fractal here. The Koch snowflake curve. And the way it works is, you start with a line segment of a certain length and you chop it in thirds, and you replace the middle third with two versions. Is that quite a third? Like this. Like this. So, we had one segment, and now we have four segments, each of length one-third. And now I do it again. This segment, I'm going to put a little kink in it. And this segment, I put a little kink in it. And this segment, I put a little kink in it. And this segment, I put a little kink in it. That's the second iteration.

Now I do it again. For each one of these line segments, I put a little kink in it, in the middle. And it's going to take some while to draw all this. But I hope you can see the kind of picture that's emerging here. And this is a fractal known as the Koch snowflake curve. Then we do it again. So, we get these other little kinks on all these. I'm obviously not going to be able to draw it, but we just keep doing this process of adding these little triangles. And you get this curve which is ever more wiggly and ever finer a scale.

And the thing about the snowflake curve, it looks more like a snowflake if you continue it around so it's a full complete shape. I've only drawn one piece of it, but I can imagine having started with a triangle, and I get this more snowflake-like picture. The thing about the snowflake curve is that it has infinite length. And you can see that it has infinite length, because every time we did the process one more time, the length got four-thirds times as big. Because I took three one-third segments and I produced four of them. So, every time I did it, it got longer by a factor of four thirds.

But I did it infinitely many times. These curves converge in a way that it makes sense to do it infinitely many times. So, it can't have a finite value, a finite length, because the length would have to be equal to four thirds times itself in order for the length to obey that generation rule. So, the length of the curve is infinite. So, that's the main thing that I need from this example. This is an instance of a kind of fractal with self-similar, it's a self-similar fractal, because the whole thing is just a scaled-up version of each leg of it. So, it's done similar to itself at a different scale.

Now, what I want to do with this thing is make a surface out of it by just simply extending it, to make this kind of corrugated, corrugated, roof kind of pattern. So, I'm going to take that so that the cross-section of this surface is exactly the snowflake curve. I just stretch it in the other dimension. And then I make a box out of it. That's my solid. It's a box, a rectangular box, and the lid is like the snowflake curve stretched in the other dimension. So, it's very wiggly this way, but here, it's all straight lines. I don't know if you can see exactly what I mean.

And now, because the snowflake curve itself had infinite length, it follows that the top, the lid of this box has infinite area. There's so many nooks and crannies in this, its very wiggly surface, so it's not really like a corrugated aluminum roof, because it's much more wiggly than that at very fine scale in such a way that actually, the cross-sectional area, the cross-sectional length was infinite. And so therefore, the area of that roof is infinite, bigger than any finite quantity. And yet, the whole thing is bounded. So, I could just dunk the whole thing in a giant vat of paint. A finite amount of paint would cover every surface to within one millimeter, so I would be meeting the paint-based criteria of finite surface area, and yet, the surface is not finite. This box with the roof has infinite area, and yet we can paint it.

And the difference between this example and the Gabriel's horn example is that, in this example, one little bit of paint can be simultaneously covering many different parts of the surface because the surface is so wiggly that if I take a one millimeter extension of this surface in all direction, I'm going to be getting a lot of overlap, so one little bit of paint counts as covering a lot of different parts of the surface. Whereas in the Gabriel's horn case, it was the opposite. The curvature was the other direction because, to cover one little surface area, I needed a lot of paint, that was wrapping all the way around that basically very thin tube. And so, it took a lot of paint to cover one tiny bit of surface area in the Gabriel's horn, the modified Gabriel's horn, but in this case, one bit of paint can cover quite a lot of the surface area.

And maybe that's an intuitive reason to understand how it could be that the paint-based account of finite surface area, it doesn't really work because of these counterexamples. So I call that the painter's paradox. So let's now do some other paradoxes of higher dimension. Actually, we can just start with curves in the plane.

## Beautiful Spirals in the Plane

There's some really beautiful curves that it's possible to draw. If you're familiar with polar coordinates, where you specify a point not by a point in the plane, not by specifying the x and y coordinates, but rather by specifying the radial and angular coordinates, the polar coordinates, then, if you think about the curve r equals e to the minus theta, where theta is the angle and this is the radius. So, I tell you theta and r, then this specifies the radius as a function of the angle. And, when you do that, you get a certain curve and it looks something like this. It spirals in because as theta increases, this number becomes very tiny, and so it spirals very rapidly into the origin. This is called a logarithmic spiral. And you can prove that even though the curve goes around the origin infinitely many times, it still has finite length. So it's a curve with finite length.

