Rating - 3: worth reading once (borrow it from a library)
As a story or a form of entertainment, this book fails. As an idea or a catalyst for interesting thought, this book succeeds. You will need to do much of the mental work yourself after the path has been sketched.
A. Square lives in a literal two-dimensional world. He is a four-sided regular polygon, the proud father of several pentagons, and a mathematician. First, he would like to tell you about his world, and how one gets along without having an up or down. Next, he would like to tell you about our world, and the wondrous revelation this third dimension is.
Let us be clear from the start, there is only the slightest plot here. The structure is much like
Gulliver's Travels, in which we meet a new world, have its interesting notions explored, then we have a little adventure there. That little adventure could be adequately summarized as "I met someone from the third dimension, and he showed me what 2-D and 3-D mean." This is not a gripping yarn or a rollicking tale.
Instead, we face the difficulties of being two-dimensional. We the 3-D think of two-dimensional space like what we see on television or in
video games. One must remember that we are looking from without, perpendicular to their world. We see a plane; they see only a line. So that is hard.
We see the society, culture, and history of Flatland, where a caste system divides people based on their numbers of sides. If you have more sides, you are a better person, and you will be educated and treated accordingly. Families are upwardly aspirational, seeking more sides and more regular angles. Women are carefully controlled, being dangerous two-dimensional figures.
"Women" must be a broad metaphor, since a line cannot be pregnant with a hexagon or such. The assignment of sex is arbitrary, although it suits the satirical purposes. Also like Swift, Flatland's society is a mirror of Britain and modern man; despite my edition's introduction, many things have not remained timeless and relevant to the modern day. Still, most of the allegories are sufficiently obvious.
There are two main ideas to play with here, which I will characterize as philosophical. First, what would it be like to be two-dimensional? A. Square notes many aspects that he skips over, and indeed many things of the story would seem impossible given the constraints of living polygons. Methods of movement, reproduction, and communication are at best non-obvious. We deal with most of it by hand-waving and suspension of disbelief, but this is a notion you can play with.
Second, what would it be like to be four-dimensional? Alternately, how would we recognize being in a 4-D universe? Despite A. Square's experience, it seems unlikely that a 2-D being pulled into a 3-D world would know what he was seeing, even if his sense organs were up to the job. If you had never seen anything, it would take you a while to connect your previous sensations to your new sights, especially if you could not interact with them.
But I go astray. How would one demonstrate being a four-dimensional being to your satisfaction? How would you deal with the influence of beings who violate your notions of space through some interaction with time or probability or whatever fourth dimension they use to circumvent your three? Practically, yes, we now believe that we live in four-dimensional space-time, but time seems not quite the same as the rest, and at worst we push it back to questions about a 5-D world. (General case: "n+1 dimensional world, discuss!")
In many ways,
Flatland accomplishes what
God's Debris does not as a primer to thought: "Here are some notions about the nature of the universe. Go play."
Wikipedia lists several books that have played with the idea.
Bonus topic for discussion: compare and contrast the King of Pointland with an omniscient, omnipresent God.
You do not need to be mathematically inclined to get it. There is no math as such in the book. If you know what a hexagon is, you should be good to go.
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