Adding extra dimensions to a problem can allow new solutions to be found that were impossible before.

Consider a piece of string tied into a knot in our ordinary 3 dimensional world (ignoring time.), if you join the ends it's now impossible to untie it. Adding a fourth dimension to this problem does now however allow you to untie the string.

Imagine, if you will, that you colour the string according to each fourth dimensional coordinate along it's length . If you stood a simple length of string along a four dimensional axis, you could colour it in smooth graduations from blue to purple through to red.

Now the tricky bit, any bit of the string can be moved through any other bit, as long as it has a different colour. All you're doing is moving through the fourth dimension.... The string starts off in the same fourth dimensional coordinate and is all the same colour. Where the string crosses, it begins to change colour, and then move through the blocking piece of string of the knot. So you can untie the knot....