Then there's another spiral, r equal theta. This is called the Archimedean spiral. And it looks something like this. And it keeps going. So, it doesn't wrap around infinitely many times at the origin. The angle starts at zero and then it gradually increases as you go, and the spacing here is quite regular. That's the characteristic of the Archimedean. The spacing here is not at all regular. It's getting much more tightly towards the origin here, but here the spacing between every time you go around, it's the same. And so this spins around at, if we go in the other direction and spin in, we only go around finitely many times, and there's finite length. Here, we go around infinitely many times, but there's finite length.

Then another kind of example would be called the hyperbolic spiral, and if I use say, r equal one over theta, and this one maybe I draw it something like this. It spirals around infinitely many times, but it's very slow, still getting to the origin, and it goes around infinitely many times, and also has infinite length. So it's different from this one, which goes around infinitely many times but has finite length. This one goes around infinitely many times and has infinite length. This one goes around finitely many times, finite length, if we go the other way with it like this. So that should have exhibiting some of the possible behavior in the plane for these one-dimensional things.

## Hypersphere Volume Peaks at Dimension 5

Let's now go to higher dimensions, and I want to ask, what is the volume of the sphere in higher dimension? Let me try to explain what I'm talking about. If I take the unit circle, the radius is one here. I know what the area of a circle is. It's pi r-squared, and if r is one, then this has area pi. That's dimension two. And now in dimension three, the unit sphere is a kind of globe-shaped thing like the Earth, and if the radius is one, then you can show that the volume of the sphere is four-thirds pi. It's four-thirds pi r-cubed but r is one, so four-thirds pi. So here, we had only pi, and here, we have four-thirds pi, which is bigger. It's one-third bigger.

And then what about higher dimension? We want to talk about the concept of the hypersphere. The four-dimensional sphere, and five-dimensional, six-dimensional. Can we go down? What about the one-dimensional sphere? What does it look like? What is a one-dimensional sphere? A circle is the set of points that are equidistant from a given point, distance one from a given point. So I can do that in one dimension too. I have the center, and the points that are distance one, it's just a line segment of length two. So this is a one-dimensional version.

And the notion of area or volume, hyper volume, that's relevant in one dimension is length. So how long is the unit sphere in dimension one? It has length two. That's the relevant notion of size in dimension one, is length, area, volume, hyper volume in the higher dimensions. That's what we call it. We could call all of them hyper volume. This is the one-dimensional hyper volume. That just means length. The two-dimensional hyper volume means area. Three-dimensional hyper volume means volume, and so on. Hyper volume means any dimension.

It's going up. Does it go up forever? What is the hyper volume of the n dimensional sphere? That's the question I want to ask. So, let's think about it. It turns out that you can derive a formula. I'm not going to derive it, I'm just going to talk about it. If V sub n is the hyper volume in dimension n, the hyper volume of the n dimensional sphere, then there's a kind of recurrence relation that you can prove, and it goes like this. V sub n is equal to two pi over n times V sub n minus two. In other words, if you know the hyper volume of the n minus two dimensional hypersphere, then this formula tells you how to get the volume of the next one.

But now we can look at this factor and see, we can apply this formula to what we already had and make a little kind of table. In one dimension, we had two. In two dimensions, we had pi, because the area of a unit circle is pi. In three dimensions, we had four thirds pi. V4 is going to be just two pi over 4 times the one, two back, so it's going to be two pi over 4 times pi, which is pi squared over 2. And V5 is, what is it? Oh, God. Eight pi squared over 15. Eight pi squared over 15, and V6 and so on, we can calculate. What is that one going to be? It turns out to be pi cubed over six if you just apply this formula. Pi cubed over six.

And it turns out that although initially it was getting bigger, and pi squared over two is like 4.9, 8 pi squared over 15 turns out to be 5.264, approximately. This one is 5.264. And so on. And V6 then turns out to be 5.1 something. 5.168. So it goes up until dimension five, and then it starts going down again. So the point is that the volume of the n dimensional hypersphere does not keep increasing forever. At some point, it starts going down, and the maximum is achieved in dimension five.

