A Brief Tour Through Basic Group Theory – Part 4

by Anna Mustata

Part IV – Relating Groups to Other Groups 

Attempting to make things more concrete leads us to developing the concept of group actions. We can now use this to zoom back out. We’ve been applying groups to geometric objects to find out how they can tell us more about the things we we’re already interested in, but can we use this to make them tell us more about themselves? 

Of course, I wouldn’t be asking this rhetorical question unless the answer was yes.  

Before we try to apply groups to themselves or to other groups, let’s establish a general way to talk about relations between groups. To see how we should look at these relations, we return to the key principle of keeping track of information. When we work in a very concrete setting – say, adding integers or looking at symmetries of a cube – the set associated to our group is very important. At a more abstract level, however, this information becomes irrelevant; the structure of the group is not determined by what the elements of the set are, but by how they combine with each other. If we were to write out the multiplication table (that is, a list of which element is formed by every possible combination of two other elements) for symmetries of the cube, then replace each function with a Greek letter everywhere it occurs, we would call the second group isomorphic to the first. That is to say, they are essentially the “same” group, but may represent a different concrete situation. 

A more interesting relation is the homomorphism. Like an isomorphism, a homomorphism equates elements of the underlying set of one group to elements in the underlying set of another without changing the multiplication table – that is, if a goes to a’, b to b’ and c to c’, then a*b = c means a’*b’ = c’. The difference is that multiple elements from one group can be equated to the same element in the second, and not every element in the second group must be reached. 

We could say that equating different elements in the first group to a single element in the second group is “gluing” them together, in the same way we glued functions together when performing a group action. The functions from the first group which are glued to the identity element from the second now perform the same role that a stabilizer performed for a group action. In this case, we call it a kernel. The elements “glued” to any other element in the second group are cosets of the kernel. 

We can imagine this gluing as “folding” the landscape of a group so that all elements in the kernel are brought to the “starting point” of the landscape. It makes sense to wonder how other symmetries are affected by this folding. In particular, we’ll look at the conjugacy symmetry where all similar paths radiate out from the starting point. If we consider the kernel as one of these paths, bringing it into the starting point requires by symmetry that we must bring all similar paths into the starting point as well. However, the kernel is defined as containing all elements that get brought into the center, so we must conclude that there are no other similar paths; in other words, conjugating the kernel by any other element in the original group returns the kernel. Subgroups which have this property are known as normal subgroups. 

Returning to the concept of groups acting on groups, we will see that examples of this have already snuck up on us and are fundamental to our descriptions of symmetries in groups. In our first symmetry, we “stepped off the path” of a subgroup by applying a function belonging to the larger group, but not necessarily the subgroup. This was, in fact, the larger group acting on the subgroup (or rather, on all cosets of the subgroup). The orbit was the set of all subgroups, and the stabilizer was the set of all elements in the subgroup itself. The same can be said for conjugating a subgroup. When that subgroup is normal, its orbit is itself and its stabilizer the entire group containing it. 

At this point, the concept of a homomorphism and the concept of a group action may look rather similar. That’s because a group action is a homomorphism. When we “apply” a function to a set, we are choosing a way for the elements of that set to be reordered amongst each other. The set of all possible such reorderings is, itself, a group known as the symmetric group of order n, or Sn, where n is the size of the set that is being reordered. Sn is an incredibly versatile group, since it can describe the reorderings of any set of size n. Our group action on a set of size n is then in fact a homomorphism from the group that acts on the set to the group Sn.  

We’ve discussed how a group can act on a subgroup of itself. However, observe that a group is a subgroup of itself. We define the regular action of a group on itself where applying an element of the group to another element of the group is the same as combining them by the group operation. This may seem like a pointless thing to do, but looking at the previous paragraph, we see that it gives us a homomorphism from any group to a symmetric group Sn where n is the size of the first group. If we examine this homomorphism, we see that its kernel contains only the identity element. In other words, no information is lost by gluing things together during the homomorphism; the group stays intact, so to say, and is merely placed directly over some path in Sn. Therefore, we can say that any group of size n is isomorphic to a subgroup of Sn. This is called Caley’s theorem. 

At this point, we have gone from thinking a group can be “basically anything” to seeing all groups of finite size as subgroups of very specific types of groups, with properties that can be understood by a person with some knowledge of permutations.  

There is one more thing we want to discuss in this section. So far, we’ve used homomorphisms to compare two groups we already knew about. However, we can also create new groups like this. We’ve already discussed “folding” over a kernel. Any normal subgroup can be used as a kernel; the process of folding this subgroup into the starting point is known as quotienting the larger group by the subgroup. Each coset of the subgroup becomes a single element in the new group. Anyone familiar with modular arithmetic can see that if we were to quotient the group of all integers by the subgroup of even integers described at the very beginning of this article, we would get a group with addition in mod 2. 

If Caley’s theorem allows us to narrow down what groups can be by embedding them in larger groups, quotienting allows us to simplify groups by breaking them down into pairs of smaller groups: a normal subgroup and its quotient group. A group that can’t be broken down any further is called a simple group (remember, these are the ones that have been classified in the Atlas of Finite Groups). Simple groups play a similar role in group theory as primes play when working with integers.  

Part V – Conclusion 

At this point, you’ve seen most of the essential ideas that allow you to work with groups. So, what now? 

