Introduction:
Hello, my fellow readers! I have a question to whisper in your ears, so come closer and try to hear me out silently. Don't you have a knack for delving into the deepest mysteries of the Universe, exploring the intergalactic dimensions? Come on, say yes, don't be shy. Everyone does! So, let us dive into exploring this together.
As you start to imagine the energies of the Universe assembling against the ultimate villain Thanos for Avengers: Endgame, and then you glance upon the enormity and vibrant realm incorporating the weirdness of outer space, it starts to give you the goosebumps of excitement and thrill, right? I get it. And I am here to enhance that feeling.
Credit: Avengers Endgame|IndiaTV News
Fascinating and super complex sounding theories like the String Theory or the M-Theory are things you must have surely heard of, right? The way these theories talk about the Universe having ten or eleven dimensions freaks everyone out. And nerds like me and you would start thinking if only we could travel across the intergalactic dimensions, it would have been amazing! And maybe in the future, one of you who is reading this article shall find a way to achieve dimensional travel. But before doing that, you must be aware of what dimensions are in reality. How do we define them? And why even bother? I will try to answer it all.
What is a Dimension?
Well, you know, “The dimension of an object is defined to be the topological measure of the size of its covering properties.” And that makes no sense, I know. So, we will try to understand the concept of dimensions at a much basic level.
Credit: Self-Made
Take a look at the image above. The first column represents a “point,” which is known to have no length, no breadth, and no height. Hence, in mathematics, a “point” is referred to be zero-dimensional (0 D). What if we place infinite such points parallelly across each other like in column two? You guessed it right; it will make a “line,” which is known to have a finite length, and hence, is considered one-dimensional (1 D). And how about if we place infinite such lines parallelly? Wouldn’t it lead to a “plane,” which has a finite length and breadth? Yes, it would. And so, a “plane” is referred to be a two-dimensional (2 D) object. Similarly, if we stack up planes one upon the other as is shown in the fourth column, we would get a three-dimensional (3 D) space with finite length, breadth, and height.
The "?" Sign and Higher Dimensions:
Now notice something interesting. The fifth column has a “?” sign. But, it should have had a picture of a four-dimensional object, right? Well, there is some mystery hiding behind. Let’s see.
Credit: Self-Made
What if I ask you to draw two lines that are ‘perpendicular’ (90 degrees) to each other? Well, you will surely say, that is easy, and the picture will pretty much look like the third column in the image above. However, if I ask you to draw two “points” that are ‘perpendicular’ to each other, can you do that? That might sound like a tricky question at first glance. Still, if you wonder about it for a few minutes, you will soon realize that any two “adjacent” points are ‘perpendicular’ to each other (mind exercise for my fellow readers). And as described above in the previous paragraph, if you keep placing points adjacent to each other ‘perpendicularly’ one after the other, it forms a one-dimensional (1 D) line. Similarly, if you start putting lines ‘perpendicularly’ one upon the other, you get a two-dimensional (2 D) plane. And this is how it goes on.
Now, take a look at the image above carefully. I will try to analyze it column by column. Looking at the first column, we can see that there is no measurable direction, and hence, we have represented it as a point (0 D). The second column has one ‘unique’ direction, which indicates that the object is 1 D. The third column has two ‘unique’ directions, meaning it to be a 2 D object. The fourth column has three such ‘unique’ directions, which makes it a 3 D object. Now, notice carefully. Throughout the paragraph, I have put two words under single quotes every time they came into the picture, the words being, ‘perpendicular’ and ‘unique’. Here is an exercise for you all again. Think about why I thought of doing such [Hint: The words ‘unique’ and ‘perpendicular’ are more closely related than you might think, the relation is so significant that it has a particular mathematical term for it, i.e., “mutual orthogonality”].
Once you figure out the reason, understanding what I will say next will become much easier. If you look at the fourth column of the image, you will see three unique directions that are all perpendicular to each other. So, what if I ask you, can you add one more unique direction that is perpendicular to all the three pre-existing directions in the fourth column? When you start to visualize this question, your mind is going to boggle up. You will see that your brain is restricting you from imagining such an object with four “uniquely” perpendicular directions. And this is the mystery lying behind the “?” sign in the fifth column of the image, the concept of four-dimension (4 D). Given our capacity of imagination, understanding the geometry of a four-dimensional object can be very perplexing. But I know you, fellow readers, very well. You wish to see how a 4 D object looks like, right? Fine, here is an image for you [A mind exercise for you all again, stare at the picture carefully for a few minutes, and try to see how there can exist four uniquely perpendicular directions].
Credit: Tesseract|Wikipedia
Conclusion and Mystery:
Finally, I believe, you understand that the concept of four-dimensions and dimensions higher than that like the fifth, sixth, and so on, are extremely difficult to visualize geometrically. But if that is so, how do we work with such higher dimensions mathematically if we cannot imagine them properly? Also, so far, we talked about dimensions like 0 D, 1 D, 2 D, 3 D, and 4 D. However, 0, 1, 2, 3, 4, and so on, all these are whole numbers. But, based on our number system, there are fractional numbers, irrational numbers, and even imaginary numbers! Does that mean we can have fractional or even imaginary dimensions? Well, that is a long story for an entirely new article. Stay tuned until then, my fellow readers. Enjoy exploring the beautiful Cosmos you live in!
References:
[1] English, Trevor. “Understanding the Fourth Dimension From Our 3D Perspective.” Interesting Engineering, Interesting Engineering, 12 Mar. 2018, interestingengineering.com/understanding-fourth-dimension-3d-perspective.
[2] Su, Francis E., et al. “Fractional Dimensions.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.
[3] Weisstein, Eric W. "Dimension." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Dimension.html