Hyperproblems

Everyone knows what squares are, right?

Imagine you have an infinitely large kitchen. If that doesn't really make sense to you, imagine a kitchen which is larger than the largest kitchen you can think of.

And now repeat that.

Suppose also that the kitchen floor needs to be tiled, with an infinite collection of equal squares. Wait, their colours may vary. But not their size: they must all have the same area. No matter how hard you will (or - more likely - will not) try, you will always end up with a kitchen floor in which every square has at least one complete edge in common with one of its neighbours.

Can you see that?

Either your kitchen floor ends up looking like a chessboard (see the picture above) – in which every square only shares complete edges, one with each of the neighbouring tiles – or it ends up like a kind of deranged chessboard, in which every row is slightly translated with respect to the neighbouring rows. Which still means that every square has two complete edges in common with its neighbours.

The upshot is that these are the only possibilities. Any other arrangement will have 'holes', which cannot be filled up with the tiles at your disposal, as they're all equally big. Think of the kitchen floor: if you try anything different from what I just described, you'll have to start breaking and cutting tiles, to fit them into the holes you created.

(the sound of a rusty rattling brain)

You still here?

Good.

Infinitely large kitchens: way too much dishes to be washed, an infinitely large cupboard underneath the kitchen sink – most likely housing an infinite collection of plastic bags – and a huge fridge.

Bigger than any fridge you can think of.

Suppose this fridge needs to be filled with an infinite amount of identical cubes. Yes, you may put different things in these boxes and they may have different colours, but they must have equal measures. Do you see it coming? No matter how hard you try, you will always end up with a configuration in which every cube has at least one complete square in common with one of its neighbours.

Either they are perfectly stacked, so that each cube shares its six squares with the neighbouring boxes (up, down, left, right, in front and behind), or you start messing around with the layers of boxes. At worst, each cube has only two squares in common with direct neighbours in one direction. Can you still picture that?

(the sound of people thinking "Where the hell is this going?")

Proceed with caution now, as this is the point where a stretched mind might end up being a strained one. Because you can repeat this idea in any dimension. 'Hypercubes?', you're asking. Yups. Hypercubes.

I will introduce you to the secrets and delights of higher-dimensional objects in a future post. As for today – and the sake of not making this post (or your coffee break) too long – you'll have to believe my word: you can define the analogue of squares and cubes in 17 dimensions. Or 85, if that suits you better. And infinitely large 17-dimensional kitchen floors need to be tiled too. The only difference however, is that hyperfloor tilings in 17 dimensions behave rather peculiarly: 17-dimensional tiles can be rearranged in such a way that each of these tiles has absolutely zero “complete sides” in common with its neighbours. Only pieces of "sides".

An abstract sense of freedom, envied by squares and cubes.

In case you're having a hard time trying to picture this: don't. It's impossible for our human brain to actually picture this, unless you're willing to trust your mathematical abstraction skills.

The weirdest thing of all, is the following: the tiling property which holds for squares and cubes holds in dimensions 4, 5 and 6 too. It does not hold in any dimension bigger than or equal to 8. Which leaves one case, right? Well, the case of 7 dimensions is still a highly non-trivial mathematical problem. People still don't know how hyperfloors in 7 dimensions behave.

(all together now)

Who cares?

Mathematicians.

Yes, they are strange people.

Never content with partial answers, always looking for problems which may or may not be in need of a solution, unable to rest before all details are completely comprehended, categorized and classified. And even if they succeed, that merely brings them back to square one.

Cube one.

Hypercube one.

You get the picture...