The following was written in the book I am reading.

Think about the symmetry of an equilateral triangle. It has two types of symmetry: reflectional and rotational. How can we describe the different symmetries without cutting out the triangle and folding it up or waving it around?

One way is that we could label the corners 1, 2, and 3,

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and then just talk about how the numbers get swapped around. For example, if we reflect the triangle in a vertical line, we will swap the numbers 1 and 3. Whereas if we rotate the triangle 120$^\circ$ clockwise we will send 1 to where 2 was, 2 to where 3 was, and 3 to where 1 was.

You can try checking that the six symmetries of the triangle correspond exactly to the six different ways of shuffling the numbers 1, 2, and 3. There are three lines of symmetry, and they correspond to swapping 1 and 3, or 1 and 2, or 2 and 3. There are three types of rotational symmetry: 120$^\circ$ clockwise, 240$^\circ$ clockwise, and the "trivial" one where nothing moves.

This shows that the symmetry of an equilateral triangle is abstractly the same as the permutations of the numbers 1, 2, and 3, and the two situations can be studied at the same time.

I can't sense but help that there's a deeper underlying phenomena going on here. So I have a few questions.

  1. What is actually the deeper thing going on here? I'd appreciate an explanation that's made as accessible to a mathematical ignoramus as much as possible, without cutting out any of the math.
  2. What might be some exercises worth thinking about (i.e. struggling with) to better grasp what is actually going on here, for someone who has virtually zero proof experience in math?



There's a number of ways of looking at this particular example, each of which yields its own insights.

One way to view it is as a "group action on a set". In this case, we have some different sets we can view as being "acted upon". As your text points out, one possible set is: $V = \{\text{vertex }1, \text{vertex }2, \text{vertex }3\}$, a set with three elements. It is not hard to see that each such "symmetry", the rotations and reflections, (these form the group that is "acting") induces a bijection of these three elements with itself. (We could also have it "act" on the one-element set: the center of the triangle, but I think a little reflection will show this to be rather dull).

In this particular example it so happens that every bijection of our $3$-element set also corresponds to a "symmetry" (a rigid motion including "flips" that maps the triangle to itself). So we have an example of an isomorphism of our symmetries (which are mappings: $\text{triangle} \to \text{triangle}$) with the possible permutations of our vextex set.

In this way of looking at this example, the "group" is a set of functions, and the "group multiplication" is functional composition (composition is associative, the identity map is the identity element, and since our mappings are bijective, they all have inverses that are also symmetries-thus the group axioms are satisfied). The correspondence of these functions with the bijections on our $3$-element set is so close, one is tempted to say "they are the same thing with different names" (this is the basic idea behind "isomorphism").

But we can also divorce the symmetry transforms from their "domain" (the equilateral triangle) and just study them as (abstract) objects. For example, we could describe the rotation of $120^{\circ}$ as $\sigma_1$, and the reflection across the vertical line as $\sigma_2$, and define (for example) a product:

$\sigma_1\sigma_2 \stackrel{\text{def}}{=} \sigma_1 \circ \sigma_2$.

Doing this, we can deduce certain basic rules, such as:

$(\sigma_1)^3 = \text{id}$ (the identity transformation of "doing nothing").

$(\sigma_2)^2 = \text{id}$.

These (and one other rule I list below) allow us to describe all six symmetries in terms of these two, explicitly:

$\{\text{id}, \sigma_1, (\sigma_1)^2, \sigma_2, \sigma_1\sigma_2, (\sigma_1)^2\sigma_2\}$

Now we have a group, in and of itself, a non-abelian group of order $6$. As it turns out, every non-abelian group of order $6$ "acts" the same as this one does, which is a lucky coincidence, here (it turns out that with groups of larger order, we often have "more possibilities", although this is not always true).

We can also imagine our equilateral triangle embedded in a Euclidean plane, with its center at the origin. We can then extend our symmetries to isometries of the plane that fix the origin, which turn out to be linear mappings $\Bbb R^2 \to \Bbb R^2$.

In this case, we then obtain the $6$ matrices:

$\left\{\begin{bmatrix}1&0\\0&1\end{bmatrix},\begin{bmatrix}-\frac{1}{2}&-\frac{\sqrt{3}}{2}\\ \frac{\sqrt{3}}{2}&-\frac{1}{2}\end{bmatrix}, \begin{bmatrix}-\frac{1}{2}&\frac{\sqrt{3}}{2}\\-\frac{\sqrt{3}}{2}&-\frac{1}{2}\end{bmatrix},\begin{bmatrix}-1&0\\0&1\end{bmatrix},\begin{bmatrix}\frac{1}{2}&-\frac{\sqrt{3}}{2}\\-\frac{\sqrt{3}}{2}&-\frac{1}{2}\end{bmatrix},\begin{bmatrix}\frac{1}{2}&\frac{\sqrt{3}}{2}\\\frac{\sqrt{3}}{2}&-\frac{1}{2}\end{bmatrix}\right\}$

which is another $6$-element group isomorphic to the other two already described.

To summarize, we have seen some different "characterizations" of this phenomenon:

1) An "abstract" group, characterized by a multiplication table (set of rules). Although I have not listed the entire $6\times 6$ table, you can derive it by using one more rule:

$\sigma_2\sigma_1 = (\sigma_1)^2\sigma_2$. This is fun to explore, and I urge you to do it for yourself at least once.

2) A structure (here, a group) operating on another (simpler) structure (here, a set). You may have seen this before in linear algebra, perhaps-there we have a field (a very rich structure) of scalars, operating on a simpler structure, an abelian group (of vectors).

3) Permutations of a $3$-element set (this is a more combinatoric view of things).

4) A set of invertible matrices (that is, a subgroup of the general linear group of the (real) Euclidean plane).

Thus, this one example ties together concepts from combinatorics, geometry, linear algebra and abstract algebra. To me, that is the "deep thing" going on, we have this "thing" that cuts across distinct branches of mathematics: a structure at once simple, and complicated. And one can use (suitably defined) isomorphisms to travel between these realms. That is part of the magic (and power) of abstracting the "particulars" away from a situation, down to the bare structure.

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The symmetries of a regular polygon are known as dihedral groups. This group is usually denoted $D_n$ for the symmetries of a regular n-gon.

The permutations of ${1, 2 ... n}$ are known as symmetric groups. This group is usually denoted $S_n$.

It so happens that $D_3$ '$=$' $S_3$. I say '$=$' because it is technically a group isomorphism, not an equation in the usual sense.

However, $D_n$ '$\neq$' $S_n$ for any $n>3$. For example when $n=4$, labelling the vertices $1, 2, 3, 4$ by going round clockwise we can see that vertex $1$ is always opposite vertex $3$. Thus the permutation swapping $1$ and $2$ does not correspond to a symmetry of the square.

As for exercises, reading up a little bit on very basic group theory will be useful, as would some of the books and articles mentioned in the comments.

Hope this helps.

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  • $\begingroup$ Some times, depending on the author, the indexing on $D$ refers to the number of elements in the group rather than the number of vertices it acts on. Therefore you see the group of symmetries of a triangle both as $D_3$ and as $D_6$, and the general group as $D_{2n}$ instead of $D_n$. $\endgroup$ – Arthur Sep 7 '17 at 12:08

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