Introduction to higher dimensions

When a space has n dimensions, it can be categorized as an n-dimensional space. For example, the surface of a sphere is 2-dimensional; longitude and latitude are needed to specify where a point is on a sphere, but a line only requires one variable. In general, the dimensionality of a space is the minimum number of coordinates required to uniquely define any point inside it.

While the objects in higher-dimensional spaces might not be easily understood, sometimes a dimensional analogy can be used to partially explain them.

The Flatland example
Imagine a 2-dimensional world lying flat before you, populated by polygonal creatures (which we may call "Flatlanders") living in polygonal homes.

If you were to ask a Flatlander "tell me all the directions" (a request I'm sure wouldn't cause any confusion), they might respond with "forward, backward, left, and right." No thought of "up" or "down" would occur to them, for those directions do not exist in their world. Up or down from their world would simply lead out of it, and asking them to move in that direction would be like asking a drawing on paper to move to another paper higher or lower in the stack of papers they're in.

Subject matter aside, merely talking to a Flatlander would confuse them, as you yourself would be outside of their world and they would be hearing a disembodied voice. If you were to try to fix this by entering their world, though, things would get even stranger for the Flatlander. From their perspective, if you were to dip your fingers into their world, they would see five roughly circular objects appear, rapidly bursting into existence from nowhere and then proceeding to undulate and move seemingly without cause.

Imagine the practical jokes one could play on a Flatlander from the all-seeing viewpoint of a higher dimension. One could see their internal organs on full display, beating and contracting as unsettling concepts are laid out to the poor creature. They could hide away in the most heavily guarded place in their flat world, only to be visited by the same bedeviling entity who somehow got around all the guards and impenetrable walls - since to us, circumventing the constructs of a flat world is as easy as lifting your pen to avoid crossing a drawn line.

As we are to Flatlanders, so would 4-dimensional beings be to us. Fortunately, we don't have reports of hovering, shape-shifting objects mysteriously appearing and vanishing... except for ball lightning.