On the concrete side, we can use these concepts to study particular groups that are of natural interest. Some such groups have been mentioned already. Symmetry groups of geometric objects tell us things about how molecules behave in chemistry and quantum mechanics. If we restrict this symmetry to rotations in two dimensions, we get complex number multiplication which allows us to find roots to equations of the form xn = 1. Groups of polynomials in multiple variables allow us to study the symmetries of these polynomials, which in turn allows us to solve polynomials in one variable of degree greater than two by looking at the symmetry of its roots. Groups of invertible matrices allows us to look at geometry – more specifically, transformations in geometry – using a simple set of algebraic rules, at a higher level than we can reach just with trigonometry and cartesian co-ordinates. The example given with colouring the faces of a cube shows that group theory can be used to solve combinatorical problems. Finally, modular arithmetic (the groups obtained by quotienting the integers by a subgroup) is very useful in cryptography. 

On the more abstract side, we now just about have the tools to start categorizing the types of groups that can exist. Based on what’s been discussed in this article, it should be possible to categorize all groups of size up to eight. Categorizing groups of larger size requires some more theorems, but still follows essentially the same principles.    

A Brief Tour Through Basic Group Theory – Part 3

by Anna Mustata

Part III – Group Actions 

So far, we have been looking for symmetries in general, vague spaces featuring things denoted by things like “g” and “*”. If you feel like we’ve lost touch from anything one might actually care to investigate, I can’t say I blame you. This is your cue to remember what I said at the beginning about the interplay between concrete and abstract, and call me back down to earth to discuss the concrete details.  

Rather than talk about symmetries of some abstract set, let’s look at something we know has symmetry: a regular polygon, say the hexagon. It’s natural to think of the hexagon as a set of points in the two-dimensional plane. We might want to try looking at its symmetry in terms of groups; however, now we hit upon a problem. We can’t use the vertices of the hexagon as objects of a group because the vertices of a shape don’t naturally interact by combining two of them with an operation. Instead, we can move in between vertices by functions such as rotations and reflections.  

So far, we’ve been seeing groups as sets of things or objects that are combined by the operation *. However, those objects can be anything, including being functions themselves. In this case, g*b simply means carrying out function b and then applying function g to the result. For such groups to have meaning, however, we must apply those functions to something. We are now working with two sets; the set S of functions in the group (for example, rotations and reflections), and the set X of objects that those functions apply to (for example, the vertices of our hexagon). Applying S to X is called a group action.  

In the hexagon example, we have the subgroup formed by rotating clockwise by 60 degrees. We can do this 6 times before returning to where we started. It has a coset formed by reflecting in the horizontal axis and then rotating. Since there are two cosets of size six, the total number of symmetries is twelve. This is formally known as the dihedral group D12.  

Much like how we find “paths” in a group by taking an element in it and applying it over and over, we find “paths” in a group action by taking one of the objects being acted on and applying the entire group of functions only to that object. We call this the orbit of the object.  

We can think of a group action as “gluing” elements of the group to the object acted on in order to compare their symmetries and find out more about either the object or the group. When we glue the identity element to an object, that object stays the same. However, you can glue multiple functions to get the “stay the same” position for a certain object. For example: a vertex stays the same if you rotate it by 0 degrees, but also if you reflect the hexagon through the axis passing through it and the vertex opposite to it. We call the set of functions that fix an object when glued to it the stabilizer of the object in X that they fix. 

A stabilizer always contains the “starting point” of the group it belongs to. Moving along the “path” of functions in the stabilizer of an object will always lead to a function that keeps that object in the “stay the same” position. Therefore, the stabilizer is a subgroup. Stepping off the path of the stabilizer means using a function that does not keep the object in its “stay the same” position, but rather moves it to a new position in its orbit. Following a path parallel to the stabilizer then keeps it in that new position. Therefore, we can “glue” each coset of the stabilizer to an object to move it to a particular position in its orbit.  

Looking back at Lagrange’s Theorem (the statement that the size of a group is the size of a subgroup times the number of cosets of that subgroup), we can say that, since each coset of the stabilizer corresponds to a position in the orbit, the size of a group is the size of the orbit of an object times the size of its stabilizer. 

Another similarity between orbits and cosets is that just like every element of a group must belong to a coset, every element of the set acted on must belong to an orbit. However, unlike cosets, not every orbit must be the same size.  

In chemistry, group actions can be used to study the structure of molecules. For a simpler example, the following question can be solved by looking at group actions and orbits:  

Suppose Alice has three colours and wants to paint a cube so that each face is a different colour. How many designs can she paint if two designs are considered the same when rotating a cube with one design makes it identical to a cube with the other? 

I won’t explain the full solution here, but the key to applying group theory to this problem is to apply the group of rotations of the cube to the set of all possible designs and count the number of orbits (since two cubes are identical if and only if they share an orbit). 

A Brief Tour Through Basic Group Theory – Part 2

by Anna Mustata

Part II – Symmetry and Subgroups 

Now we’ve established what a group is, you may be wondering why we care. That’s a perfectly fair question to ask after reading a list of rules about what you can do with some symbols? The definition of a group seems vague enough that you might think it could be almost anything, but in fact, a surprising amount of structure arises as the inevitable consequence of these very simple rules. Far from groups being “almost anything”, mathematicians have at this point classified all groups of a finite size that are “simple” (that is, the building blocks of other groups). They can be found in the Atlas of Finite Group Representations. 

