# The Hidden Complexity of the Olympics

Chad laments that we don’t hear that much about the decathlon any more, because Americans aren’t really competitive. I also think it’s a shame, because any sport in which your score can be a complex number deserves more attention.

Yes, it’s true. The decathlon combines ten different track and field events, so to come up with a final score we need some way to tally up all of the individual scores so that each event is of approximately equal importance. You know what that means: an equation. Let’s imagine that you finish the 100 meter dash in 9.9 seconds. Then your score in that event, call it *x*, is *x* = 9.9. This corresponds to a number of points, calculated according to the following formulas:

points = α(

x_{0}–x)^{β}for track events,points = α(

x–x_{0})^{β}for field events.

That’s right — power laws! With rather finely-tuned coefficients, although it’s unclear whether they occur naturally in any compactification of string theory. The values of the parameters α, *x*_{0} and β are different for each of the ten events, as this helpful table lifted from Wikipedia shows:

Event α x_{0}β Units 100 m 25.437 18 1.81 seconds Long Jump 0.14354 220 1.4 centimeters Shot Put 51.39 1.5 1.05 meters High Jump 0.8465 75 1.42 centimeters 400 m 1.53775 82 1.81 seconds 110 m Hurdles 5.74352 28.5 1.92 seconds Discus Throw 12.91 4 1.1 meters Pole Vault 0.2797 100 1.35 centimeters Javelin Throw 10.14 7 1.08 meters 1500 m 0.03768 480 1.85 seconds

The goal, of course, is to get the most points. Note that for track events, your goal is to get a low score *x* (running fast), so the formula involves (*x*_{0}–*x*); in field events you want a high score (throwing far), so the formula is reversed, (*x*–*x*_{0}). Don’t ask me how they came up with those exponents β.

You might think the mathematics consultants at the International Olympic Committee could tidy things up by just using an absolute value, |*x*–*x*_{0}|^{β}. But those athletes are no dummies. If you did that, you could start getting great scores by doing really badly! Running the 100 meter dash in 100 seconds would give you 74,000 points, which is kind of unfair. (The world record is 8847.)

However, there remains a lurking danger. What if I did run a 100-second 100 meter dash? Under the current system, my score would be an imaginary number! 61237.4 – 41616.9* i*, to be precise. I could then argue with perfect justification that the magnitude of my score, |61237.4 – 41616.9

*|, is 74,000, and I should win. Even if we just took the real part, I come out ahead. And if those arguments didn’t fly, I could fall back on the perfectly true claim that the complex plane is not uniquely ordered, and I at least deserve a tie.*

**i**Don’t be surprised if you see this strategy deployed, if not now, then certainly in 2012.

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