Many of the hottest and trendiest physics and mathematics theories nowadays refer to multiple dimensions.
But visualizing multiple dimensions is one of the hardest feats of imagination there are. Humans are born into a three dimensional world, and we always think in three dimensions. And as a core to that, we think of the three things we know as length, breadth and depth.
Visualizing other dimensions requires that we extrapolate from this: and even then, visualizing any more than four dimensions is particularly tough. A systematic approach to the visualization, then, is best.
1. 1D is simply a dimensional space with no length, breadth or depth. It is a dot that exists, but is so small you cannot see it from any direction. Think of it as a particle that is infinitely small in size, but is still there.
2. 2D is easier to visualize. 2D space is like a very, very thin sheet. That is, it has length and breadth, but no depth. If a paper could be made so thin that you could see it perfectly from up and below, but you could not see if from any other angle, that would be a bit what 2D looks like.
3. 3D is easy. We can visualize space as is. Depth, length breadth, the regular objects around us. They are all 3D. This is the easiest to visualize, because this is the dimensional space we can actually see.
4. 4D includes the 3 dimensions we normally know of, but add the time dimension. This is not a true dimension. This requires a fair bit of imagination: think of a railcar moving on a rail track. The railcar is our regular 3D world, and it moves only and only along the rail track, which is our fourth dimension, time. As with time, the railcar can only move forwards or backward, and not otherwise.
5. Higher order dimensions are even harder to visualize. Some scientists describe it as trying to imagine all of the previous order of dimensions curled up inside the next. This is a bi like 1D was ‘curled up’ inside 2D, and 2D in 3D, and 3D in 4D. So all our higher dimensions are very much like a system of Russian dolls: you crack open the 5D space to find a 4D space inside. This is not very palatable, but unfortunately, it is impossible to define it any more accurately because we cannot directly see any dimensions other than the 3.
This guide to visualizing higher dimensions is not meant to be strictly accurate in physical terms, but it should help spur the imagination.