4D

This is way harder. As humans, we don’t have a physical intuition of 4D. You could try to make analogies with our 3D world, but they never really give you a feel for higher dimensions. Visualising this is impossible, so no pictures. We’re flying blind

Fortunately for us, math doesn’t care about pictures.

So how do we even begin? Let’s start by trying to recognise patterns about what we already know:

Consider a 2d box. When “changing” from 2d to 3d, what direction is this new side? Well, all sides of a box are perpendicular to each other (that’s kind of the definition of a box), so the new 3rd dimensional sides2 are perpendicular to every possible line we could draw on the 2d box.

So what happens if we apply this pattern to the transition from 3d box to a 4d box? Well, whatever the fourth dimensional sides looks like, they would have to be perpendicular to all lines drawn in our 3d box. (stop trying to visualize it, just go with me)

Take the line along the fourth dimensional edge that touches the 3d line (that we already know the distance of). We know it has to be perpendicular to the 3d line, because of the rule above. We also know that it touches the 3d line, because the meet at the save vertex.

And what do we get when we have two perpendicular lines that meet?3 A right triangle!

Using our friend Pythagoras: