My Favorite One-Liners: Part 94

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them. Today’s edition isn’t a one-liner, but it’s still one of my favorites.

When constructing a mathematical model, sometimes certain simplifying assumptions have to be made… and sometimes these simplifications can be less than realistic. If a student complains about the unreasonableness of the simplifications, I’ll share the following story (taken from the book Absolute Zero Gravity).

Once upon a time, a group of investors decided that horse-racing could be made to pay on a scientific basis. So, they hired a team of biologists, a team of physicists, and a team of mathematicians to spend a year studying the question. At the end of the year, all three teams announced complete solutions. The investors decided to celebrate with a gala dinner where all three plans could be unveiled.

The mathematicians had the thickest report, so the chief mathematician was asked to give the first talk: “Ladies and gentlemen, you have nothing to worry about. Without describing the many details of
our proof, we can guarantee a solution to the problem you gave us — it turns out that every race is won by a least one horse. But we have been able to go beyond even this, and can show that the solution is unique: every race is won by no more than one horse!”

The biologists, who had spent the most money, went next. They were also able to show that the investors had nothing to worry about. By using the latest technology of genetic engineering, the biologists could easily set up a breeding program to produce an unbeatable racehorse, at a cost well below a million a year, in about two hundred years.

Now the investors’ hopes were riding on the physicists. The chief physicist also began by assuring them that their troubles were over. “We have perfected a method for predicting with 96 percent certainty the winner of any given race. The method is based on a very few simplifying assumptions. First, let each horse be a perfect rolling sphere… “

How many ways can you arrange 128 tennis balls?

I found this bit of computational mathematics fascinating. From

Imagine that you have 128 soft spheres, a bit like tennis balls. You can pack them together in any number of ways. How many different arrangements are possible?

The answer, it turns out, is something like 10250 (1 followed by 250 zeros). The number, also referred to as ten unquadragintilliard, is so huge that it vastly exceeds the total number of particles in the universe.

Far more important than the solution, however, is the fact that the researchers were able to answer the question at all. The method that they came up with can help scientists to calculate something called configurational entropy – a term used to describe how structurally disordered the particles in a physical system are.

Irrational / Everything’s relative

One popular (though maybe apocryphal) story from the history of mathematics involves the discovery of irrational numbers by Pythagoras and his disciples. The following quote is from the book Fermat’s Last Theorem by Simon Singh:

One story claims that a young student by the name of Hippasus was idly toying with the number \sqrt{2}, attempting to find the equivalent fraction. Eventually he came to realize that no such fraction existed, i.e. that \sqrt{2} is an irrational number. Hippasus must have been overjoyed by his discovery, but his master was not. Pythagoras had defined the universe in terms of rational numbers, and the existence of irrational numbers brought his ideal into question. The consequence of Hippasus’ insight should have been a period of discussion and contemplation during which Pythagoras ought to have come to terms with this new source of numbers. However, Pythagoras was unwilling to accept that he was wrong, but at the same time he was unable to destroy Hippasus’ argument by the power of logic. To his eternal shame he sentenced Hippasus to death by drowning.

When I was a boy, the story was told that Pythagoras could not accept irrational (i.e.., cannot be written as the ratio of two integers) numbers because their existence would mean that we live in an irrational (i.e., insane, crazy) world, and so he had the unfortunate discoverer silenced.

When I present this story to my own students, they’re usually incredulous about the story, doubting that someone so smart could act so stupidly (or irrationally). Then I’ll tell them a much more recent story, from less than 100 years ago, about how a scientific principle morphed into a statement of ethics. Einstein’s theories of special relativity and general relativity were developed in the early 1900s; his theory of general relativity explained precession in the orbit of Mercury and predicted the deflection of starlight by the Sun’s gravity, which were both unexplained by Newtonian mechanics.

Writing to a popular audience, Einstein summarized his theory as follows:

The ‘Principle of Relativity’ in its widest sense is contained in the statement: The totality of physical phenomena is of such a character that it gives no basis for the introduction to the concept of “absolute motion”; or, shorter but less precise: There is no absolute motion.

The following sentences from Paul Johnson’s Modern Times summarize the popular reaction to Einstein’s work:

But for most people, to whom Newtonian physics, with their straight lines and right angles, were perfectly comprehensible, relativity never became more than a vague source of unease. It was grasped that absolute time and absolute length had been dethroned; that motion was curvilinear… At the beginning of the 1920s the belief began to circulate, for the first time at a popular level, that there were no longer any absolutes: of time and space, of good and evil, of knowledge, above all of value. Mistakenly, but perhaps inevitably, relativity became confused with relativism.

Indeed, the modern catchphrase “everything’s relative” was spawned shortly after the discovery of special and general relativity, a moral principle that Einstein himself abhorred.

So, after telling the story about Pythagoras and \sqrt{2}, I’ll use this story to hold up a mirror to ourselves, demonstrating that the passage of time has not made us immune from translating mathematical or scientific principles into statements of ethics.

Mathematics that Swings: The Math Behind Golf

From the YouTube description:

Mathematics is everywhere, and the golf course is no exception. Many aspects of the game of golf can be illuminated or improved through mathematical modeling and analysis. We will discuss a few examples, employing mathematics ranging from simple high school algebra to computational techniques at the frontiers of contemporary research.

Design Zone at Fort Worth Museum of Science and History

If you live in the Dallas-Fort Worth metroplex or are visiting this summer, I highly recommend Design Zone, which is on exhibit at the Fort Worth Museum of Science and History until September 7. I’ve been to a lot of science museums, so I don’t make the following statement lightly: this may well be the most fun and most engaging physics exhibit for children to enjoy that I’ve ever seen. There’s all kinds of things to lift, throw, balance, and blast off so that children have so much fun that they don’t even realize that they’ve learned something.

Here’s the information posted by the museum:

Here’s a short promotional video:

And here’s a longer video describing the exhibit: