Please do not touch

In three-dimensional space, two flat, infinitely large sheets behave in a very strict way. If they are not parallel, they will eventually intersect along a line. There's simply no spare direction for one to dodge the other, but move into four dimensions of space and everything changes. These sheets gain an extra direction to tilt into, so they can slip past each other without touching. It's the same idea as two lines in a flat 2D world. If they aren't parallel, they must eventually cross. But at a third dimension, and now lines that aren't parallel don't need to touch. Sheets or planes in 40 enjoy that extra freedom too. They can avoid each other in ways that seem impossible from our 3D point of view.