We can see that from this formula directly, because once two pi over n is less than one, in other words, if n is really big, this factor is less than one, it's small, and so we're multiplying the previous value by something less than one, so it's going to get smaller once n is bigger than two pi, but two pi is like six. So if n equals seven, it's already smaller, but in fact, four is already smaller than the six in comparison with four is already smaller than five, so the maximum is achieved at five. In all the dimensions bigger than seven, it's just going down even more. And so, therefore, the situation is that the n dimensional hyper volume of the unit sphere is maximum in dimension five because of this kind of analysis, which is a little bit surprising.

Really, I want to talk about this, and I view it as related to the paradox of giants, because Galileo's argument was all about understanding the nature of giants by understanding how scaling works in different dimensions, but that's exactly what we're doing here. We're understanding how scaling works in different dimensions. He was just concerned mainly with dimensions up to three, but I don't see any reason to be limited to three dimensions only. I want to understand these hyperspheres and how they sit inside the cubes that they're naturally part of.

## Why Hypercubes Are All Corners

So for example, we have the unit circle, and we can think about it sitting inside a square. And the unit sphere, like the Earth, is sitting inside the bounding cube of it. And similarly, in higher dimensions, it's maybe a little bit harder to draw. So we have this hypercube. I'm just drawing it as three-dimensional, but we should imagine it as four-dimensional or five-dimensional and so on. And also, in the one-dimensional case, so here's one, two, three, and four, and in the one-dimensional case, the unit sphere and the unit cube are the same in one dimension. In two dimensions, it's less.

So the question is, what proportion of the volume is inside the sphere? What proportion of the cube is in, what fraction of the cube does the sphere fill out? So in two dimensions, in other words, what fraction of the area is inside the circle as opposed to the square? This is a unit circle, so the radius is one. So the area is pi r squared, pi times one squared. That's pi. So the circle has area pi, but the square is a two by two square because the radius is one, so the diameter's two. Two by two. So pi fourths of the area is in the circle. A little more than three quarters.

Here, it's four thirds pi. The volume of the sphere was four thirds pi, but the cube now is two by two by two, so that's eight. And what is that? That's pi over six. So it's gone down. So there, it makes sense because look, in the square there's only four little extra bits. But in the cube there's eight little corners that are not covered, so it's accommodating more of the area.

So what happens in the higher dimensions? So we had that formula where you multiply by two pi over n. Two pi over n. In n dimensions, the denominator is going to be two to the n because the bounding cube is two by two by two by two, n times, so two to the n. And the volume up here was v sub n, so this is the ratio that we're talking about. So the cube gets doubled each time, but the volume gets multiplied by two pi over n each time when you increase. But this fraction is getting tinier and tinier as n becomes large. So in high dimension, much less of the volume is in the sphere and the cube volume gets doubled, but the hypersphere volume gets reduced.

So therefore, the picture that we get is that as the dimension increases, more and more of the points in the cube are not in the sphere. So in other words, the cube is very corner-y. This is the phrase people use. Most of the points in high dimensional hypercube are in the corners, and very few of them are near the center. The points near the center are the ones that are inside the sphere. But the proportion of points that are inside the sphere as a comparison of all points in the hypercube is going to zero. So almost all the points when the dimension is high, almost all the points are not near the center. Rather, they're stuck into the corners.

The hypercube in high dimension, even moderately high dimension, is very corner-y. Almost all the points are in the corners, almost all the hypervolume is coming from the corners and very little of it is coming from the center. So that's a fundamentally different nature to the geometry as the dimension increases beyond the dimensions that we're familiar with. Of course, our ordinary thinking is based on dimensions one, two and three primarily, or maybe you can imagine dimension four as time or something like that. But with these higher dimensions, five and six and so on, it becomes difficult to imagine them, but we can still calculate and we can observe the nature of existence inside the hypercube has the property that almost all the points are not near the center.

So for example, if you're doing some kind of numerical simulation which involves picking points at random from a hypercube, in a high dimensional cube, then almost all the points are stuck in some weird corner and very few of them are near the center. So you shouldn't think of points centered around the origin are likely or more likely. In fact, they're very rare as a proportion of all the points in high dimension.

I want to show you some more examples of this kind of nature. So let's do it like this. Take four unit spheres and stack them like this. We put them inside a square. These are unit squares, so the diameter is two, so it's a four by four square. And now what I want to do is I want to put a little tiny ball in the middle of them. And I ask, well, how big is that ball? And then I'm going to do the same thing in higher dimension.

