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.

String theory postulates ten or eleven dimensions, only three of which we can “travel” in. Time is a fourth dimension that we experience “now” and can experience “the past” through memory, but we have no knowledge of “the future”.One could say that we traveling in only one direction on the time dimension, and cannot reliably control our transit through time. (Although when one is bored, time certainly approaches stopping!)The other six or seven dimensions are often referred to being “curled up” in a microscopic manner so that we cannot experience them. This is not something that is intuitive, and is tough to keep straight.

One alternative to envisioning these other dimensions as curled is to postulate a brane – a flat surface that is analogous to an infinitely large and thin piece of paper, but in two (or more) of the “other dimensions”. Any entity that inhabited a brane would have no or little knowledge of anything the might exist outside of the brane.

The existance of branes, facilitates string theory to describe gravity, and could be the source of missing “dark matter” and where particles in an adjacent brane reduce gravity to the weaker than expected effects of recent experiments.

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