The video game Katamari Damacy (and the other instalments in the eponymous series) puts you in control of the Prince of the Cosmos and in charge of collecting lots and lots of objects. You collect these objects by rolling around a ball called Katamari. When you roll the ball onto a small enough object, it sticks to the ball, which becomes bigger, which can be rolled onto bigger objects, and so on and so forth.

The story, like the mechanics, is a bit absurd and endearing: you are collecting these objects to transform them into stars to fill up the sky

Some gameplay spoilers ahead.

Basics of Katamari rolling

You start with a 5cm diameter Katamari. At this stage, you can only roll up thumbtacks, matches, dice, etc. When you bump into, say, a bottle of water, it might as well be a wall: you bounce back.

As you roll up those small objects with the Katamari, it gets bigger. Soon you can roll up leaves, cutlery, orange quarters, etc. The water bottle now wobbles when you bump into it.

As you roll up those slightly less small objects with the Katamari, it gets even bigger. You start to roll up whole fruits, small books, shoes, and, pretty soon, water bottles.

The game mechanics are a bit more complicated than that. And there would be a lot to say about the controls (the pushing, the directions, the intertia) and the view. But we’ll focus on the Katamari size and growth.

The sweet nerdiness of rolling the Katamari

There are several dynamics that emerge from the mechanics of rolling the Katamari that I find very interesting. Two of them very nerdy and mathematical.

First, there is the exponential growth. The bigger the Katamari, the bigger the objects that the Katamari can collect, the faster the Katamari can grow. The Katamari dynamics include a snowball effect that is almost literal.

Second, this exponential growth is tampered by the square-cube law — although in this case, it is the linear-cube law. Whilst rolling up objects increases the volume of the Katamari, the measure of its size is by its diameter. So, roughly, if it takes you 10 thumbtacks to go from 5cm to 10cm, it’ll take 80 thumbtacks to get you from 10cm to 15cm. This encourages you to seek newer and newer objects: it’ll take you only 40 chopsticks!

The balance of these two dynamics is interesting. The specific inner working of the linear-cube law are probably tweaked, the game engine probably approximates it with just enough of a lie to improve the gameplay.

The strange thoughts from rolling the Katamari

The other dynamic I find interesting is one on the philosophical side: the eventual growth of the Katamari makes the very definition of “object” a bit shaky.

You start inside a home rolling up some small things, you then go in the garden to roll up small plants and small animals, you then go into the alley to roll up buckets and tricycles, you then go into the street to roll up some benches and vending machines, etc. As you go along, anything that was once an obstacle eventually becomes something you can collect. And whilst it might sound like an obvious truth of the game, this truth applies to a greater extent than you are lead to believe in the first few levels of the game.

Indeed, by the end of the game, you can roll up the very house that you started in. You can then roll up entire neighbourhoods, and then skyscrapers and factories, and eventually islands, clouds, volcanoes, storms, etc.

There is a weird philosophical twist to the game:
What is and isn’t an object is just a matter of scale; watch what happens when you grow beyond the scale of humans!