Exponential growth in Katamari Damacy • Ross Churchley

One of my favourite video games of all time is the inexplicable Katamari Damacy. Its quirky premise involves, as Wikipedia puts it, “a diminutive prince rolling a magical, highly adhesive ball around various locations, collecting increasingly larger objects until the ball has grown great enough to become a star.” In other words, it’s the most successful game ever made about exponential growth.

Katamari makes you explore a world at many different scales, all in the same level. You might start by dodging mice under a couch; just a few minutes later, you’re rolling up the family cat, the furniture, and everything else in the room. It’s an even better playable version of the Powers of 10 video, made possible by the differential equation:

\frac{\text{d}R}{\text{d}t} = s(t)\cdot R \approx k R

You make your magically sticky katamari bigger by rolling stuff up; the bigger you are, the bigger the things you can pick up. So we would expect the radius $R$ of the katamari to grow at a rate which roughly proportional to $R$ itself. The exact rate of change is governed by some function $s(t) \approx k$ , which depends on how good you are at finding a route filled with objects of just the right size for you to pick up. The solution to this differential equation

R \propto \exp \left( \int_1^t s(u)\ \text{d}u \right) \approx e^{k t}

gives a formula for the katamari’s size at a given time $t$ .

How justified are we in saying that $s(t)$ is roughly constant? I charted the minute-to-minute progress of four let’s players on YouTube. If the exponential model is a good one, then katamari size should trace out a straight line on a log scale. And so it is:

The runs keep up a remarkably consistent exponential pace, with a couple visible exceptions — one at the end of the level, when the world starts running out of stuff, and one at roughly the ten-minute mark, when a couple of the players struggled to find items at the right scale to roll up.

I’m not sure if this proves anything other than the fact that I like to do strange things in my spare time. But if you’re a calculus teacher with a bit of time and a PlayStation 2, I suspect this would make a very interesting three-act problem for your class.