HUMANS AS CYLINDERS: A FEW THOUGHTS ON CHIA PETS & PIGEONHOLES or (alternate title) MY MATHEMATICAL TAKE ON A VERY HAIRY SITUATION
A year or two ago, my buddy Mobe and I were, no doubt, engaging in sophisticated repartee when he casually dropped an innocent comment into the conversation. As I recall, we were enjoying a snifter of brandy with our foie gras and caviar when Mobe told me that he was pretty sure he was turning into a Chia Pet. He was near the point, he said, where he was going to get his back hair braided just to change things up a bit.
Nearly everyone who knows me instantly realizes that most of the previous paragraph is total nonsense. I have never had brandy, foie gras, or caviar, and I had to do a Google search to confirm what a snifter is. So, maybe (just maybe), Mobe and I were sitting in some dive bar drinking beer and eating burritos when he brought up this Chia Pet business. Listening to Mobe talk about his increasingly shaggy person got me thinking about how hairy human bodies are or can possibly be. As I thought more about it, I realized that the trusty cylinder, a standard geometric shape known for perplexing students with its volume formula, can be utilized to help us explore this topic. So I ordered another beer, took a bite of my burrito, and decided to turn Mobe into a cylinder.
Can you imagine what this cylinder business could possibly be about? Has it occurred to you that anyone would ever have occasion to think of human beings in terms of cylinders? Did you ever wake up and think: You know what, today seems like a great day to mathematically model human beings as cylinders? That is one of the great things about being ever curious, thinking logically, and utilizing the scientific method; you just never know when such opportunities might present themselves.
Now it is time to get to work and turn Mobe into a cylinder. Actually, I want to model all human beings as cylinders. Why? I want to answer an improbable question, one that Mobe inadvertently reminded me of during our profound (cough, cough) philosophical discussion. The problem is: Are there two human beings alive that have the same exact number of hairs on their bodies? Take a few minutes and give that some thought. Now consider this: We don’t have to count the hairs on anyone’s body to answer the question. You might want to think about how that might work before you read on. After all these years, I still find the answer as surprising as it is fascinating.
This might seem like a tough problem; at the very least, it is an unusual one. I decided to write an essay about it because it allows me to introduce the concept of pigeonholing. Pigeonholing is a simple, yet robust, tool commonplace throughout science and mathematics.
A pigeonhole is just that, a little cubbyhole that you can place something in. Of course, it got its name from the fact that pigeons like to pick cozy little places to nest in. We, on the other hand, are going to take the idea of pigeonholes and co-opt it. We are going to use the concept theoretically and, instead of pigeons, files, or rubber bands, we are going to place cylindrical humans into our little niches.
So, instead of dwelling on Mobe’s inherent hairiness, let’s think of how hairy a human being can possibly be. To do this, let’s make a theoretical human and turn him (or her) into a cylinder. We can start with the assertion that there is no human 100 inches around. Admittedly, there are some big dudes out there, but I have never come across one that big. How about someone 100 inches tall? None of them running around either.
Consequently, as you might have guessed, we now have a cylinder 100 inches tall by 100 inches around. We have 10,000 square inches of surface area. Keep in mind that there is no actual human being that can come close to that size. As you will soon realize, we are overestimating for a very good reason.
Now that we have our cylinder, we can address hair density. Do you think that a person could have 10,000 hairs per square inch on their skin? I am certain that even Mobe can’t approach that figure. Clearly, that is a density that no human can ever hope (is hope the right word?) to attain. As for a Sasquatch or Chewbacca, they are (unfortunately) beyond the scope of this topic.
If we do some simple math, we find that we have a surface area of 10,000 square inches with a maximum hair density of 10,000 hairs per square inch, which gives us a total of 100,000,000 hairs for our theoretical most hairy human possible. Remember, we have modeled this person by vastly overestimating all the variables. There is no real person with close to this number of hairs, we are just trying to err on the side of caution. Any idea yet why we are doing this?
This is where the Pigeonhole Principle comes in. Let’s imagine an enormous hotel. This hotel has a long corridor with door after door after door. Each numbered door corresponds to its own room. Next, we line up every person now living and count the hairs on their body. When we get done with the first person, we arrive at a number and then give them a key and send them off to the room number that corresponds with the number of hairs on their bodies.
You should quickly be realizing what is going to happen. What is the Earth’s population? Last time I checked, it was a lot more than 100,000,000. Since the number of people alive today is far greater than the number of hairs a human can possibly have, we know there have to be two people with exactly the same number of hairs. At some point, two people will get the key to the same room. We need not count the hairs on anyone’s body to know this.
I love this little problem. It demonstrates the power of simple concepts such as pigeonholing. Also, it lets us know that a tiny bit of math and a little thought can take us a long, long way.
