# How Often Does a Butterfly Emerge?

Before you get down to the butterfly nursery on the lower level of the Texas Discovery Gardens, let’s see if you can figure out, on average, how often a new butterfly emerges from its chrysalis.

How might you go about that? Well, it would be useful to know that the average lifespan of a butterfly is about two weeks. How would that help? If we knew the number of butterflies in the room, we could reason that to keep the number of butterflies at that level, that many would have to emerge every two weeks, to replace the butterflies that would all die over the course of those two weeks. So we would just divide two weeks (or 14 days or 336 hours or 20,160 minutes) by that number to get the average interval between butterflies emerging.

So that switches our question to how many butterflies are there in the room. At first, that might seem like an intractable problem. After all, the room is huge, and the butterflies are flitting everywhere. You could never count them one by one — they just won’t sit still! And how can you tell whether you’ve counted one already or not?

So we need to use other mathematical tools. Rather than trying to get an exact count, we want to *estimate*. There are lots of ways to estimate. Here are a couple you might try. (1) Take a picture of the room and carefully count how many you see in the picture. Guess what fraction of the room is visible in the picture, and then multiply to get the number visible in the whole room. Then (here’s the most inaccurate part) guess what fraction of the butterflies are visible at any time — for example, by watching a section of the foliage for a while and seeing how many you see at first versus how many seem to turn up unexpectedly). Now multiply again to get the total number of butterflies in the room, both visible and hidden. (2) First estimate the volume of the room — maybe you can find the height, width, and length of the room from the staff. Then with your classmates, pick a well-defined section of the room that you can observe carefully and try to count how many butterflies in that section. Then measure the volume of that section. Finally multiply your count by the volume of the whole room divided by the size of that section. (3) Again, start with the volume of the room. Then estimate the average distance between a butterfly and its nearest neighbor butterfly. Pick a number of closest pairs of butterflies and do your best to measure the distance between them, using a measuring tape to compare. Include both butterflies whose nearest neighbor is close, and “loner” butterflies whose nearest neighbor is far away. When you have a bunch of measurements, average them. Finally, divide the volume of the room by the volume of a sphere whose radius is what you found for the average distance between butterflies. (Imagine the entire room filled with a spherical “bubble” around each butterfly!)

When you have your estimate, do the division to guess how often a butterfly emerges from a chrysalis. When you get down to the nursery area, you can ask about your guess to see how close you’ve come!