I know we’ve only just started, but stop reading this article and look around. Right now. You will see symmetries everywhere: in nature, in a butterfly, in flowers, in animals. There is an elegance in symmetry, which is why so many beautiful paintings and architecture use it to their advantage. There is also something innately effective about possessing a symmetry, which is one of the reasons it is of such high importance to mathematicians and physicists. In maths, finding a symmetry in an equation often simplifies it greatly.
We might all intuitively understand what a symmetry is, but to study it properly we need to model it with equations. Let’s consider a rectangle and try to model its symmetries with algebra. Alright, this is the part where you join in, grab some paper and scissors, and cut out a rectangle. We are going to figure this out together!
So what exactly is a symmetry? If I turn the rectangle upside down, it is indistinguishable from the way it was initially.
Essentially, we want to consider movements or transformations of the rectangle which make it look the same afterwards. We have an intuitive idea of what a rotation or a flip does, but we want to write down some mathematics to explain what is actually going on.
If I give two different stones you can more or less say which is heavier. However, to fully study the mass of the stone, we need a reference point. We define what 1 kg means and then measure the mass of objects with respect to it. Similarly, for our rectangle we label the corners of a side anticlockwise (don’t forget to label the opposite side with the same numbers as well) like this :
With the labelling complete, we now have a way of measuring each transformation by observing what the corners look like after transformation. For example, initially, reading from top left corner and going anticlockwise, the vertices are (1,2,3,4).
But after a rotation by 180°anticlockwise, it now reads (3,4,1,2). This way we can just read off what each transformation actually does. Let us call (3,4,1,2) the value of the rotation.
Initial configuration: (1,2,3,4)
After rotation by 180°: (3,4,1,2)
Now that we have agreed on how to represent a symmetry, let us write down all the different symmetries of a rectangle!
First, we have the identity symmetry. You can think of it as doing nothing to the rectangle. It may feel counterintuitive, but that’s exactly how we think of 0. Zero is a number that does nothing when you add it to another number! We denote this identity symmetry as R0, because it’s like rotating by 0°. Reading it off, we find the value of R0 is (1,2,3,4).
After rotating by zero degrees: (1,2,3,4)
Next up, we have a rotation by 180° anticlockwise. We denote it by R180. As we found out earlier, the value of R180 is (3,4,1,2).
The third symmetry would be a horizontal flip through the middle of the rectangle, denoted by H.
After flipping horizontally: (2,1,4,3)
Similarly, you have the vertical flip, V:
After flipping vertically: (4,3,2,1)
Summarising the symmetries we have found and their values:
I assure you, these are the only symmetries a rectangle has. The more critical readers may try to prove this (Hint: Look at values the top left corner can take. Is that enough to find what the symmetry is?).
A core idea in mathematics is finding out how two objects interact with each other. We do this all the time when we write down equations. Take the famous Pythagoras Theorem, a2 + b2 = c2 which I’m sure you have all studied in school. We can draw lots and lots of triangles with a right angle, but amidst all this chaos, we always find that the three sides of length a, b and c will be related to each other by this equation.
We ask the same question here, how do the symmetry transformations relate to each other? What do we get when we rotate a rectangle by 180° and then flip it horizontally? Let’s denote this action by R180＊H which reads ‘do a 180 degree rotation and then flip horizontally’.
Let’s apply R180, we get a rectangle. Let’s apply H on it again. Surely it will give us a rectangle as well. So the R180＊H gives us a rectangle, which means it must be one of the symmetry operations we listed earlier! Let’s find out which! Try it for yourself first before looking at the answer below…
We find that R180＊H has value (4,3,2,1), but that is the same as V! So, R180＊ H=V. Here, ＊ is called a composition which denotes what operation occurs after what. Think of it as “and then we apply”.
You might be wondering if you combine any two symmetry operations, does that always equal to one of R0, R180, H, V? The answer is yes. That’s because you get the same rectangle when you apply two operations, say a and b, in succession. So a＊b must be a symmetry as well!
Let us compile a little table showing all the possible combinations, as shown below:
|R180||R180 ＊ H = V|
Let’s start with the obvious, R0 applied to any operation X has no effect of its own because we’re essentially doing nothing. R0 ＊ X = X = X ＊ R0. So we can easily fill up the first row and first column. I encourage the reader to make their own table and fill it up as we go along.
R180 ＊R180 gives R0. Try it out with your paper cutout!
H ＊ H and V ＊V also give R0.
H ＊ V gives R180 and so does V ＊ H.
So far we have found this:
You might find a pattern in this table. It doesn’t really matter which operator you apply first, you end up with the same thing. That is a＊b = b＊a. For example, H ＊ V and V ＊ H both give R180. This is true in this case, for symmetries of a rectangle. However, this won’t always work if you take a square, for example.
Anyway, we find that V＊R180 gives H and so does R180＊V. We have successfully completed the table.
There are two important observations to be made from the results:
- Sometimes two transformations cancel each other out, hence resulting in R0. We call them inverses of each other. For example, flipping a rectangle horizontally twice (H ＊ H) gives back the initial rectangle, R0. If you look at each row, you will always find a R0 element somewhere. That means any element has an inverse (Can you see why?). We denote the inverse of A by A-1. For example, H-1 = H.
- Each row is just a rearrangement of the operations (R0, R180, H, V). All of them appear exactly once but in a different order. For example, the second row is just (R180, R0, V, H). We call this the rearranging row property. The columns are also a rearrangement, which we call the rearranging column property.
Now it is time for me to ask the same thing I asked in the first article. Is there a solution to the equation: A*X=B ?
For our rectangle, the answer is yes. I can even tell you how. Let’s say we want the solution to H＊X = R180. It might be a good idea to look at the third row, because that shows how transformations interact with H specifically. Of course we will find an entry R180 in this row, as it must occur once somewhere by the rearranging row property. We see that X = V here.
We can apply a similar reasoning to any A and B in the general equation and we will always find a solution! Try this out for yourself by finding X for the following examples:
- R180 ＊X = V
- R180 ＊X = R0
So, I think we all can agree on this result now:
A*X = B always has a solution.
This might feel similar to the fact that:
ax ≡ b always has a solution (in div n where n is prime)
As mathematicians we like to understand what is it about these systems that make them similar. Why is it that certain systems, like div 10, won’t always give you a solution, whereas other systems, like the symmetries of a rectangle always guarantee a solution?
You also might be wondering, does this property hold for symmetries of other objects as well, say that of a square or a butterfly? Can we always get a solution for X there? This is indeed a really deep question and forms the heart of this series of articles.
Mathematicians have a special name for these “nice” systems where you will always find solutions – groups. We will explore them further in the next article.
Article 1: Modular Arithmetic and calculating expenses