Turning from the shape of a butterfly’s path through the air to the shape of the butterfly itself, what do you notice about the shapes of the creatures you see all around you in the air?
First, it might strike you that there are so many different shapes! It seems as though every kind of butterfly is different. How can different butterflies produce such different shapes? It seems as though we ought to count the number of wings on different butterflies, because one way we might explain different shapes is by different numbers of wings.
However, if you look carefully all the butterflies around you, you will discover that they all have exactly four wings. In fact, every type of butterfly in existence has four wings. So there must be some other way they achieve such shape variation.
As you look at more butterflies, though, you will start to see certain patterns that recur, similarities among those differences. Can you find some examples?
A first, and mathematically important one, is symmetry. If you imagine a line through the middle of any healthy butterfly, the left and right sides are identical, just one is flipped the other way from the first. That type of pattern is called mirror or bilateral symmetry, and you will see it over and over again in the natural world.
There are other more detailed patterns in the shapes of butterfly wings. You may notice that they tend to be divided into sections by the veins in the wings, and those sections tend to have a similar arrangement in different types of butterflies. For example, you may notice that there tends to be an oval section in the middle of most wings, surrounded by fairly straight sections that proceed out to the edges of the wings.
That basic structure allows you to notice some more things. For example, many types of butterflies have “tails,” or long protrusions on the back of the wings. And those tails generally seem to come from the third or fourth sections of the back wings being longer than on other butterflies that don’t have tails.
And that observation provides an essential clue about the overall shape variation of butterfly wings: by changing the lengths of all of the different sections, the basic structure of a butterfly wing can produce thousands of different shapes. See if you can figure out the “instructions” for producing different butterfly wings, in terms of stretching or compressing different sections of the wing.