In my opinion, the best popular mathematics book ever written is "Flatland: A Romance of Many Dimensions" by Edwin A. Abbot. Designed to be as much a satire on nineteenth century British society as a book on spatial dimensions, I have seen excerpts from it in several books on multi-dimensional space. This book opens with an excerpt from Flatland, and then moves on to describe how the universe of the Flatlanders is changing due to the universal expansion of their living surface.
The social commentary continues in "Sphereland" as newborns are killed simply because their parents prefer large angles in their shapes. Dogs are either of mongrel or pedigree stock depending on which mirror image they are. Finally, Burger also satirizes the intellectual segment, when the findings of the knowledgeable experimentalist are almost rejected out of hand as they do not conform to current physical theories. The sphere continues to make periodic appearances in "Sphereland", but it is also subject to emotional frailty, becoming quite angry at some of the suggestions the humble Flatlanders make about the world he inhabits.
While it does not match the high quality of Flatland, Sphereland is still an excellent description of the differences in the properties of spaces as additional dimensions are added. Being restricted to three dimensions, humans can only conceive of space with more than three dimensions with mathematics or by analogy. Both are used here and it is done well without resort to sophisticated mathematics. The satire also is of a high quality, although it must be read with an eye to the fact that it is an extension of the social commentary of Abbot. He was a strong advocate for the emancipation of women, working very hard to promote womens rights, and Flatland was only one part of that struggle.
Sphereland is a worthy sequel to a book that sets the standard for popular mathematics and social commentary. As long as you read it as that, it will be very instructive and amusing.