So how do we find structure in what it seems should be a fairly uniform, amorphous landscape? Well, just pick an element and start walking. To take a simple example, consider the group formed by the set of all integers and the operation of addition. If you start at the element two and keep adding it to itself, then go backwards and keep subtracting it, you will eventually reach all the even numbers and only the even numbers. In other words, if you follow a path that starts from two, you can do everything the rules of groups allow you to do while only using half the space available to you. Since this path is self-contained, we can see it as a full landscape in its own right, contained in the larger group, so we call it a subgroup. Because we formed it simply by starting at two and adding or subtracting that from itself again and again, we call two the “generator” of this subgroup. If we had needed another element to reach all the corners of the landscape that we wanted to explore, the subgroup would have multiple generators – this is allowed (though it will never happen with the integers – thanks, Euclid’s algorithm!). 

This doesn’t seem to have advanced our cause much, but in fact, it gives us a strong foothold into the structure of groups, because subgroups can’t just be any set of elements from our group. Earlier I called the even numbers “half” of the integers. This does not, in fact, technically make sense – how do you define half of infinity? But it made intuitive sense for me to say this, because there is a clear correspondence between the even and odd numbers. If you imagine the integers as a sheet of paper, you could “fold” it in half so that over every even number 2n you have the odd number 2n + 1. The odd numbers do NOT form a subgroup, because the starting point – in this case 0 – is not among them. Instead, they form what we call a “coset” of the subgroup of even numbers.  

Cosets of a subgroup can be found by moving off the “path” of the subgroup (taking an element that is not in the subgroup) and then tracing out a parallel path starting from this element. Notice that we could start from any odd number; the coset formed by stepping off the path by 1 step and then moving in even steps is the same as that formed by taking three steps then moving in even steps, so there are exactly two cosets (the even subgroup and the odd coset) in the group of integers. Importantly, every element is in a coset. This gives us the kind of symmetry – that is, the recurrence of a pattern – we are looking for: whenever we find a path through the origin, we can arrange all other points in parallel paths. In particular, if the size of the group is finite, it is equal to the size of a “path” (subgroup) multiplied by the number of “parallel paths” (cosets). This is called Lagrange’s theorem.  

Already, we can see how our options for what a group can look like are limited. If the size of a group is prime, there is space for only one path. Since we have seen that starting at the identity element and doing the same thing over and over gives us a path, we can conclude that a group whose size is prime can be generated by a single element. Furthermore, we can say that any element (other than the identity) will generate the entire group. It’s also possible to prove that if a prime divides the size of a group, the group contains a subgroup of the size of that prime. 

If we are to try to imagine this type of symmetry, it would be like a wallpaper pattern where a shape or sequence is repeated side by side. However, another type of symmetry is like a flower or a snowflake, where all symmetric components radiate out from the same starting point. This is a type of symmetry worth looking at because the idea of a starting point is so important to us that it was one of the basic requirements for something to be a group. We can find such symmetries by taking advantage of our ability to retrace our steps. One thing: for me and I guess many more people the term “symmetry” appears in no more places than Geometry and literature, and so although I can intuitively understand what you are referring to in the paragraph before but it will get a bit more confusing as you get more abstract later on in the article. Do you think it would be a good idea to maybe find a place and explain a little more what does symmetry mean in algebra? Is it a kind of isomorphism between the properties of numbers? Is it the shift of position of a certain function (for example)?

For every element g of our group, we will write its inverse as g’. Then, we define the conjugate of another element h by g as g*h*g’. Intuitively, we can see how this will help us “loop” back to our starting point.  

 If, and after moving off the path of a subgroup by an element g not in it and taking a parallel path, we move back by g’, we once again get a “parallel path”, this time containing the conjugates by g of every element in the subgroup. Remember that putting a g and g’ together makes them disappear, so we can prove this symmetry by observing that (g*a*g’)*(g*b*g’) = g*a*(g’*g)*b*g’ = g*a*b*g’ – in other words, conjugating two elements and then combining them is the same as combining them and then conjugating the result, giving us a correspondence between the two paths. 

If we realize that conjugating the identity element just gives us the identity element again (g*e*g’ = g*g’ = e), we see that all these paths start from the same point, hence giving us the type of symmetry we were looking for. Since all paths lead back to the identity element, the conjugate of a subgroup is itself a subgroup. In this way, it is a stronger symmetry. However, it is “weaker” because not every element is guaranteed to belong to one of these paths. This is the payoff for repeating the same element (“e”) in multiple paths, thus not guaranteeing a “clean split” of all elements into either one path or another. 

A Brief Tour Through Basic Group Theory – Part 1 (Introduction)

by Anna Mustata

Part 0 – Disclaimer 

This article is a rather experimental attempt at explaining how I think about mathematics in an aesthetic and intuitive way, and hopefully help the reader see the purpose of studying certain things in mathematics and the reason behind what is done. Essentially, I want to portray mathematics in a way that is appealing and that shows more than the final results and technical proofs. Please keep in mind that none of the explanations I give in this article are fully-fledged proofs, just intuitive motivations for things being a certain way. Also, quotation marks denote terms which are NOT the official or technical term for what I am describing, but are a good intuitive way of thinking about it.  

Part I – Introduction 

Everything happens in a context. Sometimes, in mathematics, this context is obvious: Euclidean geometry happens on a flat, infinite plane. Hyperbolic geometry happens in the interior of a circle. 

