This is a seminal little book. Edwin A. Abbott‘s Flatlands, A Romance in Many Dimensions is what allows thousands of us to be able to visualize higher dimensions.
Flatlands is the story of a two dimensional person who has become aware of the existence of three dimensions.  He tells us, from a two dimensional perspective all about his world – its features, science, society, social classes, intriques.

What the story achieves within the first few chapters is to expose us three dimensional beings to what it means to live in a world constrained by the dimensions we inhabit. He lives in two and has learned about three.  We live in three dimensions universe and can be be aware of four or more additional dimensions by extrapolation.

For many, this is a difficult task – even with my hands free I cannot describe a four dimensional square, or tessaract.  Abbott has done this in an easy reading romp through our two dimensional friend’s world.

The world he describes is bizarre but understandable.  The first several chapters set up a framework to visualize higher dimensions, and these chapters should be required reading for every student planning to study solid geometry.

Abbott explores, in a matter of fact way, the social structure of his flat world.  Our flatlander friend’s description and opinions about his society also provide a framework for thinking about the society of our world – by extrapolation.  To understand this concept it is necessary to read the entire short book.  I am sure that his intention was to show that his flatlander’s class structure was just as arbitrary as Victorian society.