Back when I was in CÉGEP, I spent my summer months working as a gardener. It was a radical change from my usual routine of doing mathematics and physics to planting flowers and cleaning flower beds in the heat of the summer. It wasn’t bad by any means, but it was quite different from what one would normally expect myself to do. In essence, it was a job of convenience. I didn’t hate it, but I did look forward to doing something else. In fact, I remember telling my coworkers for multiple summers that I was working as a gardener only until I could finally work in my own domain of interest.
Fast-forward to now, and I finally have that job I’ve wanted. I’m working as a summer research assistant for one of the professors at my school, and it has been an interesting experience so far. I wanted to post updates on the blog about the things I’m learning, and how I’m finding the job.
So far, I’ve been playing catch-up for the last few weeks. I’m attempting to learn the basics of general relativity, and it hasn’t been a simple task. The reason is that I’ve only taken my basic physics and mathematics courses, so while I do have the mathematical tools of calculus, differential equations, and linear algebra, I don’t have things like calculus of variations, Lagrangian and Hamiltonian mechanics, and so on. As such, it hasn’t been easy to generalize my mathematical tools to curved spaces.
I’ve been learning by going through a textbook, which has its ups and downs. The book is Sean Carroll’s Spacetime and Geometry, and I like the book and the author, but I feel like restricting oneself to only a textbook makes it difficult to learn the material fully. The book packs in a lot of information, but it’s often at the expense of carefully going through each result. I’m sure for those who have more background to the subject, the book would flow well. The trouble is that I can get stuck on a single line for a long time, simply because no explanation is provided and I don’t know what to do. That’s where learning the subject with someone else would be helpful. The ability to ask a question right when it comes up could make learning the material so much easier. This is exactly what I try to do as a tutor when I help younger students. I want them to feel confident in what they’re doing, but be able to ask a question whenever it comes up.
Despite the difficulties, I do like what I’m doing. Learning is something that I truly enjoy, so I am happy to do the difficult job of learning a subject on my own. The ideas are interesting, and the mathematics that I’m learning is also useful, so I know that whatever I learn will indeed be useful. I think the key is to keep on pressing with the material, and by working with the ideas, I’ll get more familiar and comfortable with the subject. I’m enjoying the start of my summer work, and I can see myself doing this kind of stuff for a long time to come.
The Friends/Strangers Problem
I recently came across a pretty interesting puzzle that used a different kind of mathematics – called Ramsey theory – in order to tackle the problem.
Here’s the problem:
Suppose you have a group of six people. Show that you will necessarily have at least three mutual friends or three mutual strangers.
This is a sort of pigeonhole problem, which sort of means that we have to make a binary choice for each person. But what I love about this proof so much is that it’s entirely visual and fairly easy to grasp.
First, let’s imagine we have six people, who we can each represent as dots.
From there, we want to show the relationships between people. To do this, we will use lines to mark the type of relationship two people have. We will use orange to represent friends, and blue to represent strangers.
To begin, we need to start drawing lines. The first thing that you need to know is that there will always be at least three lines of any one colour that goes out from one person. The reason that this is true is because if we look at one person, the five others have no choice but to be friends or strangers. As such, every person will have at least three orange or blue lines connecting to other people, so we can simply choose one scenario of having three friends, which means three orange lines. We’ll connect them to other people at random, and so we get the following scenario.
Now that we’ve got three lines down, we know that each line represents a friendship between the people. But notice that, to show what we’ve set out to do, we only need to have one of those three friends also be friends. Put differently, if we can find a friendship such that we get a triangle of a single colour, we will prove our statement. However, it’s trivial to simply draw a triangle of one colour and say that we’re done. Instead, we need to show that we’ll be forced to make a triangle of one colour, no matter how much we don’t want to.
Suppose we consider one of those friends who were touched by our original rays. If we look at Person 4, they are friends with Person 1. Additionally, we see that Person 1 is friends with Person 3 and Person 5. Therefore, if we draw a friendship line between either Person 4 and Person 3, or Person 4 and Person 5, we will create a triangle of one colour, which means three people are friends. Since we’re trying to avoid that at all costs, we will draw two blue lines between these people, making sure that no orange triangles are created. This give us the following:
But what, we now have a problem. Look at the line we will need to eventually draw between Person 3 and Person 5. If we draw an orange line between them, then we will create an orange triangle between Person 1, Person 3, and Person 5. But on the other hand, if we draw a blue line between Person 3 and Person 5, we’ll be creating a blue triangle between Person 3, Person 4, and Person 5. Therefore, we’re stuck!