In algebra, this is harder to see, but it can be visualized. Real numbers can be added on a number line. Integers can be added similarly if you restrict the points on this line that can be used. Complex numbers can be added in a plane with two axes, real and imaginary. But what happens when we try to get more abstract? Is there a space where we can add, say, polynomials in the variable x? At this point – as tends to happen when things get abstract – our simple geometric intuition seems to fail us. What we can’t show in pictures, we can describe mathematically through a set of rules. Such landscapes where mathematics can happen are known as groups. 

As with most areas in mathematics, diving headfirst into group theory without a guide will quickly leave you tangled in minutiae and wondering why you bothered approaching it in the first place. Therefore, let us outline a few key philosophical ideas to keep an eye out for in order to ensure that we are moving forwards in our attempt to study groups. 

The first and most straightforward is to remember to think of groups as a landscape. Even when we switch from literal, geometric landscapes to a list of rules, we can still think of the metaphorical “shape” of a group as something that determines how we can “move” through it when we do mathematics in that context.

Another thing that is true throughout mathematics is that everything is about information. Specifically – and this is likely to be best appreciated by computer scientists – it is about making sure that the amount of information needed to describe a useful idea is not greater than the amount of information that can be generated from it; otherwise, our idea is useless. Combining this with the concept of landscape leads us to look for symmetry; that is, for patterns that allow us to examine a small area of this landscape and deduce something about the shape of the rest of it. 

Finally, in order to progress, we need to recognize how we might get stuck, and how to become unstuck. Generally, there are two extremes where progress tends to peter out: too much abstraction, and too little. Too much abstraction eliminates all details we might care to know and leaves us with a meaningless, hollow outline. Not enough leaves us repeating the same process indefinitely, never catching a glimpse of the larger picture that might allow us to improve our measure of “information in – information out”. For the best effect, we must combine the two perspectives so that they fuel each other, zooming in and out much as one might when looking at a fractal: first, finding a detail, then using that detail to extrapolate a larger outline, then once more looking closely in order to fill in that outline.  
 
Keeping those things in mind, let’s dive in! 
 
Hopefully at this point you have a fairly good intuitive idea of what we mean by a group; however, this is mathematics, so let’s get a rigorous definition in place. In order to have a mathematical “landscape”, you need objects, and you need to be able to do things with them. Therefore, a group will be defined over a set S and an operation * (which can be anything that combines two objects to give a third). We don’t want to be able to walk off the edge of this mathematical landscape, so applying * to two elements in S must produce another element in S. There are three more axioms that need to be satisfied for (S, *) to be a group. 

1) * must be associative: that is, brackets don’t matter, or a*(b*c) = (a*b)*c. This is saying that any ordered sequence of elements in S uniquely define a “path” to another element in S.  

2) We also want our landscape to have a “starting point” (this would be 0 on the number line, for example). We will call this the identity element (usually written as e or Id). Moving by the identity element should not affect your path, so for any x in S, x*e = x = e*x. 

3) In order to actually make computations and do things, we need to be able to retrace our steps; so, every element x in S must have an inverse y such that x*y = e. 

A few algebraic manipulations will show that every element has exactly one inverse and that if x*y = e then y*x = e. (Note: it is not always true that x*y = y*x).

IMO Diary Part 1: training camp

Bit of introduction:

IMO (International  Maths Olympiad) is an international event hosted for pre-college students, it’s an annual event of 6 hard maths problems that required a lot of creativity to solve. The test is spread out over 2 days, each day with one paper, each paper consists of 3 questions and students are given 4.5 hours to solve them individually. Each participating country will send a team of 6, plus a leader and deputy leader. IMO 2018 (2nd of July-14 of July) took place in Cluj-Napoca, Romania.

2nd of July

12:00 p.m.

This is the big day! I have no idea what awaits in the next magic two weeks, and l’m sure I’m gonna love it!

On the plane: nothing much happened, we did problems and discussed together, and I finished reading the handouts from the training camp. In general no one disturbed us.

We got to the hotel, where we would be training with the team from Trinidad and Tobago, Everyone’s exhausted, yet I didn’t fall asleep immediately, I was stuck in this half-dream-half-awake state in which that I have to satisfy 3 equations in order to be allowed to sleep…

3nd of July

Very long day of training! We had our training session in the hotel dining room, the glass window above turned the room into a real greenhouse, and we’re really toasting while doing the math problems. We had three sessions, one on Number Theory, one on combinatorics and one on geometry. I got really stuck on a geometric question at the end of the day, and at one point I thought I got a solution. During supper, I was on the process of writing it down when I realized the solution doesn’t work and that I assumed something is true. I got really frustrated, as you can imagine, and I really could not taste the food in my mouth… It was the most surreal supper I ever had, and I wasn’t be able to sleep that night!

4th of July

I GOT IT!

I lay in bed thinking about the math problem last night, and I thought I had an insight, so this morning when I woke up, the first thing I did is to scribble down my solution. I completely abandoned my method from yesterday, but it took me very little time to get it out once I was on the right track. I was so happy, but at the same time bewildered that this question could take me so long.

We went to the Babes-Bolyai university today to train where our team leader managed to get a room. We trained the entire day, and in the evening we went to the national botanic garden to relax. In this process, we had loads of fun, and we got to know the Trinidad and Tobago members really well. They were all super nice, and I found a surprising shared passion with one of the teammates in our love of singing Disney songs and practicing Spanish. We would burst into spontaneous songs and dances and acting in the middle of the street, which caused everybody to stay away from us in order to save their faces.

