The dialogue of Book VII continues to explain why we must use only our intellect and the power of logic to understand the true nature of things, for only the dialectic goes to the most basic principle and works up from there without any use of hypotheses. "When a person starts on the discovery of the absolute by the light of reason only, and without any assistance of sense and perseveres until by pure intelligence he arrives at the perception of the absolute good, he at last finds himself at the end of the intellectual world as in the case of sight at the end of the visible." (Plato, translation by Jowett) This is a wonderful way of thinking about how we may come to understand higher dimensions, for it shows that the fact that we cannot see them with our eyes is purely superfluous since we can come to understand things even more truly and accurately when we use the dialectic rather than perception.
The allegory addresses the issue of what happens as one adjusts to viewing beyond the limited scope of one's senses--it is disorienting, perplexing and seems more illusory than the more limited reality which was previously thought to be limitless. Eventually, one adjusts and can understand either the higher dimension or the higher good that is out there to be known by intellectual reasoning and then the opposite transition is equally difficult--going back into the more narrow world can be something so undesirable that one may not want to go back and the original society may ostracize the enlightened. This ostracism was experienced both by A Square after journeying to spaceland and by Gulliver after his fantastic travels.
One last topic I will touch on in this short paper is Plato's beautiful explanation of why mathematical understanding is infinitely valuable on its own, not only 'even' when one neglects physical applications but precisely when you neglect the 'real world' uses of it. "I must add how charming the science is! And in how many ways it conduces to our desired end, if pursued in the spirit of a philosopher, and not of a shopkeeper! [...] I mean, as I was saying, that arithmetic has a very great and elevating effect, compelling the soul to reason about abstract number, and rebelling against the introduction of visible or tangible objects into the argument." (ibid.) It is all too easy to confuse a real-world representation with the mathematical ideal--especially when one thinks of mathematical objects which are easily or commonly represented such as numbers, arithmetic operations or two- and three-dimensional shapes. We can never see any mathematical ideal with our eyes (or other senses, for that matter); we must only strive to know them dialectically. This is good to keep in mind in general, and useful in particular to reflect upon when imagining how fourth dimensional objects 'look'.