What Does a Snowflake, Rubik´s Cube And Donald Trump Have In Common?

Groups formalize the idea of symmetry. What the heck does this mean? Bad news is symmetry has diferent definition in diferent areas of mathematics.

Good news is, all the definitions are connected by similar intuition and, as I am going to show you, you probably already know what symmetry is before realizing it.

Many seemingly different things form a group.

Quiz time! What do all the items below (Donald Trump, snowflake etc.) have in common? I give you three possibilities to make it easier.

A) They are all difficult to understand

B) They all want to build a wall and make Mexicans pay for it

C) They are somehow symmetric

Own graphics. Including Canva graphics, Donald Trump, Elliptic curve and Integers

I wont keep you in suspense — C) is correct.

You can see that Donald´s face looks the same to the left of the line in the middle as to the right of it. We can say that the face is “symmetric along the line” (or generally “along an axis”).

The snowflake is also symmetric like this. You can try to draw a line somewhere on the snowflake and see that you can fold it along that line such that left side of the snowflake flips over to the right side.

This property of objects is just one particular type of symmetry. It is called reflectional symmetry, meaning we can find an axis and flip one side of the object to the other so that these two sides overlap.

But the snowflake has one more kind of symmetry — rotation symmetry. This means you can find a rotation such that the object looks unchanged under this rotation. Here, if you rotate the snowflake by 60 degrees, you just map one spike of the snowflake to the next one. (60 degrees, because snowflake has six side and 360 degrees divided by 6 is 60). Similarly, if you rotate it by 60 + 60 degrees or 60 + 60 + 60 degrees or — 60 degrees and so on, the snowflake still will appear to have the same position in space to you.

For the rest of the examples on the picture: The integer numbers on the axis on the left and on the right from zero are also somehow similar (symmetric). Similarly with the elliptic curve which is symmetric along the horizontal x axis. And finally, you can move each face of rubik´s cube and then reverse the move and such pairs of moves are also somehow similar — one is like the inverse of the other.

Own graphics. Including Canva graphics, Donald Trump, Elliptic curve and Integers

So what exactly is symmetry?

Another wonderful article describes them as:

What can we change in the triangle, to where we get a copy back, and we can’t tell the difference between that copy and the original?

From symmetries to groups

What we have learned is that the idea of similarity of parts of an objects or of transformations made on that object is formalized by the mathematical notion of symmetry, which, in turn, is formalized by a mathematical structure called group.

As indicated in the previous examples, we, as humans, are able to think about certain shapes or patterns as being more or less symmetric than others. A square is in some sense “more symmetric” than a rectangle, which in turn is “more symmetric” than an arbitrary four-sided shape.

Now we make it more formal and define a group, being aware of this intuition.

To define a group, we always start with a set of elements.

What does “group” mean in an ordinary, non-mathematical sense? Well, just a set of (not further specified) elements.

Elements can be numbers, points, moves, rotations, permuations… But also croissants on your table, . In common language, you might describe bunch of stuff as a “group”. In mathematical language, group is also a bunch of stuff, but with the difference that this “stuff” must have some additional properties. To speak formally, mathematical group is a set of elements with some extra structure.

If you need to understand more what a set is, see this.

You have probably came across many sets, like sets of natural numbers, set of your friends, set of bonbons in a bonboniere etc. Sets can indeed have tangible elements (like the bonbons) or abstract elements (like numbers or moves).

Source: Own graphics (using canva), Rubik´s cube moves

Now group is a mathematical structure that a set can have. Imagine it as a set of rules that the elements of a set obey. Not all sets can have such structure defined, but some can.

The snowflake example

Here the elements of our set are rotations, reflections and other transformations that we can perform on a snowflake. We will later show how they form a group by adding three rules that these transformations have to obey.

Group is a set of elements that follow some rules.

Consider snowflake just as some kind of “underlying object” for this group. In mathematics, we say that a group (of elements) acts on an object, but that is out of the scope of this text.

This is a visualization of the elements of the group of transformations of a snowflake.

