Ok, this is probably a lot to take in with one picture! I’ve had a look at the last 12 months worth of marathons and training data and tried to capture it on a plot showing how training has improved my performance. Firstly I’ve focussed on the relationship between heart rate and speed. This is a well established linear relationship and as such you can easily plot out the line for yourself with just a few runs at different steady speeds. Now the goal of training can be achieved with a number of approaches, but the end result will be the same – your heart rate will get lower at a given speed, in-fact at all speeds! This means it can be used as an excellent indicator of fitness. The gradient of this line has an interesting unit, beats per kilometre, and as your body adapts to more training load the gradient will start to go down (and vice-versa I’m afraid!). So whatever your approach to training, you want your beats per kilometre to go down. Still with me? Good!

So bearing this in mind I’ve plotted the fit of four weeks worth of running before the taper of each of the 6 marathons I’ve run in the last year. The fit gives me a line that should then in-turn give me speed estimates for different heart-rates. Now after running several, evenly-paced, marathons I know I can sustain a speed corresponding to an initial heart rate of around 168-170 bpm (this number will be specific to each individual but I’m finding it’s in the range 20 – 28 beats less than your maximum). The progression of the lines from each marathon shows that the increased training load in each bout (averages over 4 weeks shown in the legend) does indeed improve my beats/km and with it all my race times. Putting in half marathon and 5k results would clutter this plot even more so I’ve stuck with marathons, plotting points corresponding to the average speeds I managed to finish each marathon with the half-split noted for reference. As you can see they don’t quite match the interception with the heart-rate/speed lines at 169-170bpm but they come fairly close and the trend follows the lines nicely. The end result is that with even a small dataset of runs including heart rate you can get a pretty accurate idea of your speed and fitness and track its improvement (and decline too which I know all to well from last Christmas, I’ll save that particular plot for another time!).

The most surprising number

The most surprising number I’ve seen in mathematics is deeply connected with the notions of randomness and dimensionality. The first two numbers in the series that lead to this number are: 1, 1. So far so easy, one is a pretty straightforward number. To begin, I’ll reveal where these two ones come from.

Imagine a line. Imagine that line divided into equal segments. I now place a pin in the middle segment. Now I’m going to start moving this pin. Where I move it to depends on the flip of a fair coin; heads I move it left, tails I move it right. That’s the setup. Now the question: what is the probability that the pin will visit a particular segment? The answer is that, given enough coin tosses, whichever segment is selected the pin will visit it with certainty, that is the probability equals one. This probability is the first number in this series of interest. The the next number in the series, also a one.

This second number is derived from the same problem as the first with the difference that we are no longer dealing with a line, but a plane divided into equal sized square segments. Instead of a coin we randomly pick a direction to move in. Pick a starting point on a flat surface. Pick an end point.  The probability that a pin, moving randomly, will eventually reach the chosen end-segment is one. No matter which end-segment you pick. Another way of putting this is that a pin moving randomly will eventually visit every single segment possible. This result is the same as for the line.

The trend in the series is becoming apparent, the number in the series at position N is the answer to the question:

What is the probability that a random walk on a lattice in N dimensions will visit an arbitrary point x?

So for one dimension, this probability was one. The same for two dimensions. Any guesses for three?

The answer for three dimensions is: 0.3405373296 . . .

This is the most surprising number I’ve ever seen. It may mostly be surprising to me as I knew and appreciated why for one and two dimensions the value is unity. On thinking about what the number could be for higher dimensions I imagined it would either remain one or decay with some series that involved a relatively simple relationship with π. The expression for this number in three dimensions is not straighforward:

In truth I find it difficult to explain just why this seems so odd to me. The most striking feature to me is that it happens in the transition from two to three dimensions, the space in which we naturally find ourselves. Could it be that random collisions in two dimensions does not allow for much fluidity, and collisions in four or more are far too infrequent to allow for many interesting reactions? Just a thought…

For those that are interested the value for four dimensions is: 0.193206 . . .

and this series of numbers is known as Pólya’s Random Walk Constants.