The fastest possible mile
Anyone who has ever tried to get in better shape knows that there is a law of diminishing returns for exercise. The harder you work, the better you get. But the better you are, the harder you have to work to improve.
This law is particularly evident to me lately, as I try to train for a marathon this summer. I’ve watched myself get progressively faster ever since I started running a year and a half ago, but it’s getting harder and harder to improve my times. It makes me wonder: what’s the fastest I could ever hope to be? For that matter, what is the fastest any human could ever hope to be? In this post, I’ll try to answer those questions. I’ll focus specifically on the mile run, where data from the world record over time provides surprisingly compelling evidence that the fastest possible human mile time is extremely close to the current world record of 3 minutes 43 seconds.
First off, I should justify my assumption that there is a “fastest time”. Why can’t the world record for the mile get progressively faster without limit? Well, if you want to take the argument to an absurd limit, you can say that no one will ever run faster than the speed of light, meaning the mile world record will always be faster slower than 5.37 microseconds. More seriously, though, it’s probably safe to assume that no one will ever run as fast as Usain Bolt for an entire mile (16 times longer than his 100m race), which puts an upper limit of about 2:36 on the fastest mile time. The world record stands at 3:43 right now.
My goal is to find this lower limit on the fastest possible mile time, both for my own self and for humans in general, by looking at the data for progression of the record. In order to do so, I’m going to have to make an assumption that can be translated into a mathematical statement. I’ll assume that halving the difference between your current mile time and your hypothetical best takes a constant amount of work. For example, if your hypothetical best is a 4:00 mile, then improving from 6:00 to 5:00 takes the same amount of work as improving from 5:00 to 4:30. It’s an unjustified assumption, but it’s fairly easy to believe. Moreover, it’s easy to check. If the assumption is true, then we should see the mile record decaying exponentially in time to some characteristic value.
My own personal best
I’ll test out my method first using data (as I remember it) for my personal best mile times when I was on the track team in high school. There are only five data points, but the data does seem to resemble exponential decay (see the graph below). My apparent asymptote (fastest possible time) is about a 4:28 mile, with a “half life” of about 1 and a half seasons. Not too bad! I can even predict that I would have run about a 4:43 if I had stayed for one more season of track. Of course, the data also suggests that I had no real hope of making the state championship meet, which had a qualifying time of 4:25.
The fastest runner in the world
If we can figure out my personal fastest possible mile time, why not do the same thing for the world as a whole? Data for the world record time in the mile is readily available, so it should be possible. When you first look at the data, however, one major problem appears. The world record in the mile has been decreasing pretty much linearly ever since 1913, when such things were standardized by the IAAF. There is no real lower limit in sight.
This would seem to contradict my original assumptions: either there is no lower limit for the mile, or there’s no way we can get it from the data. However, there’s something that the data is not showing. The population in the year 2000 was over 6 billion, while in 1913 it was less than 2 billion. There were many more runners in the 1990′s than there were in the 1910′s, so the fact that the world record was not falling more quickly in the 90′s suggests that it was getting significantly harder to beat. In other words, we have hope of finding a “leveling off” in the world data, the same as we saw in the data from my own personal bests.
What we need to do is redefine time. What we want is a plot of “world record mile time” as a function of “amount of work”. To figure out my own lower limit, I plotted my mile time as a function of seasons on the track team. To find the world’s lower limit, I need to plot the world record mile time as a function of the number of “person-years” that have elapsed since the mile world record was kept. A “person-year” is like a “man-hour”: it’s the number of people that were alive at a given time multiplied by the number of years they lived. So the year 1913 represents about 2 billion person-years, whereas the year 2000 represents about 6 billion.
Plotting the world record vs. the number of person-years since 1913 gives us exactly what we were looking for: a beautiful exponential decay to a particular value: 3 minutes, 39.6 seconds. (Note that in the plot below the y axis is in log scale, meaning that a perfect exponential should end up looking like a straight line).
The fact that all the points in this graph lie so close to the fit line is very strong evidence that the world record in the men’s mile is decaying exponentially to a particular value, just like our hypothesis. More importantly, we can deduce the fastest possible human mile time: 3:39.6. If you want to translate back from “person-years” to real time, this is what it would look like:
A few comments: have we reached the end of the exciting days of distance running?
The prediction of the last section looks exceedingly bold. The mile world record has improved by more than 30 seconds in the last hundred years, but I’m saying that we should never expect it to fall more than 3.5 more seconds, not in the next hundred years or in the next thousand. That would mean that when Hicham El Guerrouj ran 3:43.16 in July 1999, he was running to within 1.5% of maximum human capacity. Is it really possible?
Well, the short answer is that I wouldn’t have written this post if I didn’t think so. If you don’t believe me, maybe you can consider this for evidence: the current world record has stood for almost 10 years. That’s the longest the mile world record has ever been unchanged, despite the fact that the world has more runners now than ever before. So maybe we are “levelling off”; maybe it is becoming almost impossibly hard to beat the world record. I think it is. The early days of track and field were full of record-breaking distance running performances, and I think those days are coming to an end. If my predictions here are correct, then the world record will pretty much stop being broken within my lifetime (or, at least, it will stop being broken by more than tiny fractions of a second — much less exciting).
It’s sort of a sad thought. But for me, it’s also sort of an awe-inspiring one. When I watched El Guerrouj break the world record as a wide-eyed teenager on the track team, I was watching something extremely close to the greatest physical exertion of which mankind is capable. That’s pretty amazing.
UPDATE: Apparently someone already did this kind of analysis in 2005 and reached the exact same conclusion. Interestingly, they also predicted a fastest possible marathon time of 2h03:38. Three years later a 35-year-old Haile Gebrselassie ran it in 2h03:59, only 21 seconds slower than their predicted “impossible time”!