I heard that the friendship between the two countries dated back a long time ago, when one of the IMOs was held in South America. We have been in touch since, and often trained together year after year. I think this kind of connection is really touching, how people from different time zones could be connected by a single event.

5th of July

Training! It was intense but I like it. At the same time though, it just taught me that I have a super long way to go. Ego losses occur.

Also, a very mind-opening conversation occurred between me and my teammates. We talked about what subject we are going to study at university. A lot of people’s are, without a single slight trace of doubt, mathematics. I, however, was still wavering between physics and maths. My interest in physics was peaked by a brilliant teacher, who was a deeply philosophical person and connected his teaching of laws of refraction with the philosophy of life and scientific methods. I was fascinated with the idea of exploring and figuring out the true law of nature. I never really considered maths a possibility, because it had a reputation of being hard and I never thought of exploring it any further than school work.

They shone some unique light into the difference between science and maths.”They have completely different ways of thinking, science is external coherent, and maths is internally coherent.”

I asked what this meant, and they explained that in science, you observe something that happens, and develop a theory to interpret it. Hence external coherent, you wrap your system around your observations. In maths, however, you build up you theories from the most basic axioms, and you never, never for the world contradicts what you already have with what you are developing, unless there is a desperate incoherency which forces you to break the limits, like what happened when people started to question the Euclid fifth postulate and developed projective geometry.

I found this an acute angle of looking at the difference between science and maths. This triggered great mental conflicts in me, and causes me to consider: is maths maybe for me?

6th of July

As a break, we decided on embarking on a great journey to Cheile Turzii (at least, that’s what Google map told us where we were).

We had a trip across the valley, looking at the beautiful mountains and the pairs of butterflies on the meadows. We crossed a brook barefooted to avoid having to walk all the way to use the bridge, and met up with a friendly shepherd and his flock of sheep. This picturesque country scene made me fall in love with the country immediately.

It was a great excursion, not to mention our great picnic under the trees and Romanian traditional dessert while doing math in the shade on the hilltop. (and feeding the shepherd dog that was always hungry) Oh, that moment was such a colourful moment, one that I would never forget.

And just after we finished our lesson on spiral similarity, however, there came the sound of thunder. A large clot of grey cloud hang heavy across the sky, and seemed to be coming at us. The shepherd started herding the flock of sheep over the hill back home. Some of us were a bit worried while others were unconcerned and insisted that staying under the trees will protect us very well. While we were debating, thunder can be heard rumbling down the hill and the Irish team would not stand it any longer. We raced down the hill, just when raindrop started to fall. We found a restaurant at the foot of the hill, and took cover under an umbrella. The minute we got settled, the rain started splashing down, blurring all our vision and even angrily invading the space under the umbrella. The rain covered the world in a thin sheet of downpouring water. About 10 minutes later, we saw the people who decided to stay slowly walking across the hill soaked in rain, a bitter smile on their face… We couldn’t stop laughing!

Looking back, it was a great adventure but I had to thank my luck. To think, I had my passport in my bag and everything. If that got soaked…

7th of July

We had a review session in the morning, we trained in a dark underground lobby, and because there weren’t any chairs everybody sat on bean bags, REALLY comfy!

We went over what different types of questions could come up in different topics of olympiad maths, and what to do to meet the different challenges. The session is really structured and helpful. Next, we headed to our official hotels where we will be staying with the other teams. Our two teams arrived a week early because we would like to train together for a week before the actual contest starts, but actually, most of the countries arrive on this day. Turned out, though, all the countries are spread out across 4 different hotels, and our team and the Trinidad and Tobago team were separated cruelly. This was a pity, given that a huge part of IMO is socializing, and now we won’t get the opportunity to talk to a lot of other teams.

The best part, though, was the goody bag! A grey, neat and smart looking backpack that was stuffed full with stuff, two T-shirts, one of which have the “dance of the functions” on it (Which I LOVE! I want to dance every time I see someone wearing it) an earphone, a water bottle, a tray (yes, a small breakfast tray, to my great bewilderment…we still haven’t figured out what is the meaning the committee is trying to convey) and the Mascot, a cute little teddy bear with a small head and big body called ”MIMO”. We tried out the games room, which is a common feature of IMO, a room in which people can play games and play chess and relax and socialize. However, this year’s game room only had a handful of people in it, and they are from the US and Canadian team, so after 5 minutes mt teammate Anna and I went back to our room, feeling a bit intimidated (after all, these are IMO gold-medalists!). The boys stayed, though, and we were very curious about their result in their fearless contest against those amazing mathematical minds.

8th of July

The boys told us what was the result of yesterday. They did lose a few rounds of chess, but in the end they developed a strong bond with the New Zealand team by a game of Monopoly, which, as it turned out, later became a bit chaotic and no one knows who is the winner.

Today was the opening ceremony, so we changed into fancy clothes and went to the big arena which will be our exam room the very next day.

We looked around the find the Trinidad and Tobago team and found them seated very much to the back. We had like an hour before the actual thing starts, so we started chatting around. I chatted with the girl sitting in front of me who was from Iraq. This was her first year here and will be her last as she was in her final year in highschool. She was really nice and we exchanged contact numbers took a selfie together.