This means that no matter what, a group of six people will have at least three mutual friends or three mutual strangers. Using this technique, we’ve pigeonholed, or forced, ourselves into this situation.
What I like the most about this proof is that it’s visual and easy to understand. It’s what happens when you apply logic at each step, until you arrive at an undeniable conclusion. Furthermore, it’s a glimpse into a type of mathematics that wasn’t shown to me in school, so I thought it would be interesting to share. As I mentioned, I first saw this from a video by Raj Hansen who explains this problem in the introductory video to his Ramsey theory series. You should definitely check it out if you find this proof interesting. Personally, I’m excited to see what other kinds of applications we can find for Ramsey theory.
Introduction to Vectors
If you were talking to a friend about the drive you made this morning, you might tell them something along the lines of, “I drove from my place to hear in about an hour, going east for the entire time.” Your friend, who knows where you live and has a rough idea of the route to get to where you both are, understands implicitly that you don’t really mean that you went and drove in one direction only for the entire hour. After all, you first had to exit the property of wherever you were parked, then you probably had to navigate through the neighbourhood of where you live, before finally getting onto a highway or a direct route to your destination. It’s only at that point that you actually began going east. Before then, you may have driven in all directions, for varying amounts of time.
In physics, we call this route a trajectory. It’s how you move from one place and get to another. In order to describe the specific trajectory of your drive, we would have to do a lot of specifying, since your trajectory was complicated. But we’ll start off much more simply. Bearing in mind the impossibility of it all, imagine you just had a single block moving in a straight line, like so:
In this case, it’s quite a bit easier to describe how the route of this block will change over time. You don’t even need to be that great with the mathematics of it all for common sense to tell you that the block will move a distance d from wherever it is now by multiplying the speed at which it is moving and the time it travelled. If the block was moving at $5m/s$ and went for $3.5 s$, we would expect for it to move a distance of $d = (5m/s)(3.5s) = 16.5 m$.
This is a simple, but profound idea. We can figure out how far something has travelled by knowing its speed and multiplying it by the amount of time it has travelled for. Simple, right?
Unfortunately, that’s not quite the whole story. As you will see, a lot of learning mathematics is about understanding the special case, before moving on and getting more general. It’s the same story here. In the above example, it was easy to figure out where the block will be after a certain amount of time. Just multiply speed with time! But what if I asked you to look at this:
It isn’t so easy to figure out how far the cannonball will travel in a certain amount of time, even if you know how fast it is to start with. Common sense will once again tell you that, no matter how hard the cannon fires upwards, the ball will eventually fall back down to the ground. As such, the speed the cannonball is moving will change over time. Therefore, we will need a more sophisticated tool to analyze situations like these.
That tool is the vector.
What is a vector? Well, it’s a question that means something slightly different depending on if you’re a physicist, a mathematician, or a computer scientists. For our uses, the most practical way of thinking about a vector is as an arrow. The straight arrows you saw in the above pictures can be considered vectors.
The two main parts of a vector are its magnitude and direction. Direction means exactly what you think it means. In the above examples, the vectors I drew were pointing in the general right direction. So when we were talking about the block and I said it was moving $5m/s$, it would have been more accurate to say, “It’s moving $5m/s$ to the right.
The magnitude of a vector simply means the absolute value of the quantity we’re talking about (we always write magnitudes to be positive). In the first example, the speed of the block was $5m/s$, so that was its magnitude. However, these quantities don’t have to be fixed. Consider again the cannon scenario above. As soon as the cannonball leaves the chamber of the cannon, its speed is changing all the time (and, for that matter, so is its direction). As such, it’s a good idea to remember that while we will start with vectors that don’t change with time, there will come a point in which it is useful to use time.
But how is this helping us with the mathematics?
Fine, I told you about vectors, but how do we describe these situations with vectors? Isn’t it awfully vague to talk about something like a direction?