## The Blue Sphere That Escapes Its Box

The process is we take unit circles stacked and put them inside a four by four square, and I look at the tiny little blue circle that fits in the middle of them. We can just calculate how big this circle is, because if I think of the origin as in the center here, that's the origin, these are circles of radius one. So I can think of this as going one over and one up, so this is one, one, the coordinates, and this is one minus one, and this is minus one minus one, and this is minus one one. And so I can think of myself as having this little triangle here. This is distance one, and this is distance one, so therefore this distance is square root of two, so if the blue radius is r, then r plus one is equal to the square root of two, and so therefore r is the square root of two minus one. So the square root of two is like 1.4 something, so this is like 0.4 something, 0.414.

Let's do it now in three dimensions. I'm going to take spheres like this and pack them, stack them on top of each other, and another one, and then behind, so there's going to be eight altogether. Oh my god, let's see. Can I really draw it? Let's see. I've got eight balls, like billiard balls, and they're touching and they're aligned in a totally orthogonal manner, and then I'm putting them inside this cube. So they're stacked inside this cube, and then I'm going to fit inside, in the center, this blue sphere. It's occluded by some of them, so you can't quite see it, so let me shade in here, it's shaded. There's a little blue sphere inside there.

And I want to know, how big is it? And what proportion of the area is it, of the whole cube and so on? Well, I can do the same kind of analysis here, because if I put the origin at the center of the cube, then again, this is a four by four by four cube, and I'm going one over, so the coordinates here are just the coordinates one, one, one, or one, one, minus one, or one, minus one, one, and so on. And so I can look at the line that connects the center of the blue sphere with the center of one of the larger spheres, and I've got this triangle, so this is r, this is from r2 to, here now we have r3, the three-dimensional one. Plus one, so if I draw the line from the center of the blue sphere to the center of one of the bigger spheres, then that radius is one, so the radius of the blue one gets me all the way, so r3 plus one is equal to the square root of one squared plus one squared plus one squared, so it's going to be the square root of three. So here we get that r is the square root of three minus one. So in three dimensions, the blue sphere has size square root of three minus one.

And the same exact kind of reasoning shows in the general case, that we get the square root of n minus one. The size of the blue sphere in dimension n, the one that fits exactly in between all the hyperspheres, is the square root of n minus one. And now let's just think about that a little bit. The square root of n minus one, because I can make n really big, so when n, say, when n is nine, the square root of nine is three, so the radius is two, which means that in dimension nine, the blue sphere is bigger, twice as big as the other spheres, because they have radius one, but it would have radius two. And in particular, it means that in dimension nine, the blue sphere touches the walls of the hypercube, because its radius is two, which gets me all the way from the center to the edge.

So in dimension nine for the first time, the blue sphere that's sitting in the middle of all those other hyperspheres is actually touching the walls of the, it's hard to imagine if you only think about two dimensions and three dimensions, in these higher dimensions, that blue sphere is much, much bigger than you think, because this number can be enormous, so we can see when n is four, then the square root of four is two, so this is two minus one, so in dimension four, the blue sphere that fits inside the other four dimensional hyperspheres has the same size as those hyperspheres. So in dimension four, when you put hyperspheres in that rectangular pattern, you can fit another one inside in the middle, even though in these smaller dimensions it's much smaller.

And then in dimension nine, we can see, the square root of nine is three minus one is two, in those cases, the blue sphere that's in between the nine dimensional hyperspheres is actually twice as big as the other spheres, and in fact touches the walls of the hypercube in which it's residing. And then as the dimensions get bigger than nine, then the square root of n minus one is bigger than two, and therefore the blue sphere, the blue hypersphere in dimensions bigger than nine actually sticks outside of the hypercube that's bounding the hypersphere. It's hard to imagine, because it's just totally different from the situation in dimensions two and three, and so we really have to stretch our minds to understand, well, what is the nature of this kind of geometry in these higher dimensions?

And dimension nine isn't even that big, it's only nine, that's not such a big number, but what about a dimension of a million or whatever? Then the blue sphere is so enormous, that it's difficult to grasp, but that's exactly what the mathematics shows us. So I hope you enjoyed that account of the paradox of giants which led us from Galileo and his analysis of the way dimensions scales, the way volume and strength scales in dimension into these other paradoxes of dimensions, so hope to see you next time.