Like many people, I was a bit caught off guard when the woman's 100 meter freestyle swimming event at this year's Olympics resulted in a two-way tie for the gold medal. And I was even more surprised when the following night it happened again—this time the men's 100 meter butterfly ended in a three-way tie for silver.
How is this sort of ambiguity possible in the modern high-tech world? After all, we humans have developed truly astonishing technologies for calculating teeny-tiny intervals of time, so it seems like we should be able to simply use more precise timers and add significant digits to each swimmer's time to break ties. While the math behind this line of reasoning is sound, problems like this in the real world don't always cooperate in making themselves neat and tidy and easy to solve.
This leads us to today's big question: Why is it that these ties in swimming persist? Clearly, there's no way that two swimmers who "tie" actually tie to an infinite degree of precision—certainly one person must win at least by an itsty-bitsy bit. So why is it that we can't simply increase the number of significant figures and break ties? Let's find out.
How Precisely Can We Measure Time?
Before we get to the specifics of Olympic swimming, let's talk a bit about the numbers and concepts behind measuring intervals of time. As I mentioned earlier, humans have the ability to measure time incredibly precisely. In fact, physicists have developed a clock, which "ticks" upwards of 1 quadrillion times per second—that's a million billion times every second! In principle, this means we can measure time intervals down to the nearest quadrillionth of a second—aka the nearest femtosecond.
In practice, we don't yet have practical timers deployed in the wild that can reach this level of precision, but it's relatively easy to find a timer that can measure time intervals down to the nearest millionth of a second. While that's way less precise than the femtosecond timer, it's way more precise than the timers used in Olympic swimming which measure only to the nearest hundredth of a second. But before you conclude that Olympic officials are simply behind the times and unaware of the existence of these new and improved timers, you should know that there's a very good reason they've chosen to limit precision: fairness.
Why Are There Ties In Olympic Swimming?
The logic behind why it's more fair to measure time less precisely in Olympic swimming is fairly straight-forward. To begin with, swimming pools are not idealized mathematical objects—they are actual objects in the real world that are engineered and built by humans. As such, they are not perfect. In fact, if you look closely at the official rules and regulations of how Olympic swimming pools must be built, you'll find that each lane must be 50.00 meters in length with a tolerance of 0.03 meters. Since 0.03 meters is the same as 3 centimeters, this says that each lane must measure exactly 50 meters plus or minus up to 3 centimeters (which is just a little over 1 inch). Which means that the length of each lane in an Olympic pool—and thus the distance each swimmer swims—is slightly different!
The length of each lane in an Olympic swimming pool is slightly different!
Why do the lengths of lanes differ? The short answer is: physics. To begin with, a pool can only be built so precisely. There will always be natural variations in length across the width of the pool due to how perfectly (or imperfectly) it's made. Making matters more complicated is the fact that pools are filled with water, and this water weighs a lot—around 4,000 tons for an Olympic-sized pool. The weight of all this water pushes on the walls of the pool and can actually distort its shape and change the lengths of lanes by small but potentially significant amounts.
So why does this matter? How does it relate to the limited precision with which Olympic swimming events are timed? Remember, as I said earlier, it has to do with fairness to the competitors. In particular, as we calculated last time, it has to do with the fact that elite Olympic swimmers move through the water at speeds up to about 5.3 miles per hour. And thus, in 0.01 seconds (the smallest time interval measured in Olympic swimming competitions), an Olympic swimmer may swim up to around 2.4 centimeters. Which, as you've hopefully noticed, it very close to the 3 centimeter tolerance allowed for the natural variations in each lane.
Significant and Insignificant Digits
The point is that if the powers that be decided to add another significant digit and record times in Olympic swimming races down to the nearest thousandth of a second, they would be measuring differences in distances of just a few millimeters (that's how far an Olympic swimmer moves in 0.001 seconds). But the various lanes in an Olympic swimming pool can be not just a few millimeters but actually tens of millimeters different in length.
Which means that one swimmer might finish a few thousandths of a second faster than another simply because they were in a slightly shorter lane. Obviously that's not fair to the athletes. Which is precisely why Olympic swimming events are measured only to the nearest hundredth of a second. If one swimmer finishes 0.01 seconds ahead of another, that means they finished at least a few centimeters ahead of that swimmer. And, according to the tolerances allowed in building Olympic pools, that means that the winning swimmer in this case finished first because they were actually faster … not just because they were in a shorter lane. In other words, it means they won fair and square.
Wrap Up
Okay, that's all the Olympic math we have time for today.
For more fun with numbers and math, please check out my book, The Math Dude’s Quick and Dirty Guide to Algebra. Also, remember to become a fan of The Math Dude on Facebook and to follow me on Twitter.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
Swimming pool image from Shutterstock.
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