One of the amazing things about science is that what you learn when you study one thing often applies to something else that you wouldn’t think was related at all. In today’s adventure, Daniel, Peter, and Mary discover that spaghetti, earthquakes, animals, and authors all have a secret connection.
When Peter’s mother walked into her kitchen, the last thing she expected to see was pieces of spaghetti lying all over the table with Mary and Peter laughing over them. But that’s exactly what she saw. Peter and Mary were sitting on opposite sides of the table, watching each others’ hands very closely as they took turns picking up pieces of spaghetti and snapping them.
“Hey,” Peter’s mother said. “What are you two doing?”
“We’re doing an experiment,” Peter explained. “I read somewhere that spaghetti always breaks into two pieces when you snap it and Mary didn’t believe me. So we decided to see if it was true.”
Looking at the pile of broken spaghetti, his mother shook his head.
“I know that you are supposed to replicate your results, but why so many times? ” she asked. “You should have been able to answer it with the first few pieces.”
“Well,” Peter said. “I can get the spaghetti to break into two pieces pretty consistently but Mary almost always gets three pieces except when she gets four. We’re trying to figure out why.”
“Well, it certainly looks as if you’ve done a pretty thorough job. But have you even decided what the variables are in your experiment?”
“Well, no, ” the two admitted sheepishly.
“So what you are really doing is making a mess,” his mother admonished. “If you don’t know what you’re doing, then you can hardly know if you are doing it right. Tell you what. If you’ll sweep all of this mess into a bowl so we can use it for dinner tonight -”
“Oh, boy! Spaghetti!” Mary interrupted.
“-then I’ll show you a real experiment with spaghetti. Deal?”
By way of agreeing, Peter and Mary quickly jumped up then gathered up the spaghetti pieces into a bowl. Once the table was clear, Peter’s mother put five pieces of whole spaghetti in front of each of them along with a penny.
“Here’s the experiment,” she explained. “What you’ll do is pick up each piece of spaghetti in turn, working from left to right. The other person will flip their coin. If they get heads, then you have to snap the spaghetti into two pieces of equal length by holding it in the middle; you then put the two pieces at the far right side. If the other person gets tails, then you put the piece of spaghetti at the far right and you don’t snap it. Got it?” At their nods, he continued. “You’ll each flip your coin forty times. So you know the variable – the coin flips. Here’s the question: How many pieces of spaghetti will you have at the end?”
“Well, we start with five pieces,” Peter said. “So each one will get eight flips. If they get broken half the time, then each piece should get broken four times giving us two pieces each time and we’ll have five times four times two or forty pieces.”
“I don’t think that’s how it will work,” Mary said. “Remember that this is random. So some of the pieces will get broken more than once and others won’t get broken at all. So it will probably be less than that; I think it will be closer to thirty pieces.”
“Good,” Peter’s mother said. “And notice that you both added an extra prediction. Peter seems to think that the pieces will all be about the same size but Mary, you think that they’ll be different sizes.”
“That’s right,” Peter said.
“Well, there’s only one way to find out,” Peter’s mother said. “Start flipping!”
What do you think will happen? Do the experiment!
For the next ten minutes, the kitchen was filled with the sound of flipping coins and breaking spaghetti. Soon, some of the strands looked less like spaghetti and more like toothpicks while other strands stayed long. At the end of all the flipping, they quickly counted up their pieces of spaghetti.
“Hey! I only got twenty-five pieces!” Peter exclaimed.
“Yes, and one of yours is still whole,” Mary added. “I have thirty pieces but none of mine are longer than a half.”
“So what gives?” Peter asked. “I kept track – we both got the same number of heads.”
“Ah, but did you get them in the same order?” Peter’s mother asked. “Because you kept cycling through the pieces in a strict order, getting heads or tails at a different time meant that you’d get a different length. But, if it makes you feel any better, if you had done more flips on more strands of spaghetti, then you would both have gotten almost identical results. Yours are actually pretty close, given how few times you flipped the coins.”
“How many would we have needed in order to get the same result?” Mary asked.
“An infinite number,” she replied. “But if you’d flipped your coins about 1000 times and used ten strands of spaghetti each, then you two would have been much, much closer.”
“But why didn’t we get forty pieces each?” Peter asked.
“It is just like Mary said – you forgot that you’d be flipping on individual pieces and not the whole strand each time; there’s a fun little math equation that describes what will happen. But what’s really cool about this is that you can do the math backwards and find missing things. Take asteroids for example,” she explained. “We know how many we have found and what their approximate sizes are. By applying that equation, we can predict how many we haven’t found.”
“Cool! Is that why they say that we know where 90% of the big ones are?” Mary asked.
“Yes. But the math doesn’t just apply to rocks in space; it also tells us what the distribution of rocks in an avalanche will look like, and how the number of earthquakes will change with size, and even the number of fish of a given size that are hiding in a lake. Alfred Lotka even used it to predict the number of scientific papers that would be written in any given year.”
“What? How did he do that?” Peter asked.
“He just looked at the number of papers that were written by an scientist who had already published a paper. Let’s say that in one year, there were 500 scientists who each published a paper and that the next year just 125 of them published another paper. Then he knew that in ten years there would only be five of them who wrote a new paper. If you then add up all of the people who were doing research and assumed that they stopped after ten years, you’d just add up the number of scientists publishing for the first time, the second time, and so forth, and discover that there would be 774 papers published that year,” Peter’s mother said. “The neat thing is that it isn’t just papers that you can do that way; it also tells you how emails would be written in a given year and how many blog posts or how many television shows would be filmed.”
“That is so cool!” Mary said. “I didn’t know that one equation could do so much.”
“Well, there is one thing that it can’t do,” Peter’s mother said. “It can’t clear the table so I can make spaghetti!”
With a quick grin, the two hungry young scientists turned to the table and started cleaning up their experiment.
About the scientist
In 1918, an influenza pandemic swept across the globe and killed one out of every twenty people alive at the time. It was the greatest health disaster since the Black Death and inspired many scientists to find new and better ways of predicting disease outbreaks. One of these was Alfred Lotka who was trained as both a biologist and a mathematician. He had been working on ways of predicting malaria outbreaks and soon discovered that his work was more widely applicable than he had expected.
He started looking at the number of papers published by scientists, At first, he was just curious about why so many scientists seemed to write fewer papers as they got older but he was surprised when he found that he could actually predict how many papers would be published using a simple formula. His result turned out to be a special case of a more general equation that governs everything from animal populations to gravel size to the number of sunspots.