I was on the edge of my seat trying to get up and chat with the Chinese team. I come from China, and of course, I have long heard of maths olympiad team in China which held endless mystery to me. I was even strangely nervous when I saw them seated not too far from us. “Go!” my teammates urged me, and after hesitating a while, I went over to say hi.

They were a bit shocked when I approached and greeted them in Chinese. The two teachers started to chat with me while the 6 team members eyed me curiously at a distance. I felt a bit guilty, actually, to think that back in China I guess I won’t even be able to get in city-level selection, yet here I am, standing together with the Chinese team who have struggled for so long and trained for years to be here.

The teacher discussed the difference between western and Chinese education systems with me, and he thought that the western education system indeed holds many possibilities for students to explore their full possibilities. Like me, for example, never in the world will I go and poke around the realm of mathematics if I were in China. Whereas here, I was already starting to think of choosing maths as my future field of study. This is stuning change that I could never see coming.

Anyways back to the Chinese team. The 6 teammates looked the kind of quiet student you might think you could find anywhere in your district, but wait until you hear their history… In the 6 of them, 3 are accepted into Tsinghua University, 3 are accepted into Beijing University, the top universities of China! And by “accepted”, it was not the ordinary admission, it was the type that guarantees the university entrance, no exam results needed (ok, well, you’ll have to pass, but you really don’t need to bother about it much).

While we were doing all these socialising, the team leaders of different countries are sharing the problem sets from their countries. Turned out, this is a “ritual of exchange”, if you will, between countries. Each country would bring a booklet of the problems in their country and share them around with other countries. An exchange of knowledge and resources.

The opening ceremony itself was not all that exciting, loads of speeches, and we tried in secret to do maths problems. The parade of countries was interesting, though, we had our ”three minutes of fame” as we walked on stage with our Irish flag.

After the ceremony, we started taking pictures with other countries, and we also got to walk around a little and socialize with other countries. I found a lot of friends from EGMO (European Girls Maths Olympiad). Friends from Ecuador, friends from Ukraine…

In the afternoon, we had the choice of going to the arena again to see our seatings for tomorrow. However, we preferred instead to go to a street food festival just next to the arena. Yum!

After dinner, we went to the supermarket to buy food. The test would be long and people would get hungry. It was always recommended to buy food that you like and would give you energy, as long as the food was not, according to IMO instructions 2018, crunchy and noisy.

The Benefits of the European Girls’ Mathematical Olympiad

The European Girls Mathematical Olympiad, or EGMO, is an annual competition similar to the IMO (International Mathematical Olympiad) but with only female participants. It was set up to encourage more girls to have an interest in mathematics. However, so far there has been no correlation between a country’s participation in EGMO and the number of girls on their IMO team. (http://www.egmo2018.org/blog/some-statistics-for-girls-at-imo-2017/)

The IMO is heavily male-dominated. At IMO 2017, 10.1% of the participants were female – hence almost 90% of the people there were male. There were 115 teams with a grand total of 619 competitors. That works out at about 556 males, an average of about 4.8 males per team(note: not every country brings a full team of 6 people).  The EGMO teams consist of the top four females, and on average the IMO teams consist of the top five males.

So, has the EGMO been a failure? So far it has not made an impact on gender balance at IMO. However, getting on the IMO team is not the be all and end all, the only marker of success in mathematics. Many people go on to have a successful career in mathematics without going to the IMO. The reason we need the EGMO is not to prepare girls to get on the IMO team. According to the website from EGMO 2016 in Romania, ‘[EGMO] was initiated with the purpose and desire to stimulate and motivate girls and young women to pursue their passion for mathematics’. EGMO provides friends with similar interests, role models and inspiration. Many of the girls I met at EGMO didn’t get on their country’s IMO team, but they did develop a love of mathematics.

Olympiad maths as a whole is immensely beneficial. Solving challenging problems, having to think outside the box, be creative and most importantly, persevere. It ignites a passion for mathematics in young people – for many it is their first experience of maths that isn’t boring and easy. Friends for life are made at classes, camps and competitions. It ignites a passion for maths and problem solving in many. EGMO brings all of these benefits to hundreds of young women every year.

Personally, I know that my experience at EGMO had a profound effect on my life. I really liked maths beforehand, but EGMO really sparked my love and passion for maths. This wasn’t just from the training and the competition itself, it was from spending a week surrounded by amazing people who all loved maths. We spent a week having the most fascinating conversations about mathematics. I have done far more maths since EGMO 2018 than I did before. Another thing is confidence- before EGMO I was convinced I would get 0 points there. While I didn’t do particularly well, I soared far above my expectations. The friendships I made at EGMO are lasting – this blog is one of the results of the friendships made at EGMO 2018!

Tianyiwa, who was also at EGMO 2018, agrees:

‘That was the first time I ever went on a maths olympiad, so I gained a lot of new experiences. I learnt the structure of maths olympiad contests, and it helped me get a true perspective into the difficulty of maths olympiad. And of course most importantly, the friendship! I met the most amazing people in EGMO from all around the world, and developed amazing friendships with my teammates. It made me more curious in maths, thanks to the mind-blowing questions our teachers threw at us in long train rides and on breakfast tables. For example, the question ”What is an area?” shook our very foundations of our understanding; The explorations we made into 4D geometry were also such a magical experience. EGMO opened up endless opportunities as well, just to list a few: We later formed a study group where we post interesting questions and study together; We got in touch with gold medalists around the world; We got invited to take part in interesting conferences… and in my personal case the curiosity on 4D geometry persisted long after the EGMO trip and later on this led to a maths project on higher dimensional geometry… and it is the first maths project I have done and through it I learnt so much once again about higher dimensions.’

