Exponential growth in Katamari Damacy

YouTubers' progress on the 'Make the Moon' level of Katamari Damacy

Katamari Damacy is a wonderful game: simple, fun, delightfully bizarre, and deceptively mathematical. Katamari and its sequels follow the tiny Prince of All Cosmos as he rolls a magical sticky ball (called a katamari) around Japan. As things stick to the katamari, it becomes bigger, enabling it to pick up larger and larger objects. Eventually, the Prince builds up a massive enough katamari to roll up people, cars, buildings, islands, rainbows, and just about everything else in the game.

Katamari’s core game mechanic is the exponential growth model. As long as the stage has plenty of objects to pick up, the katamari grows at a rate roughly proportional to its size. Katamari delivers an aesthetic experience that conveys the essential intuitions behind exponential functions, similar to short films like Powers of Ten.

In the above chart, I explore how closely katamari size tracks an exponential curve. I watched five Let’s Play videos of different YouTubers playing the final level of Katamari Damacy and plotted their progress. Sure enough, each run traces an approximately straight line on the logarithmic scale, indicating exponential growth.