How to think about dimensions

OK, so I've finally resolved how to understand dimensions (it's only taken me about 15 years ;)). The key is asking "How do I make two points equivalent?".

So if we start with a line (1 dimension), we can define a point on the line by a single number. To make two points appear at the same place, we need to bend the line and make it cross itself (travel through a second dimension).

Similarly, if we have a 2-D plane of co-ordinates, to make two co-ordinates the same we need to bend the plane through a third dimension.

Here is the fun one. Let's say we have a 3-D space of points (x,y,z). To make two points equivalent we now need to bend through a fourth dimension. Generally we refer to this as time but then it is not measuring a distance like the others. What we really need is 'ct', time multiplied by the speed of light. This is equivalent to saying 'time stops when you travel at the speed of light'. In other words, if you want two points in space to be at the same place, travel between them at the speed of light. We could conceive of a 4-D wormhole which carries us from one point to another at the speed fo light.

So let us now say we have a 4-D space (x,y,z,ct) which defines a point in space at a point in time. To travel in time and space (like the TARDIS) and make two points equivalent, we need to fold 4-D space through a fifth dimension. We now need a 5-D wormhole which connects two points in space and time. This is pretty much where our brains stop because of a lack of everyday experience of such things.