Have you ever attempted to invent a new color? I don’t mean a different shade of green or a blend of oranges and reds. I mean a color not previously conceived: a color that would completely alter the color wheel.
As a child, I often attempted to complete this task, and of course I always met frustration. It seems somewhat clear that if we “invent” a new color, that new color must be formed from the colors we have already, but do we really know this to be true? Edwin A. Abbott’s Flatland gives us clarity on this subject.
Abbott’s Flatland describes a world of no height; that is, a two-dimensional plane whose citizens are of the form of polygons. Abbott, from the perspective of a square, extensively describes basic functions, the culture, societal constructions, and brief moments in history of this strange world. Included in his notes on Flatland anthropology (Abbott describes these living polygons as people, so we’ll adapt the term “anthropology” too), the square indicates that a man’s status is determined by the number of his sides, women are lines (a satirical jab of the society of the time), and to be irregular is nature’s way of declaring one a villain.
After describing the inner workings of Flatland, the square begins a narrative which begins with him having a dream of visiting another world: Lineland. As the name suggests, this world consisted of just one line, and the people on this line are either line segments or points fixed on this line.
Upon meeting the king of Lineland and a after some dialogue full of confusion, the square feels a sense of duty to enlighten the king to the concept of “left and right”. But the king’s world consists only of north and south. There is no east and west. There is no left and right. The king’s only sense of vision is the point in front of him. The king cannot even conceive of the notion of a square; from his perspective, the square is just a point that disappears using some sort of trickery (As the square can move left and right, but the king only sees the point where the square intersects his vision.). The square’s interaction with Lineland ends in fruitless discourse.
Upon awakening in Flatland, the square finds himself haunted by a strange visitor… I don’t want to give away the story, because I would like you to find out for yourself what happens. Suffice it to say, though, that the square has a revelation of his own existence and the limitations of living in Flatland.*
This semester, I assigned some excerpts of Flatland for my Linear Algebra class to read and to discuss the concept of other dimensions. A principle theme of Linear Algebra is the mathematical structure known as a vector space. You are familiar with certain vector spaces if you ever learned grade school arithmetic (R_1 – the set of real numbers with standard operations), drew lines in the Cartesian plane (R_2 – the set of (x,y) coordinates consisting of real numbers with standard operations), or felt the wind blowing (R_3, standard three dimensional space).
Among the properties of vector spaces is the property known as “closure”. Basically speaking, the closure property states that adding or multiplying objects in a vector space results in an object that is also a part of that vector space. For instance, if you add two real numbers, you get another real number, and it is impossible to get a complex number through the addition of two real numbers. (On the other hand, the collection of odd integers is not a vector space, because 3 + 5 = 8, and 8 is not an odd number. This collection does not have closure.)
I hope, dear and patient readers, that you are not disinterested due to some math-speak. I bring this up for a reason, and I believe Abbott was trying to say the same thing. The king of Lineland could not conceive a square, and he should not be condemned for his limited thinking. Why? Because his existence is a one-dimensional existence, and the square is a two-dimensional object. Because of closure, the thoughts of the citizens of Lineland are limited to things of Lineland.
The wise King Solomon wrote in his book Ecclesiastes, “What has been will be again, what has been done will be done again; there is nothing new under the sun.” Why can I never conceive of a new color? I am a creature of the Earth and its universe, and for that reason my thoughts are limited to this universe. We have the visible spectrum, the rainbow, and thus we have our colors.
In some sense, we are stuck. We are stuck to length, width, and height. We are subjected to physics, cause, and effect. But we are stuck in a very large environment. Though there may not be anything new under the sun, I believe there is still much to be discovered. To borrow another term from vector spaces, we have not yet spanned the universe we live in, and that is encouraging.
Flatland, though, is to challenge us to acknowledge that there are things that we cannot conceive. Many people, myself included, believe there is a spiritual interpretation to workings of Flatland. It is amazing that this 100-page, easily-read book can so eloquently challenge our thinking about who we are and our limitations. I feel that everyone should encounter Flatland at some point in their life.
Pages: 103 FOA Pages: 7,562 (The total number of pages reported upon by the Friends of Atticus)
* I did not really touch on the climatic movement of the story, because I don’t want to give anything away. This book is a journey that each reader should enjoy for himself (herself).