1. ## Symmetry

I have been perusing these forums and I came across another question that was a bit helpful in clarifying this question. The question is:
With pictures and words, describe each symmetry in D3 (the set of symmetries in an equilateral triangle).
I think I could start with drating a point in the center of the triangle and draw a line to the midpoint of each side of the triaangle. I know that the answer involves three rotations (0, 120 and 240) but I do not really know what that information really means.

Thanks in advance for any help!

2. So you know for a 2D polygon there are two types of symmetries, rotations and reflections.

Label your triangle's vertices counterclockwise $\displaystyle ABC$

So firstly, how can you rotate the triangle? The rotation must take place about the centre of the triangle and after you have performed each rotation, the triangle must remain where it is.

So clearly a rotation of 0 degrees works. As you correctly stated, roations of 120 and 240 degrees work aswell (a rotation of 360 degrees is the same as a rotation of 0 degrees).

If you look to see what this is doing to your labelled vertices you will notice that a rotation of 120 degrees sends $\displaystyle A \rightarrow B$ $\displaystyle B \rightarrow C$ and $\displaystyle C \rightarrow A$

Now look for yourself how a rotation of 240 degrees acts on the letters.

Now for reflections. You correctly asserted that a line of symmetry must pass from the midpoint of an edge, through the centre and through a vertex. As there are 3 edges (and 3 vertices) there can only be 3 different reflections of the triangle.

Now observe what one of these would do to $\displaystyle ABC$

Imagine reflecting in the line passing through the vertex A and the midpoint of BC, then $\displaystyle A \rightarrow A$ , $\displaystyle B \rightarrow C$ and $\displaystyle C \rightarrow B$

Again, repeat this for yourself with the other lines of reflection

Now you are finished you can observe that $\displaystyle |D_3 | = 6$ (3 rotations and 3 reflections)

It should also be clear now that your rotations and reflections are just reshuffling how you labelled your vertices so as you have demonstrated, the size of you group should be 3! = 6

Hope this helps, I can't think how to explain the group $\displaystyle D_3$ any more.

Liam.

3. As a quick side note, the order of $\displaystyle D_n$ is not $\displaystyle n!$ in general (it's 2n) as some reflections can be obtained by a series of different reflections and rotations, thus making the size of the group smaller than n!

For example, with a square, a rotation of 180 degrees is the same as a reflection horizontally and then another reflection vertically.

Hope I didn't confuse things too much.

4. Thanks for the great help in clearing up my confusion. The concept is fairly straightforward and concrete. Thanks again!

5. Originally Posted by mathatlast
I have been perusing these forums and I came across another question that was a bit helpful in clarifying this question. The question is:
With pictures and words, describe each symmetry in D3 (the set of symmetries in an equilateral triangle).
I think I could start with drating a point in the center of the triangle and draw a line to the midpoint of each side of the triaangle. I know that the answer involves three rotations (0, 120 and 240) but I do not really know what that information really means.

Thanks in advance for any help!
Just a couple of quick points:

The symmetries of your shape are, essentially, permutations on 3 points, and so $\displaystyle D_3 \leq S_3$ (more generally, $\displaystyle D_n \leq S_n$). Although this is not true for higher $\displaystyle n$, here it is quite easy to see that $\displaystyle D_3 = S_3$ as labelling the points of the triangle 1, 2 and 3 we have that (1 2 3) and (1 3 2) are the rotations and (1 2), (1 3) and (2 3) are the reflections (they keep the point that is on the axis of reflection constant).

That is to say - instead of trying to find what the group itself looks like we can say that it is a subgroup of $\displaystyle S_3$, then look at $\displaystyle S_3$ and see that every element acts on the triangle as either a reflection or a rotation and so they are the same group.