You’re absolutely right. To help, we need to use a tool that began with Descartes and is invaluable useful today. It’s a way to visualize what is happening in a system, so we don’t have to resort to only writing down equations. I’m talking about, of course, the Cartesian plane, where you’ve undoubtedly seen how to graph things like $y = 6x$ and the like. But now, you’ll find that it’s a fantastic tool for studying vectors.
Let’s go back to the example with the block, but this time, I’ll overlay a grid near the block so we can describe the situation more accurately:
From now on, you’re going to start thinking about space as a sort of grid system. Here, there are two directions: the $x$-direction, and the $y$-direction. Of course, you can go in a combination of these two directions, as the cannonball was doing before. But now that we have this grid system, we easily describe both the magnitude and the direction of any vector. Notwithstanding my bad drawing, the vector above has a magnitude of $5m/s$, and its direction is along the positive $x$-axis. In general, we measure the direction (or angle) of the vector by starting on the positive $x$-axis and rotating counterclockwise until we get to the direction that the vector is pointing.
This is all well and good, but I know that many of you are observant, and might ask, “If we have a vector going diagonally, can we describe it by talking about how much it moves in the $x$ direction and how much it moves in the $y$ direction?”
The answer is yes!
This highlights an important part of vectors: we can decompose them into their constituent parts. While it might be difficult to think about a certain motion in three dimensions, it’s much easier to think of the motion with a component in each of the $x$, $y$, and $z$ directions. Then, we can calculate things along each of those directions, before finally packaging everything up in the end to give a vector description of the motion.
(I want to take a second to mention that, of course, you can generalize the notion of what a vector is and what kinds of things you can do with them. This generalization usually leads you to the area of linear algebra, but we will focus for now with regular vectors that you can visualize as arrows, either in two or three dimensions.)
We will slowly build up to some interesting interesting results with these vectors, but I want to specify that I’m going to try and bring the logic and thinking behind a lot of these concepts from the perspective of someone who is looking to do physics. It can be all well and good to take a mathematician’s or computer scientist’s approach, but we’ll leave that to the side for now.
At this moment, you might be wondering something along the lines of, “Are these vectors really going to be that useful in physics?” The answer is a resounding yes. It may seem like vectors are only useful for describing the paths of bodies being launched, but the truth is that vectors are used all over the place. We use them to describe the equations of motion of bodies, the way electric and magnetic fields work, and basically anything that you can describe with the notion of forces. So basically all of physics, really.
For now, I just want you to take this initial point away: vectors specify magnitudes and directions, and we can add and subtract them component-wise in order to figure out how vectors interact with each other.
Can You Teach What You’ve Learnt?
Teachers are usually regarded as people who are experts at their subject. They may not be literal experts (such as those who teach scientific courses), but they have usually enough experience to make them experts. This expertise is usually forged through many years of teaching and working with students.
However, I’ve found that it has been extremely helpful for me to revisit content through the lens of a tutor. By switching roles from the person who is learning to the one who is explaining the concept, I’ve found that it helps reinforce the ideas in my own mind. When I can teach a concept to another student, it shows that I know what I’m talking about.
For me, the main reason I find this helpful is that teaching another student can be tricky. Often, the way you see something in your mind is not at all how another person sees it. Therefore, when teaching you have the challenge of not only transmitting an idea to the student, but of translating it as well.
This is particularly pertinent for the students I help in secondary school. Since that period of my life is several years old, the way in which I attack problems or situations demanding the skill set from that time has cemented in my mind. However, I need to sometimes put that aside when working with other students and focus on how they see the situation.
It’s a challenge that I think is worth thinking about. When learning a new concept, ask yourself: “Am I comfortable enough with this concept that I could actually explain it to someone else?” Trying to answer this question in the affirmative is a good way of assuring yourself that you’ve studied well.
It’s easy to have enough knowledge that you “pass” through a test without being noticed for your lack of knowledge. However, it’s much more difficult to do when you need to explain the entire concept to someone who has no experience. There, you can’t fall back on the chance that you won’t be asked a question on the one thing you aren’t sure about.
You can pass through a test without being discovered, but your goal should be to be capable of explaining a concept.