Math is Beautiful: Symmetry

Feb 26, 2016 | Schertz

Many factors come into play when humans assess aesthetics and determine which objects are pleasing to the eye. One of these is symmetry! We say that a figure has symmetry when it has certain repeating features—essentially, it’s the property of being “exactly the same” on both sides of a line or plane or about a center or an axis. From familiar shapes and man-made objects to patterns and structures observable in nature, you’ll see that symmetry is everywhere!

A figure has rotation symmetry when its appearance remains unchanged when you rotate it by less than 360° on a center point. A polygon’s angle of rotation can be found by dividing 360° by the number of sides or vertices (points where two or more lines meet to form an angle) in the polygon.

Now, when you fold a heart shape in half vertically (up and down) so that the edges match up exactly and unfold it, you’ll notice that the shape on one side of the fold is an exact mirror image of the shape on the other side. Because of this, we say that a heart has reflection symmetry—half of it mirrors the other half. The fold or line down the middle is called a line or axis of symmetry. Now fold the heart in two horizontally (across). Do the sides match up? The shape on one side of the horizontal fold is very different from the shape on the other side. This means that the horizontal line is not an axis of symmetry. While hearts only have one line of symmetry, other shapes have more.

Look in the mirror and imagine a vertical line running down the middle of your face. Is your face symmetrical? Your face and features may appear perfectly symmetrical when you’re looking at both sides at the same time. Actually, all faces are slightly asymmetrical, that is, one half is a little different than the other. While no one has a face that’s perfectly symmetrical in the mathematical sense (and if you did, you’d probably look pretty funny and not at all like yourself), studies have shown that the closer to symmetrical a face is, the more likely it is to be described as "attractive" or “beautiful”!