Of course, in an ideal world we wouldn’t have the EGMO. However, we have to face the reality that there is a massive gender imbalance in maths and that needs to be addressed. I believe that the EGMO is one of the necessary steps to address this gender imbalance. Mathematics and other subjects such as science benefit from these young women developing interests in maths – many of these women are very talented and by not encouraging them the world would be missing out on some fantastic mathematicians and problem solvers. The EGMO has had a big influence on many young women and plays an important role in introducing more women to the beauty of maths and problem solving. I hope in ten years time we don’t need it, but for the present it has many benefits.

-Laura

Women in Mathematics Conference

On the 29th of August, the Women in Mathematics 2018 conference, hosted this year by the School of Mathematics and Statistics at University College Dublin took place. The event was an amazing experience, we got to meet up with interesting people across the mathematics field, and the audience was able to learn so much more about mathematics. At the beginning of the talk, Minister Mitchell O’Connor talked to us about the importance of bringing more girls into maths, and joy one can find in exploring the field of STEM. After the motivational talk, the audience gained amazing insight into the life and work of Sheila Tinney, an Irish mathematician, physicist and educator. She was also the first Irish woman to achieve a PhD. And there were also introductions from other women involved in mathematics about their work. Works involving the use of mathematics that ranges from health statistics to physics, including a project involving the building of a database for public libraries. An element of pure maths was introduced as well, which involves a bit of intuition and brain cells to understand, but awe-inspiring at the same time. It was fascinating to hear about where mathematics can take you across the world and across different branches of study.  It was very motivating to hear about their work, and for me, deeply reassuring. Being a girl dreaming about going into mathematics, to hear about the real-life achievements in mathematics of other wonderful women makes my choice seem less daunting to me.

I think there is a power in conferences and talks like this. It has a deep effect on the audience, whether they are involved in the field already or a member of the general public. It give insights into other people’s work, it opens up our eyes and can lead a change in our minds. Connecting with other people who have the same interest and passion, and clash your idea with them can spark creativeness and friendship you never expected to happen.

In the morning, there was a group of Transition Year students who caught an early train just to attend this conference. I think this was the best part of the conference, spreading dream of STEM in the hearts of students of the next generation…and who knows? The next Sheila Tinney might have been in the audience that very day.

BT Young Scientist Exhibition

BT Young Scientist Exhibition (henceforth abbreviated into BTYSTE) is a large scale science exhibition in Ireland, and it is a platform for high school students to innovate, create and explore. As the BTYSTE website described, it is an unforgettable experience of a lifetime for the students who take part. And I completely agree with in this statement, in the two times I participated, I went home each time with a completely different perspective to the STEM field, as well as the potential young people can have.

The exhibition has two parts, in the fist part students interested has to enter a one page proposal and some more information onto the website, and those projects that qualifies would be able to compete in the final round, where students would be able to display their project and have the opportunity of communicating with other students with their unique ideas.

In the preparation stage one has to write a project report (Hard work there! So time consuming and my experience is that it is even easier to do the project than writing out the report), a project diary and design a poster. The day before the exhibition, students go to the big exhibition arena to set up. The time is needed as some projects require large machines or different demonstration props. and then comes the hardest and most exciting part: the exhibition itself!

The most daunting part, of course, is the judging. There are at least three rounds of judging for each project, and the presentation of your project to the judges can have equal importance as your project report. The project book shows all the details and your scientific knowledge, but the presentation gives an outline and shows your passions. A good presentation can be absolutely crucial to the evaluation of the project, and I personally found it so much harder than I thought it would be because you will have no clue where to start and if you does not have it planned beforehand the result can be a quite disorganised speech!

The exhibition is also a test for persistence. I have chatted to a participant who, every day after returning home from the exhibition, would carefully analyse how he did in his presentation today, and consider if there are any flaws to his project and if there is room for improvement. Then he would do more research and add in more details to his project. He did this to the very last day, and his hard work also paid off in the end, he got home on the last day finally relieved and holding a wonderful prize.

Of course, the prizes aren’t everything, the most exciting part is to be able to chat with other people. I cannot describe the motivation I felt after talking to amazing people my age, seeing their ability, and getting to know what I can work towards. It is also eye-opening to hear about their creative ideas.

Talking to the people who came to visit is also a tremendous joy, there are all sorts of people who come around and ask you the most amazing questions. There are people who are experts in the field and would throw you off guard with a detailed technological inquiry. There are people who are genuinely interested and would listen to you talk without end, and there are people who understand you and would discuss in detail with you the potential of the project. It is wonderful to get your ideas heard and appreciated. Sometimes, even crazier things happen, such as a job proposal, or a trade of name cards. These conversations can sometimes lead to more amazing adventures, and for ambitious people like us, there’s nothing more thrilling!

All in all, it is a wonderful experience that leaves you smiling when you remember it, and it cracks opens doors into worlds you have never explored before. I cannot but think what a marvellous thing it is, for a high school student to be able to take part and experience such an event. It might just change a life or two.

4D Cube Explained-part 2

Hello again, this is the promised follow up, in which we dive deeper into the secrets of the 4D cube!

