- This video is sponsored
by Curiosity Stream. Hi, there, welcome to
Up and Atom, I'm Jade. There's a type of object
that has finite volume, but infinite surface area. This seems to lead to paradoxes like being able to fill it with paint, but never finish painting it. Today we're going to look at the math behind one of these objects and talk about what it means. The first of these objects was discovered in the 17th century, by a student of Galileo,
Evangelista Torricelli. It was given the name Gabriel's horn after the archangel Gabriel, who blew his horn to
announced Judgment Day, perhaps hinting at what it was
about to do to mathematics. Torricelli had discovered an object that was infinite in surface
area, yet finite in volume. That was infinitely long, but not infinitely big. This launched him to mathematical fame because it intuitively doesn't make sense. It also connected the finite,
which we can experience to the infinite, which we can't. This blurred the line between the mathematical
and physical worlds. How is it possible, and what does it mean for mathematics? The math behind Gabriel's
horn is pretty complicated. So in this video, we're going to look at a
mathematically simpler object with the same weird property. Well, it's actually a series of objects. Here are a series of cubes
that get smaller and smaller. The biggest measures one foot,
by one foot, by one foot. The second biggest measures, one over square root two feet, by one over square root two feet, by one over square root two feet. The third biggest measures one over square root three feet, by one over square root three feet, by one over square root three feet. This pattern continues
with the nth biggest cube measuring one over squared n feet, by one over square root n feet, by one over square root n feet. Now imagine that you have
an infinite number of cubes. This is the series of objects
we'll be working with. Cubes have six square faces. So the surface area is just six
times the area of each face. The area is just one side
multiplied by the other to give us the equation, six S squared. To calculate the total
surface area of all the cubes, we just add all of their
surface areas together. This gives us an infinite series, the sum of an infinite number of terms. We can get a better sense
of what this series is doing if we represent it graphically. Naturally the surface area
of each cube gets smaller as they get smaller. As we add them together, the total surface area
approaches infinity. It does this pretty
slowly as at 100,000 cubes we're only at 72.5 feet squared. But when we put it on a log linear graph where the X-axis goes up in a log scale, we can clearly see the trend is up. (soft chiming) This isn't too surprising. It's kind of what we'd expect
from infinitely many cubes. Now let's calculate their volume. The volume of the cube is just the height
times width times depth, which gives us S cubed. Again, to calculate the total volume, we just add all the
individual volumes together. Let's see what happens
when we sum them up. It's increasing at a much
slower rate this time. At 100,000 cubes, the volume
is still only 2.6 cubic feet. Even when we put it on a log linear graph, the curve flattens out. Rather than going off to infinity, the volume of an infinite number of cubes approaches a finite number. The fact that an infinite
sum adds to a finite number might sound weird, but it's not a new idea. It was revolutionary back in its day, but now our entire theory
of calculus is based on it, and math students don't
think twice about it. The name of these kinds of series are called convergent series
as they converge to a number as opposed to divergent series. But that's not what's
weird about these cubes. What's weird is that they
have an infinite surface area, but a finite volume. This just sounds wrong. It seems to imply that they
can be filled with paint, but never be painted. That doesn't jive with
our intuition at all. Everyday experiences show
that feeling of volume usually takes a lot more
paint than painting it. So it sounds weird to
have their surface area be so much bigger than their volume. Secondly, instead of looking at the outer surface area of the cubes, let's look at the inner surface area. Doesn't filling the cubes automatically coat their
inner surface area? Covering all the inside faces should take up the same amount of paint as covering the outside faces. So how can I fill a cube without also coating its surface area? Put another way, if the
cubes were see-through wouldn't they look
painted from the outside? This is called the Painter's Paradox. And I actually like it
best in its original form in Gabriel's horn. The inner surface of the horn is infinite, therefore an infinite amount of paint is needed to paint the inner surface. But the volume of the horn is finite, so the inner surface can be painted by pouring a finite amount
of paint into the horn and then emptying it. So can we paint the inner surface or not? This stumped mathematicians
of the 17th century too, making them doubt their
entire theory of mathematics. But we'll see that there's nothing wrong with the math at all. The paradox lies entirely
in our interpretation. Let's address the first point first. Why does the math tell us
that their surface area is bigger than their volume? To understand, let me ask you a seemingly unrelated question. What is the length of this line? Try to describe it without
referring to another length. Notice that saying something like, it's so and so centimeters long, is just describing it in terms of the length of a centimeter. Pause the video now if you'd like to try. And let me know your
answer in the comments. If you did manage to answer, you've just undermined the entire field of dimensional analysis, the study of the relationship between different physical quantities. It's key principle is that
measurement is comparison between like objects. This means that one, absolute
measurement doesn't exist, only relative measurement does. And two, for a measurement
to be meaningful it needs to be compared to
