[A little cartoon story in 1, 2 and 3 dimensions – inspired by Edwin Abbott’s novella ‘Flatland’ – exploring how perspective shapes what we can see and know.]
You and I are encased in time and space. The extents of what we can see and know are controlled by this fact. This theme was explored in Edwin A. Abbot’s 1884 novella Flatland, to which I am indebted for providing the inspiration for this short allegorical tale.
[Note: because 0D points are infinitely small and therefore hard to see, points are shown here as small circles.]
Imagine that you and I exist in a 1D universe. You and I are points, because points are all that exist in this universe. Our entire existence consists of movement up and down one dimension, one line. We can travel along the line in either direction, but we cannot escape it. We cannot see beyond it. It is our entire universe.
When you look down the line at me, all you see is a point, because that is all I am. No matter how near or far away I am, you simply see a point.
Now imagine that another being joins the line at a place between you and I. This third being is also a point, a red point. It has to be a point, because only points can live in the 1D universe.
All you and I can now see is the red point. Neither of us can see beyond it. The appearance of the red point on the line has shut out our vision of each other.
Now imagine that the 1D line, the universe in which you and I live, actually exists within another universe. This second universe is a 2D universe: a plane. You and I and our 1D line exist within this 2D universe, but we are still constrained to our 1D existence. We cannot move outside the line, and we can see nothing except what is on the line. This means we have no solid concept of the 2D universe. We cannot see it, and we cannot understand it.
Now imagine that outside the line, in the 2D universe, is a third being. This 2D being is called Bob. Bob is constrained to the 2D universe, but he has an extra dimension to you and I. Bob is not a simple point; Bob is a 2D star.
The limits of Bob’s 2D universe are described by the black rectangle. Unlike you and I, he is not constrained to the 1D line. Within the limit of his 2D universe, Bob can move and act freely.
This 2D freedom gives Bob some powers that you and I in our 1D universe do not have. For example, from his perspective, Bob can see how far apart you and I are on our line. Being stuck on the line, you and I have no way of telling how far apart we are from one another. No matter what the distance between us, we always see each other as identically sized points. Compared to us, the vantage point of the 2D universe gives Bob understanding. There are things Bob knows about us that we cannot know about ourselves.
Bob’s 2D freedoms also let him interact with the 1D line on which you and I exist. Bob has power over us, because our line is entirely contained within his 2D universe. Imagine now that Bob draws a green line across our 1D universe.
From our perspective, a green point has suddenly appeared in our universe. Constrained to our line, we cannot see Bob, or the full extent of the line that he has drawn. All we see is the single point where his green line crosses our 1D universe. However long or short a line he draws, this green point is all we see.
It is actually impossible for us to imagine the shape of the line that Bob has drawn. In our 1D universe, all we can ever see are points. Though we ourselves live on a line, we can have no concept of what a line actually looks like. We have no vantage point to see the line on which we are fixed. Bob’s shape, and even Bob’s existence, are unknown to us. All we can see when someone or something in the 2D universe crosses our 1D line is a point. The point we see is only a shadow of who and what that being or object really is.
But drawing shapes is not the only way that Bob can interact with our universe. Bob could actually cross the line to make himself known to us.
But we will not see Bob as a star. As it was with the green line, all we would see a shadow of Bob: a blue point.
We could never know that Bob is in fact a a star. If we believed on the basis of what we could see that Bob was a blue point and nothing more, we would be wrong. Bob’s identity cannot be constrained to the limited perspective we have of him. The blue point communicates something true about him, but it is not the whole truth. How could we, who only ever see and know points, begin to understand Bob’s 2D ‘star-ness’? What we do know of Bob is not false knowledge. We see that Bob exists, we see that Bob is blue; neither of these are false observations. But we would be in error if we thought that Bob can only be a point simply because we ourselves are points. Bob’s full identity remains outside our understanding.
Now let us imagine that Bob is joined by two friends, Amelie and Tim. Like Bob, Amelie and Tim each have a 2D shape and exist in the 2D universe.