In the last chapter, we covered the basic ideas of a 4D cube as an introduction. In this chapter, we would explain further into the idea of 4D geometry, as well as answering the questions proposed in the last chapter.

https://scienceangles.wordpress.com/2019/01/04/4d-cube-explained-part-1/

So, we have derived what a 4D cube look like, and to view it from a 3D perspective, we have two ways of representing that image:

Graph 1
Graph 2

                                                                                  

Wait a sec, those two look quite different, are they both correct? What’s the difference?

As we can see, in Graph 1, the two 3D cubes that are supposed to be opposite each other are of different sizes. However, in Graph 2, the two 3D cubes opposite each other are the same sizes. In the previous chapter, we explained how to intercept Graph 2, but what about Graph 1? How do we understand Graph 1?

Let’s go back to the relationship between 3D and 2D. What is the 3D equivalent of Graph 1?

We get a picture like this:

Two opposite sides of a 3D cube are two squares. One is bigger, one is smaller. They are connected at all vertices, to create the picture above. Where have I seen this picture before? Think back on the museums you’ve visited, the Renaissance paintings you’ve seen. Think about the study of perspective. Or better yet, think about the room you are in. Go to one end of this room, place your back against the wall, and look across the room at the other end. You get a picture kind of like the one below.

Is it not exactly what we drew just then? When we place ourselves inside the 3D cube, Given a variable distance, we can see the two sides opposite each other at the same time, only of different sizes.

Artists know this. They use this skill to create realistic feeling of a 3D world on a 2D plane. Take a look at this painting, School of Athens. In this famous painting, the people far from us seem smaller, whereas the people near us seem bigger. This creates the feeling of 3D world on the 2D panel.

School Of Athens by Raphael

So, if we have a 4D cube that looks like Graph 1 it means that we are placing ourselves inside a 4D space, the smaller cube farther from us and therefore looking smaller, the bigger cube nearer us, looking bigger. I’m using the term ”farther” “nearer”, but these adjectives are the adjective of a new variable, a variable that does not exist in our dimension, the 4th variable in space. What might that be? It could be anything, colour, temperature…anything will do. But for our convenience let’s consider the 4th dimension as time, which is the easiest thing to understand in our 3D world, but the one thing that we cannot control. Ok, so now our variable is time, and the cubes are just two cubes through different time lapse. We are standing at a starting position “Now”, and the smaller cube we see in the distance is the exact same cube, labelled ”a year from Now”. That cube is considerably farther from us, so it seems smaller.

hat is how we interpret the difference between Graph 1 and Graph 2. Also, we gained a brand new understanding of a cube in 4th dimension. It is a cube, consisting of several 3D cubes that are spread across a time span, but also tangible at the same time.

Check out these photographs from photographer Steven Wilkins, showing the same landscape at different times on the same panel. In these pictures, the same thing at different times coexists. These pictures amazingly share the same idea as the structure of a 4D cube.

https://www.stephenwilkes.com/fine-art/day-to-night/52fa9a11-c5e8-4048-8071-0b560af4b6c2

So through these two articles, I hope I have given you a better understanding of a 4D world, and I would be very glad if you would be more fascinated in the mysteries of the higher dimensional space. There are endless magic to that topic and still a lot to explore!

The Math Olympian – Richard Hoshino

Rating: 9/10

Suitable for: Everyone

The Math Olympian is certainly one of a kind- a book about Olympiad maths, yet it is a fictional novel. The novel spans the three hour paper of the Canadian Mathematical Olympiad, which the protagonist, Bethany, is taking, detailing her thought process while answering the questions and using flashbacks to weave together a picture of her journey.

Hoshino succeeds in creating an engaging, relatable, entertaining and inspiring story of how creativity and persistence can allow you achieve your dreams. Bethany, the main character, is likeable yet imperfect. The hurdles she faces will be familiar to both girls in maths and anyone involved in olympiad maths. Bethany has to deal with being in a male dominated environment and with the various challenges that are part of olympiad maths.

I really liked how Hoshino described some of the difficulties young people who are training for maths olympiads face. Bethany learns to deal with failure, a vital lesson that mathematical olympiads teach you but a difficult one to learn. Other challenges Bethany faces include self-doubt, bullying and not fitting in.

Bethany’s love of learning and mathematics is infectious. Hoshino portrays mathematics as the subject it is, full of creativity, innovation, problem solving and beauty, relevant in every aspect of life. It illustrates how different maths is from how it is often perceived, dull, dry and boring, like it usually is in school.

From a mathematics point of view, I thoroughly enjoyed the descriptions of how Bethany solved the problems. It reminded me of the Thinking Out Loud articles from www.egmo2018.org. Hoshino himself solved these problems when he was training for mathematical olympiads, so the accounts are authentic, hence believable. The solutions are well explained, I had no trouble understanding them and I think that most people who have experience with school maths could get the main ideas of the problems.

My only criticism is I felt that Hoshino gave a bit too much attention to the issue of faith. For me it seemed a bit irrelevant to the rest of the storyline and too much time was spent on it. At first it was interesting but it drew on a bit, becoming a distraction and detracting from the book as a whole. This is only a minor issue that I personally didn’t really like but it didn’t have a large impact on my overall enjoyment of the book.


Overall The Maths Olympian is an insightful, inspiring and entertaining book that I would highly recommend. A must read for any young people (especially girls) who are interested in maths!

-Laura

View on Amazon: https://www.amazon.com/Math-Olympian-Richard-Hoshino/dp/1460258738