Source: The Mathematics Behind Snowflakes
Source: 3Blue1Brown — Group theory, abstraction, and the 196,883-dimensional monster

Notice that the groups determined by transformations of snowflake consists not only by single moves (reflections, rotations etc.), but also of their cominations. It is easy to realize that if you dont change the snowflake under one rotation, you dont change it no matter how many times you repeat that rotation. This also applies for rotations in the opposite direction (clockwise vs counterclockwise).

We can also count the “zero” move as one of elements of this group. It may seem absurd, but if you dont perform any transformation, the snowflake stays unchanged. The “zero move” is often called “identity of the group”.

Making it mathematical

We have seen one example of a group where the operation acts on its own elements (integers). And one example of a group which acts on a different object! A group of symmetries that acts on a snowflake.

To make it formal here may be confusing. So let´s summarize what we know

  • Group is a set of elements
  • Group always has also an operation
  • The group operation must follow the three rules (axioms)
  • Operation takes two group elements and maps them back into that set.

Definition — Group: A group G is a set of elements, combined with a kind of group operation (which we denote *). The operation must follow three rules which we explain here on the example of the snowflake. Recall that operation can be multiplication, division, subtraction, but also something less “natural” such as in the case of the snowflake, a composition of rotations or reflections of the snowflake.

Group rule 1: For all elements a, b, c ∈ G: (a*b)*c = a*(b*c).

Explanation: In the snowflake case, this means it doesn’t matter if we compose two rotations a, b together as one and then compose it with another rotation c, or make a rotation a and then add two rotations b, c combined. Similar for reflections. If you could think this property is pretty straightforward and you are right. But it needs to be explicitly said, because not all operations in math have this property. (For example subtraction. (a-b)-c is not the same as a-(b-c).)

Group rule 2: There exists an element e such that e*g = g*e = g*e*g = g*e = g for all g ∈ G.

Explanation: Recall that elements of the group acting on a snowflake are the reflections and rotations. Then the mysterious element e in this case is just “doing nothing” to the snowflake. If we do nothing and then make a reflection or rotation, it is the same as if we first make reflection or rotation and then do nothing. Again, this seems really obvious, but this “neutral element” is really important.

Another example is the group of integers with the operation being addition, where the identity element is simply zero. You can try yourself that no integer number changes if you add 0 to it.

Group rule 3: For any g∈Gg∈G there exists an element g−1∈Gg−1∈G such that gg−1=g−1g=egg−1=g−1g=e.

Summary

Now we put it all together.

  1. We have shown that some objects have symmetry, which means they stay unchanged under certain very specific transformations (such as reflections or rotations). Not all objects have symmetries of course.
  2. In mathematics, the transformations under which the objects (that have symmetries) stay unchanged, form a structure called group.
  3. Group is a structure, meaning it is a set of elements and an operation. The operation behaves according to three rules. A group is any set of elements for which the three rules are true.

A group structure indicates a degree of symmetry in an underlying object.

In the end, I provide a standard mathematical definition of a group. The three rules are often called axioms and the operation has to be binary, meaning it takes two inputs. The three group axioms are refered to as associativity, existence of neutral element (also called identity element) and existence of inverses.

J.S. Milne — Group Theory

Regarding the examples at the beginning, the group acting on Donald Trump consists of reflexions along the line cutting Donald in the middle and just one rotation — around full 360 degrees. These are the transformations that leave the object as it is. For the case of the Rubik´s cube, the group alements are also moves and you can read more about them here. For the case of elliptic curves, it is more complicated to tell how the group elements look and behave. I added that example, since this has interesting application in cryptography and you can study the group properties of ellipses for example here.

Thank you for reading this. This article aims to put rigorous mathematical theory into something easy and intuitive, which works only to some extent. If you are interested in studying group theory more systhematically, I added a few interesting resources to the list below.

I am grateful if you provide feedback for this story in the comments!

List of References and Resources

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Tereza Tizkova

Tereza Tizkova

Mathematics Graduate | Blogging about Science (mainly Math) & Technologies (mainly Blockchain) & My Life (mainly Fuckups) | Grateful for your feedback