Flatland, the Fourth Dimension and Our Imagination

Imagine if we are to live in a world of two dimensions. We are all absolutely flat.  That is, we only know about the dimensions of length and width. Our world is flat. Our movement will be limited.  We only know about moving forward, backward or sidewards. We are living in Flatland. Everything has two dimensions.  Perhaps our fellow creatures will be a square, a rectangle or any other regular or irregular polygons. Others can be a circle or any other two foci planar objects. All of us are two dimensional objects existing in a flat world.

This is Flatland (topview), I am that circle. lolx

And obviously at first glance, we will not recognize one another. Which one is a square or a circle. For our vision will be limited as well.  Imagine. Everything we see around us will resemble a line. Now, pretend you are a thin square paper (like that dark red square) carefully placed on top of a table. And next to you is another thin sheet of paper resembling a circle (like that yellow circle). And you look towards your neighbor, you’ll just see a line. For you can’t go above it to see that it is circle.  Nor can you go below it. Because, you are both living in a flat world (which is the table in this case). We are living on a plane. Everything is coplanar. And the horizon will be monotonous. We cannot comprehend a third dimension, which is height, because we have never heard of it. Our existence is confined only within two dimensions. We will never know what is up or down. What is tall. What is above or below. We never know because we have never seen it. Because we can’t. We go by our everyday routine in flatland and never can we imagine that a world having three dimensions is possible. Unless someone living from a three dimensional world comes and takes you to his world. You will be amazed and astounded to see a three dimensional world. That there is a world beyond your existence. You will be able to distinguish, what is above, what is below, and what is high and low. You will see a new world you have never seen before. You will see another dimension. It was utterly impossible to have thought of such a world with an extra dimension, but there it is, right in front of your eyes. And you try to remember everything. You are just amazingly astonished! And you will be allowed to go back to Flatland. And now, how can you explain the concept of a third dimension to your fellow flatlanders? Height. Which way is up? Which way is down? What is a solid cube? You’ll explain that it is composed of numerous similar squares placed on top of one another. But then, what do you mean by on top of one another? What do you mean by height? When all of your world’s history you have known only two dimensions, just length and width.

This is the same dilema an entity living in a three dimensional world has when faced with a question if there indeed exists a fourth dimension. And so goes also with the fifth, sixth, seventh..etc. dimensions. This is like explaining the concept of tall buildings to a humble bacteria.

Imagine also that there is a world called Pointland. All  the inhabitants of Pointlant are points.  And how about a world called Lineland, where all creatures are just lines.  And their world is just one long long line. How is that? lolx.

Let us stretch our imagination even further. If you grab a book and place it below a source of light. The book is three dimensional having a width, a length and a height. The shadow it casts, because of the light projected to it, would simply be two dimensional. This three dimensional object can be encoded into a two dimensional entity. Like a photograph. Imagine, if you were a photograph. Could you even believe that you were made out of three dimensional objects shot out of a camera. Perhaps it will be outworldly to think like that.

In contemplating our own existence, we can always rely to the figments of our imagination. What world is beyond the three dimensional world that we have? Sometimes to think outworldly is sensibly necessary.

The best way to relate to us the concept of a fourth dimension is by using our knowledge of the hypercube. In geometry, a hypercube is an  n-dimensional analogue of a square (n = 2) and a cube (n = 3) n stands for the number of dimensions. It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space’s dimensions, at right angles to each other and of the same length. A point is a hypercube having one dimension. When a point is connected to another point, which is exactly one unit length and forms one line segment, it becomes a hypercube having one dimension. If a line segment is moved in a perpendicular direction from itself with the same exact length, it will form a square. This is a hypercube having two dimensions. If one moves on square unit in a direction perpendicular to the plane it lies, it forms a cube. A cube is a hypercube having three dimensions. The same goes with the cube, it you move a cube with the same unit length to the fourth dimension it will form a tesseract, a hypercube having four dimensions. Imagine cube is extended one unit length of itself having opposite parallel line segments aligned in each of the space’s dimensions, at right angles to each other and of the same length. Whew. Keep thinking. In geometry, the tesseract, also called an 8-cell or regular octachoron, is the four-dimensional analog of the cube. The tesseract is to the cube as the cube is to the square.

Now I too am getting dizzy. lolx. Let’s relate this sequencing of dimensions to casting shadows. Say these objects are  positioned parallel with the direction of the source light. A three dimensional object, like a cube for example, casts a two dimensional shadow. A two dimensional object, like a square casts a one dimensional entity, a line. And a one dimensional object, like the line will cast a zero dimensional shadow, a point of non-light. This is a clear analogy on how a four dimensional object in a four dimensional world will cast a three dimensional shadow.

Dimensional analogy will help us perceive the basic properties of higher dimensional objects.  Now, can you imagine what would a four dimensional world would be?

This is for our imagination. We humans, have always been fascinated with the idea of higher dimensions. But somehow, let me point out that what I am referring to  now is the fourth spatial dimension. The fourth spatial dimension is totally different from the fourth Spacetime dimension which is the temporal dimension, Time. In Theoretical Physics, Spacetime is composed of three spatial dimensions – length, width and height and one temporal dimension which is Time.

Our physical universe can be defined using these spatial and/or temporal dimensions. But the question is still open as to how many dimensions is needed to describe our universe. Who knows an entity from the fourth dimension may appear suddenly infront of us and take us to their world and show us how a four dimensional world looks like. lolx.

Well, perhaps these kind of out-of-this-world realities can explain some physical events in the universe only that us humans still have to figure them out.

We always have to realize that when it comes to the unthinkable, in this world of multiple dimensions, everything is possible. Everything.