Mathematics is a funny subject. At some point during your high school education, it splits off into a two-pronged road. On one path, students learn about ‘everyday’ maths: simple and complex interest, transactions, basic summing and multiplication. Some would argue that indeed this path is more useful to the individual. The other path ventures into a thick forest of pure mathematics: trigonometric exploration, complex numbers, polynomials and high-level calculus. This is the stuff they tell us that is vital to the progression of society. This maths lets us build rocket ships, design bridges and allows our world to function as it does.
It is inevitable that one eventually asks oneself, what is the culmination of what I am learning? For me, I chose to head down the road of pure mathematics, despite having little desire to use maths heavily in the future. I realised in one of my classes today that the pinnacle, penultimate moment of in the glorious sun of mathematics had come. It did not come in the form of a universal equation for all the known laws of physics, nor did a prodigious breakthrough occur which would have cemented my name as one of the greats of mathematics for eternity. Rather, it was a quiet moment of satisfaction that occurred when my teacher explained what he called the infinite trumpet, and what the rest of the internet calls Gabriel’s horn. It is, without a doubt, the world’s greatest party trick.
The concept, while not involving any eyeball-melting maths, is fairly startling. I will attempt to explain it as simply as I can, and if you do indeed manage to pop this out at a party sooner or later, I take my hat off to you. If you want to skip over to the initial maths, just scroll down to where it says, “Now for the fun bit.”
Most of you would be aware of the Cartesian plane. Essentially, if you have any graph on this plane, what we refer to as a function, then you can actually find the area under this function by a process of integration:
This involves finding the area under the curve between the x-value of a and the x-value of b. This is a fairly simple process and many people cover it before the end of high school. However, in a slightly more complicated process, you can also find the volume of the shape if the curve is rotated around a particular axis. The curve above is actually y=1/x, and as a very rudimentary approximation we can actually find a rough estimate for the volume of the solid if we rotate the curve around the x-axis. We can do this by finding the volume of the cylinder with height b and length b-a:
However, this approximation is so far off it’s “f****** laughable,” in the words of my teacher. So, instead we divide the volume into two cylinders, giving us a better approximation:
And then we do it again, getting even closer to the actual volume of the solid:
As the width of the disks gets closer to zero, we eventually get closer and closer to the actual volume of the shape formed:
Now, we find an expression for the volume of one of the discs, and sum it in a series which we can then convert to an integral to find the area. The radius of one of the discs will simply be f(x), or the height of the function at that particular point. The width of the disk is often described as delta-x, and with this information we can use the formula for the volume of a cylinder to find the volume of each disc and then sum them:
From this we get the actual volume of all the cylinders together. We do this by examining what happens as delta-x approaches zero (that is, the thickness of the discs approaches zero) and summing an infinite number of these discs together:
Now for the fun bit. So we know the function above is f(x)=1/x. What if we were to find the volume of the solid formed if we rotated the function above around the x-axis from x=1 to x=infinity? Well, if we plug the values into the integral above, we get:
Simplifying further gives us:
So, despite the actually solid being infinitely long, its volume is actually just ∏ units cubed. How crazy is that? An infinitely long solid can have a volume of about 3.14 litres. But the fun is just getting started. Your party trick isn’t over yet. It turns out that if you want to find the surface area, all you need is the function’s derivative:
Now you just plug and chug in the surface area formula:
And, because your maths teacher told you so, you can then determine that:
(Credit goes to http://www.dummies.com/education/math/calculus/how-to-find-the-volume-and-surface-area-of-gabriels-horn/ for that little bit of maths. My teacher’s method was a bit too nifty for use here.)
So, here’s the party trick, for those of you who’ve stuck around for long enough: assume you’ve explained all the maths. You ask someone to paint the inside of the shape formed above (we’ll call it the trumpet). They start painting, and after a couple of billion years they realise that the surface area is infinite and they’ll never actually finish. So how do you paint the inside? Well, you can do it in a couple of seconds. You just grab 3.14 litres of paint (or there-about), turn the trumpet on its side, and tip the paint in. The trumpet’s volume is only∏ units cubed, and consequently the trumpet immediately fills up with paint, and your party friend is still labouring away for eternity painting an infinite surface area.
Maths doesn’t have to change the world for it to be fun.