Bob and his friends are all constrained to the universe of the 2D plane. This means that all any of them can see in any direction is a 1D line. This is similar to how you and I, stuck on our 1D line, can only ever see an individual ‘0D’ point. Having one more dimension available to them than you and I do, Bob, Amelie and Tim see a single 1D line. As they look out, they see the black line that marks the edge of their 2D universe. They can see each other, but only as coloured 1D segments on that line. So, this is what Bob sees when he looks at Amelie and Tim.
Bob has no way to see what is inside the 2D shapes of Tim and Amelie. He only sees how wide they are, as defined by the length of the coloured segment. If Bob thought that on this basis he knew everything about Tim and Amelie, he would be wrong. He has no way of knowing that Amelie is a 2D sun, and that Tim is a 2D hexagon. Nor can he see that Tim is broken inside.
Just as Bob was able to modify the 1D universe of our line, he can make other changes by drawing in his 2D universe. Imagine now that Bob draws a red line around Amelie.
Now Amelie can no longer see Bob or Tim. The only thing that Amelie can see in every direction is a 1D red line.
Bob and Tim are still there of course, but Amelie cannot see them, and they cannot see her.
Of course, the narrator and the reader of this story can see things that you, I, Bob, Tim and Amelie cannot see about ourselves or each other. They can also see things that none of us can see about the 1D and 2D universes in which we live.
That’s because narrator and reader exist in 3D space. The narrator is able to view both the 2D universe, and the 1D universe within it. He sees both of them from an external viewpoint. The third dimension of the reality in which they live allows the narrator and the reader to stand back and see the whole.
Just as you and I on the 1D line could not see Bob, Amelie and Tim; none of us in either the 1D or 2D universes can see the 3D narrator. He exists outside the 1D and 2D universes, and is able to see them without their citizens being able to see him. The narrator sees everything clearly. He sees our shapes. He sees all of us, and how we are related. He sees all of us, even though we can’t all see each other. He sees Tim’s brokenness. He sees that Amelie is trapped, and he sees what is trapping her. He sees the 1D line, and he sees you and I separated by a green line. But he sees more even than these things. He can understand why you and I in our 1D universe can’t see or comprehend the 2D universe, and he can understand why the citizens of the 2D universe can’t see or comprehend his 3D universe. Everything is open to the narrator’s gaze.
But this doesn’t mean that the narrator is unable enter or change the 2D universe or the 1D universe within it. Here, the narrator has placed an image of himself into the 2D universe, to introduce himself to Bob, Tim and Amelie.
Of course the image is not his full likeness. It is only a 2D image, whereas the narrator is a 3D being. And only a 3D being standing outside the 2D universe could possibly see what the narrator truly looks like. The citizens of the 2D universe can now see something of the narrator, but it is only see a shadow of his true being. They see a 1D line, with coloured segments. They see the brown of the narrator’s hair, and the pink of his skin.
If Bob, Tim and Amelie believed on the basis of what they could see that the narrator was a brown line segment connected to a pink line segment and nothing more, they would be wrong. The narrator’s identity cannot be constrained to the limited perspective they have of him. The brown and pink segments communicate something true about him, but it is not the whole truth. How could Bob, Tim and Amelie, who only ever see and know 1D lines, begin to understand the narrator’s 3D form? The narrator’s full identity and the reality he inhabits remain outside their understanding.
The narrator can make himself known, but only as a shadow of his true identity. He is unseen until he wants to be seen. He has power to change both the 2D universe and the 1D universe within it. He can surround their universes, stretch them, shrink them, fold them, add and remove beings and objects from them. He can draw and he can erase. He is omnipotent and omniscient over all. Yet, he can come to meet them, placing himself in their realities, condescending to their limited perspectives. He can reveal a shadow of himself.
We are apt to dismiss things we cannot easily understand. We discount things that cannot easily be fitted into our conceptual frameworks. What things might we have discounted as unlikely or absurd, just because they were beyond our full seeing or understanding?