It is difficult – if not near-impossible – for human beings to visualise higher dimensional space. It is like explaining colours to a man born blind, as the protagonist PW said in his lecture in Chapter One. We are not able to visualise higher dimensions because of lack of evolutionary opportunities, i.e. our ancestors had no need to think what a 4-D space looks like, because savage beasts did not jump out at them from the higher dimensions.
Also, our relationship with our world is built on what we can see, and as we can only see 3-D, so we are stuck in a 3-D reality. Most of us go through life without missing the ability to see higher dimensions: we are content with the colours of the seasons and the shapes of modern architectures.
But imagine if you can…..
Early one October morning, when we were walking the dogs on the beach in Sandown on the Isle of Wight, I came across this structure under the old pier. If you stand at the right spot, you will see an infinite number of cubes receding into infinity.
For me, it is a lot like a a four-dimensional hypercube, known as a tesseract. Basically, the tesseract is an extension of the 3-D cube to one more perpendicular axis, and it is the simplest form of higher dimensional structure. Imagine being inside a tesseract! It would like being in a room with each window opening to a different time and space, like in the film Interstellar. If you want to watch a pure tesseract movie, here it is: http://www.imdb.com/title/tt0285492/
If you want to develop your spatial abilities, try drawing a tesseract. It is just a cube within a cube:
And here is a lovely youtube clip about unwrapping a tesseract: https://www.youtube.com/watch?v=BVo2igbFSPE