Geometry: Symmetry

Symmetry

Just what is symmetry? Symmetry is one of those things that you can recognize but cannot put into words. Many words or phrases have a meaning similar to symmetry; balanced and well-proportioned are two that immediately come to mind. However, those words don't help us when we are trying to come up with a mathematically precise definition of symmetry.

For example, take a look at the three triangles in Figure 25.7. The first triangle is a scalene triangle. The angles of the triangle are all distinct, as are the lengths of the sides. The second triangle is an isosceles triangle. It has more symmetry than the scalene triangle, because two of the sides and two of the angles are congruent. An isosceles triangle has more balance than the scalene triangle. The third triangle is an equilateral triangle. All three sides are congruent, as are all three angles. It is the most symmetrical triangle in the group, but why?

Figure 25.7Three triangles with different levels of symmetry.

Imagine that you are small enough to stand at a vertex of these triangles. Suppose you start by standing on the vertices of the scalene triangle. Think about the views if you stand on vertex A, vertex B, and vertex C. Your view will change depending on which of the three vertices you are standing on. The segments that you look out on have different lengths. The included angles that the sides make have different measures in each case. There isn't much in common with the three perspectives, and that can be disconcerting. Each view involves something different.

What if you do the same thing with the isosceles triangle? Your view from vertex B and your view from vertex C are the same. The segment that you look out on has the same length in both views, and the measures of the angles formed by the included sides are the same. So there's some familiarity in what you see from vertex B when compared to what you see from vertex C. Things look the same from vertex B and vertex C, and the symmetry of this triangle is greater than the symmetry of the scalene triangle.

Moving on to the equilateral triangle, imagine your view from each of the three vertices. Because all sides are congruent and all angles are congruent, the view is the same from all three vertices. Identical views go hand in hand with maximum symmetry.

Now look at things from a different perspective. Instead of shrinking down to the triangle, imagine that you are all-powerful and are in complete control of the triangle. You can rotate it and flip it. If you rotate the scalene triangle, it looks different until you have gone full circle. Any way that you flip the scalene triangle it looks different. If you rotate the isosceles triangle, it also looks different until you have gone full circle, but if you flip it (interchanging vertices B and C) it looks like you haven't done anything. Whenever you can rotate or flip a figure and it looks like you haven't done anything, the figure you are playing with has some symmetry. The more ways you can move a figure and have it look like nothing has happened, the more symmetry the figure has. You are getting closer to how a mathematician describes symmetry.

A symmetry of an object is an isometry that moves the object back onto itself. In a symmetry, once the movement is complete it looks like nothing has been done to it. All objects have at least one symmetry; the identity isometry can always be applied. But this is a boring symmetry, hardly worth mentioning. But because the identity isometry is an isometry, it must be mentioned. The symmetries of an object follow the isometries of an object, so the names of the symmetries will be similar to the names of the isometries. There is a translation symmetry, a reflection symmetry, a rotation symmetry, and a glide reflection symmetry.

• Example 1: What are the symmetries of a square?
• Solution: A square looks pretty symmetrical (from a nonmathematical perspective) so there are probably lots of symmetries around. Figure 25.8 will help sort out the symmetries of a square.

Figure 25.8The symmetries of a square.

First of all, look at reflections. If you reflect the square across l₁, you'll end up with what you started with. The same holds true for reflection across l₂, l₃, and l₄. So there are four reflection symmetries for a square.

Next, consider the rotations. If the center of the square is the point O (where the diagonals intersect), you can rotate the square 90º, 180º, 270º, and 360º. Realize that the last rotation is just the identity rotation.

So there are eight symmetries: four reflection symmetries and four rotation symmetries. That's a lot of symmetries, but you shouldn't be surprised, because the square is pretty symmetrical.

This is just the tip of the iceberg. I could write volumes on symmetry alone! The applications to art, music, construction, science, and nature are endless! And the mathematics behind symmetry are as elegant as the symmetry of a snowflake. I hope this brief introduction inspires you to learn more.