We live in a universe with four observed dimensions (3 space and 1 time). This universe is capable of harboring things (and even beings) of complex shapes and sizes. Some are cubical and some spherical, some are of irregular shapes. Atoms, planets and black holes.
A uni-dimensional universe
I assume that the universe we are taking about is a finite universe. We can think about at least three kinds of finite uni-dimensional universe:
- Open flat universe: can be represented by a straight line with a beginning and end.
- Closed circular universe: can be represented by a circle with no beginning and no end.
- Open curved universe: can be represented by a curve with beginning and end.
In this article, I will focus on the first type of universe: a finite open flat universe. I also assume that the universe I am talking about also enjoys time as we experience it in our three dimensional space. This universe is a straight line with a beginning (A) and an end (B). A model representation of such a universe is shown in Figure 1.
What kind of universe is this? In this universe, things or beings can exist anywhere between A and B. Nothing exist outside of this line AB. The universe AB as we can see above can never be observed like this by the beings inside the universe. This view is possible only for us who are observing from third dimension, the dimension that does not exist in the universe AB.
What kind of things (or even beings?) can potentially exist in a uni-dimensional space. What do they look like from second and even third dimension? How do they look at each other? What do they see on each other. How can they behave with respect to the universe and with each other? These are the questions I want to try to address in a purely mathematical sense with some anthropocentric narration.
Things or Beings
The only possible things/beings that can exist in this universe are either points or straight lines. Every organ or component of the point has to be the point itself. If it can hear, its through the same point. If it can see, the point itself is its eye. The line can have a little bit more space for its organs. A line is a locus of a point or collection of points. So everything in this universe is either a point or a collection of points.
Suppose each of the points in this universe that can ever exist is opaque for light as well as sound. Nothing can cross a point.
When a point sees another point, it will see a point. When a point sees a line, it again sees a point. A point can never see line as a line in a uni-dimensional universe. It can see of its coming closer, moving away or standing still by sensing how fast light beam bounces back and forth. The maximum the point can learn about other beings in this universe is two other points in its immediate neighborhood. All other points will not be observable for this point. Even if there are a million points, each point in this universe will die assuming that there are only three points: itself, one on the left and the other on the right. That is the maximum (yes maximum!) that the points will ever be able to observe. Even if there are some lines, the points will lever be able to know that its a line unless the line can communicate it being a line with the points in its neighborhood.
The two beings (points or lines) in this universe can never cross each other. If there is a race between two points towards B, then points closer to B will always win because the points who are behind this point can never overtake due to the lack of a second dimension.
If a circle from another dimension enters into this universe, it enters as two points such as X and Y in Figure 2. If an observer at point A looks at this alien entering its universe, it will suddenly see a point appear on its right side at X. This observer will be shocked as this kind of event is known to happen only in Hindu scriptures.
As the circle moves down, the point X will come closer and closer to the observer until the universe AB becomes the diameter of the circle. Then X will be seen as a point running away from A. Finally, the point X will disappear in a similar way lord Mahadeva disappears.
What an unlucky observer is this? The universe offers him a circle and he is able to see just a point appearing, coming closer, moving away and suddenly disappearing from the scene altogether.
Suppose the circle enters into this universe at exactly the point where the observer is standing. Like a point between X and Y. At first, the observer will be crushed by a point on its head. The point is X as well as Y; they coincide at the moment when the circle enters the AB universe. Provided that this observer survives the heavenly arrival of the circle on its head, which is itself a point.
Then, it will see two points X and Y on its two sides moving away from itself until a section of AB universe coincides with the diameter of the circle. Finally, the points X and Y will come closer and closer to the observer and would disappear as soon as it again hits its head.
Without calculus, the observer might not be able to say whether its a circle that entered its universe or its a tilted rectangle. If a rectangle enters this universe perpendicularly, the observer would see a point appear and then disappear forever.
Similarly, observers in a two dimensional flat universe (two dimensional universe can be of many types), can see one more things other than points. They can see lines also. But from a distance they will not be able to distinguish between a circle and a rectangle. Everything will be seen as a straight line of varying lengths.
If a sphere enters into this flat 2D universe, the observer will see a straight line suddenly appearing and increasing in length, then decreasing and suddenly disappearing from the scene.
Of course, I am assuming that the points and the lines have non-zero width and height (in 2D universe, this is not possible though) for things to be seen.
Imagine if we, in a three dimensional universe, are like the single point observer in the AB universe. Maybe there are other higher dimensions and the beings and things enter into ours out of nowhere and disappear in a blink of an eye. Maybe all the magical stories are true observations made by lucky observers. What a pity to those observers and us.
Before we have any solid evidence, its just a mathematical conjecture. And I am not indicating any mathematical proof of a god or a ghost in a finite dimensional universe we live in.