something of a similar dimension. For example, saying that one kilometer is longer than two yards is fine because they're both measuring a length. But saying that one hour
is longer than two yards doesn't mean anything because they're not
measuring the same thing. Sounds obvious. But now let me ask you, which is bigger, the surface area of this cube
or the volume of this cube? Hopefully you've realized that this question doesn't make any sense. Surface area measures a
two-dimensional surface and volume measures
three-dimensional space. What does it mean to say a chunk of space is bigger than a chunk of area? It means about the same as saying a piece of time is longer
than a piece of length, nothing at all. We get tripped up because
of everyday situations like needing more paint to
fill a cube than coat it. And this makes us think that volume is somehow bigger than area. But these numbers aren't saying we need a bigger amount of
paint to coat the surface area than to fill the volume. The surface area of all the
cubes measured in feet squared is larger than any finite
number of square feet. It's a comparison to other surface areas. The number 2.6 feet cubed is
a comparison to other volumes. They aren't a comparison with each other. Mathematically, there is no paradox with a shape having infinite
surface area and finite volume. There are other shapes that
are infinite in one dimension and finite in another
like the Koch snowflake, which may be easier to visualize. It's a fracture which follows
a simple iterative process. You end up with an
infinitely long perimeter, but a finite area. These objects exist in
the mathematical world. And when I say exist, I mean that they're consistent with the laws of that world. So that was the technical
resolution to the paradox, that there is in fact no paradox. By trying to apply paint to an abstract volume and surface area, we went inadvertently comparing
two incompatible quantities. Basically don't try and paint
infinite mathematical objects. But what if we did? I feel like this video
wouldn't be complete unless we at least talked
about what might happen. This is one interpretation. Now I say interpretation because we're applying unspecified paint to a mathematical infinite object. Depending on what assumptions you make, the outcome is going to change. This next section is not
mathematically rigorous. It's more speculation
based on mathematical and physical ideas. Feel free to share your own
interpretations in the comments. That's part of the fun of these kinds of questions after all. So looking at our questions from before, doesn't filling something with paint automatically cover its surface area? Covering all the inside faces of the cubes should take up the same amount of paint as covering the outside faces, right? If the cubes were see-through wouldn't they look
painted from the outside? When we asked these questions, we were assuming that you can have a pure
surface area of paint, but this is of course not true. When you finish painting
your walls, the can is empty. Physical paint has volume and thickness. When the cubes are big, this
thickness is negligible, which is why we kind of forget about it. But when the cubes becomes
smaller, it becomes significant. At a point, it will take more paint to
coat them than to fill them. So to answer our questions,
filling the volume doesn't automatically
cover their surface area, as the surface area uses separate and sometimes more paint than the volume. And the cubes wouldn't look
painted from the outside as they're missing their
entire layer of thickness abstract mathematicians
watching might be thinking, "Paint doesn't have to have thickness. What is this, physics?" And sure, that's another interpretation. We can use paint that
gets infinitely thin, but I'll leave that one up to you guys to speculate on below. Some people might say
that it's a waste of time to even think about
these kinds of questions. But it's questions like these that opened the eyes of
mathematicians of the 17th century and led to a lot of the modern
mathematics we enjoy today. Apart from the fact that it's fun, it exposes our flaws in thinking and makes us realize
that even simple ideas like area and volume are more complicated than we might think. To end, I'd like to leave you with a quote that I think perfectly captures
the spirit of this paradox. "One then discovers a striking
truth about mathematics, that it is not trapped in
what is called reality, a tangible, intuitive life. Based on logic, mathematics can go farther than our imagination can sometimes go, and leave us in awe when
it systematically oppugns the knowledge that we have so
long considered as the truth." To have a purely mathematical object that can't exist in the real world, raises some interesting questions. Does math exist only in our minds? Is it something we've invented to make sense of the world around us? Or does it exist outside
of us and we discover it? The question of where the
math is invented or discovered fascinated me so much that I made a feature
video about it on Nebula. Nebula is a streaming platform made by a bunch of us
educational YouTube creators, where we can explore making different and experimental content. It's been nominated for a Streamy Award. Curiosity Stream, a streaming service with award-winning documentaries is so supportive of our new venture. They're offering Nebula completely free when you sign up with them. There's a promotion going on right now where you get a 26%
discount off the bundle. So that's two streaming
services for just $14.80. In fact, there's a really cool
documentary by Hannah Fry, about the same question of whether math is an
invention or discovery. It's full of trippy visuals and loads of mathematical ideas, but presented in a very
easy-to-understand way. I watched the entire
three-hour series in one go. If you'd like to watch both mine and Hannah's take on this question, sign up with the link in the description or go to curiositystream.com/upandatom and start discovering today. That's it for me. And I'll see you in the next episode. Bye. (upbeat music)
