It is worth thinking about when we are asked to consider that there may be more than three dimensions of space.
Well, in part at least. Author Edwin A. Abbott (1838-1926) clearly intended quite a lot of social commentary as well.
Of course, the subject of two-dimensional thinking - or even one-dimensional thinking - obviously suggests a chance for an author to condemn short-sighted social policy. But still, in addition to his two-dimensional world, Abbott also offers us a geometrical glimpse into a one-dimensional world, dominated inevitably by a supreme egotist. Needless to say, it is not a very interesting or promising place.
Having read this short book, available on line, I think it is a good way to understand dimensions. It may help us grasp what additional dimensions in our own universe would be like (assuming there are any).
For example, Abbott makes the two-dimensional Flatlander in his story confront his three-dimensional companion,
But, just as there was the realm of Flatland, though that poor puny Lineland Monarch could neither turn to left nor right to discern it, and just as there was close at hand, and touching my frame, the land of Three Dimensions, though I, blind senseless wretch, had no power to touch it, no eye in my interior to discern it, so of a surety there is a Fourth Dimension, which my Lord perceives with the inner eye of thought. And that it must exist my Lord himself has taught me. Or can he have forgotten what he himself imparted to his servant?Well, in the story, the three-dimensional human being obviously enjoys his superiority and can simply not easily confront that idea.
In One Dimension, did not a moving Point produce a Line with two terminal points?
In Two Dimensions, did not a moving Line produce a Square with four terminal points?
In Three Dimensions, did not a moving Square produce - did not this eye of mine behold it - that blessed Being, a Cube, with eight terminal points?
And in Four Dimensions shall not a moving Cube - alas, for Analogy, and alas for the Progress of Truth, if it be not so - shall not, I say, the motion of a divine Cube result in a still more divine Organization with sixteen terminal points?
Behold the infallible confirmation of the Series, 2, 4, 8, 16: is not this a Geometrical Progression?
Go here to buy the book, with Ian Stewart's notes. Most readers will benefit from the notes, I expect.