People who study science and engineering discover at some point the idea of dimensions. Not parallel dimensions, although that a fascinating (and absolutely scientific!) topic as well, but space dimensions as height, length, and width (depth), also known as spatial dimensions. At that point, one of the best exercises one can do, and normally natively does is to start wondering about higher or lower dimensions. How would a four-dimensional world look like? Could it be that our world is four- or even five-dimensional? And what about a two-dimensional world, what limitations it will have?
Higher or lower dimensions take a serious place in all aspects of our culture: In mathematics, physics, biology, and engineering; In Science-fiction, ancient fables, cinema, and computer games. And of course, in art. Today we start a new mathematical multi-dimensional journey. No one knows where it will take us and what surreal and magnificent lands we will discover on its way, but surely, we will document it all in our future articles in the Dimensions series.
But first thing first, a proper introduction is needed, and what introduction can be better than a novella by Edwin A. Abbott, named “Flatland: A Romance of Many Dimensions”. Edwin Abbott Abbott (1838-1926) studied classics, mathematics, and theology and became the headmaster of City of London School when he was only 26 years old. In 1884 he wrote Flatland, which later became his best-known work. The novella tells the story of creatures that live in a two-dimensional world, and have an absolutely different point of view, literally, on life.
The story is told from a point of view of a square, a Flatlander that describes a hierarchical and satirical Victorian-like culture. The square explains the limitations of his two-dimensional world: how its habitats see each other, how a house in Flatland should be built, why line-segments are practically invisible for him, how gravity, color, rain, and fog work in Flatland, and more. The square also describes his transcendental journey into a one-, zero- and three-dimensional worlds, sharing along the way his thoughts about a four-dimensional world.
Without further ado, let’s dive into the first chapter of the book:
I call our world Flatland, not because we call it so, but to make its nature clearer to you, my happy readers, who are privileged to live in Space.
Imagine a vast sheet of paper on which straight Lines, Triangles, Squares, Pentagons, Hexagons, and other figures, instead of remaining fixed in their places, move freely about, on or in the surface, but without the power of rising above or sinking below it, very much like shadows—only hard and with luminous edges—and you will then have a pretty correct notion of my country and countrymen. Alas, a few years ago, I should have said “my universe”: but now my mind has been opened to higher views of things.
In such a country, you will perceive at once that it is impossible that there should be anything of what you call a “solid” kind; but I dare say you will suppose that we could at least distinguish by sight the Triangles, Squares, and other figures, moving about as I have described them. On the contrary, we could see nothing of the kind, not at least so as to distinguish one figure from another. Nothing was visible, nor could be visible, to us, except straight Lines; and the necessity of this I will speedily demonstrate.
Place a penny on the middle of one of your tables in Space; and leaning over it, look down upon it. It will appear a circle. But now, drawing back to the edge of the table, gradually lower your eye (thus bringing yourself more and more into the condition of the inhabitants of Flatland), and you will find the penny becoming more and more oval to your view; and at last when you have placed your eye exactly on the edge of the table (so that you are, as it were, actually a Flatlander) the penny will then have ceased to appear oval at all, and will have become, so far as you can see, a straight line.
The same thing would happen if you were to treat in the same way a Triangle, or Square, or any other figure cut out of pasteboard. As soon
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as you look at it with your eye on the edge of the table, you will find that it ceases to appear to you a figure, and that it becomes in appearance a straight line. Take for example an equilateral Triangle—who represents with us a Tradesman of the respectable class. Fig. 1 represents the Tradesman as you would see him while you were bending over him from above; figs. 2 and 3 represent the Tradesman, as you would see him if your eye were close to the level, or all but on the level of the table; and if your eye were quite on the level of the table (and that is how we see him in Flatland) you would see nothing but a straight line.
When I was in Spaceland I heard that your sailors have very similar experiences while they traverse your seas and discern some distant island or coast lying on the horizon. The far-off land may have bays, forelands, angles in and out to any number and extent; yet at a distance you see none of these (unless indeed your sun shines bright upon them, revealing the projections and retirements by means of light and shade), nothing but a grey unbroken line upon the water.
Well, that is just what we see when one of our triangular or other acquaintances comes towards us in Flatland. As there is neither sun with us, nor any light of such a kind as to make shadows, we have none of the helps to the sight that you have in Spaceland. If our friend comes close to us we see his line becomes larger; if he leaves us it becomes smaller: but still he looks like a straight line; be he a Triangle, Square, Pentagon, Hexagon, Circle, what you will—a straight Line he looks and nothing else.
You may perhaps ask how under these disadvantageous circumstances we are able to distinguish our friends from one another: but the answer to this very natural question will be more fitly and easily given when I come to describe the inhabitants of Flatland. For the present let me defer this subject, and say a word or two about the climate and houses in our country.
Indeed, life in Flatland isn’t easy. We won’t spoil it for you, but it gets much more interesting. We wrote about math in the art (The Infinite Painting) and literature (in our Alice series – part 1, part 2) and if mathematics and literature are your cup-of-tea, we strongly recommend you read the book. In it, mathematics takes the main part, and yet it’s absolutely readable and understandable to the most common reader – yes, math can be that easy, or as Isaac Asimov described Flatland in his foreword to the Signet Classics 1984 edition: “The best introduction one can find into the manner of perceiving dimensions.”
Ironically, the novel wasn’t a great success upon its release. In the entry on Edwin Abbott in the Dictionary of National Biography, Flatland is not even mentioned. But it was re-discovered after Albert Einstein’s general theory of relativity was published. In a sense, Abbott was ahead of his time, a visionary due to his intuition of the importance of time to explain certain phenomena. The simple and intuitive way he described the idea of dimensions guide us to these days. Definitely, a must-read.
Next in our Dimensions series, we will share with you fables, games, related stories, and more. Stay tuned.
Featured image by 95C from Pixabay.
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