The group of symmetries of the square

Diagonal axis flips of a square

Groups are a basic mathematical algebraic structure that I presented in this blog post, where I also gave the example of the group \mathbb{Z}_n . Here I want to expand on this and give another more visual example.

Symmetries are a big part of group theory and its applications. You can present every group as symmetries of something. That all might sound very abstract, so in the following I explain what that means using the example of the symmetries of a square. Below is the basic square with colored corners.

The original square

The elements of this group

The elements of this set are the movements you can perform on this square that leave it in a position symmetric to its original position – so not skewed, but again upright. The colors in the corner are just for visualization, they are not part of the shape, but they help us keep track of the corners, so we see that the different movements do different things.

One family of movements would be to flip along the horizontal or vertical axis as in the following figure.

Flips along the vertical and horizontal axes

Another very similar family of movements is to flip along the diagonal axes.

Flips along the diagonal axes

And finally we have clockwise rotations by 0, 90, 180 and 270 degrees. Obviously rotating by 0 degrees does nothing, so that is the identity element of this group. We could also choose the counter-clockwise rotations for the same effect but not both, because rotating by 90 degrees counter-clockwise is the same as rotating 270 degrees clockwise. If we included both, we would have two elements that are the same and that is not possibly in a set.

Clockwise rotations by 0, 90, 180 and 270 degrees

The binary operation of the group: composition

The operation on those movements is composition. We can first rotate by 270 degrees and then by 90 degrees, which is the same as rotating by 0 degrees. This means that Rotation-270 and Rotation-90 are inverses of each other, because together they result in the identity element of Rotation-0 (just leaving the square as it is).

Another composition would be to first rotate by 180 degrees and then flip along the horizontal axis. The end result here is the same as if we had flipped the square along the vertical axis. You can try this out for any combination of the above movements and you will find that each has a inverse so that the composition results in Rotation-0 and that each composition is indeed an element of the group that is presented in the pictures above.

And that’s all already, groups in their definition are quite simple. In the following posts I will start presenting some statements about groups and algebraic objects related to groups to expand on the topic.

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