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What I actually want to deal with here is size and not in the sense that we attribute to it, such as the size of a house or a city, but rather extra-dimensions and extra-dimensional worlds. So, what are dimensions, in this case? Let us simply say that they represent the number of directions in which an object can extend. A house, for example, extends in length, width, and height. It is a three-dimensional object. Can you imagine a house in two dimensions? What about four?

Let us literally start from zero: what is a zero-dimensional world like? It is a dot. There is not much else to say. Nothing can move, there is no extension of anything. There are no sentences like ‘this object is this big or that big’. To inform someone else on our whereabouts would not require any geographical coordinates. All the inhabitants of this world would be plain dots, identical to each other and piled on top of each other – assuming we accept that several dotted inhabitants of the zero-dimensional world could be piled one over the other in a single point.

Therefore, let us proceed to a world in which the inhabitants can distinguish themselves a little bit more from each other, and in which there is also a little more dynamism: the one-dimensional world. The whole world would be on one line (or a line segment). The inhabitants would be segments, so at least we could distinguish ourselves by length, and we would spend our lives moving left and right. Our position would be identified by a single geographical co-ordinate: ‘I am at point one hundred, join me’.

New step, new possibilities: the two-dimensional world. The inhabitants would be 2D geometric figures: circles, triangles, squares, hexagons and really any other shape you can draw on a sheet of paper. We could move left and right, but also up and down. To let a 2D-friend know where we are, two numbers would be needed, as in the Cartesian plane: the pair x,y unambiguously defines a point.

I think, the three-dimensional world come as no surprise, at least for those who are reading this article without LSD effects. The 3D world is the world in which we live in. We move right and left, up and down, back and forth. We even need altitude to tell us exactly where we are. It is true that Einstein taught us that our world is a four-dimensional world in which the fourth dimension is time, but in this article, we are completely ignoring the matter of “time” – our point of interest is solely space and geometry. So, for today we are going to neglect Einstein.

One could go on like this ad infinitum: 4D worlds, 5D, 100D, *etc*. The problem is that I cannot describe the 4D world to you. When I was a freshman in college, I had two particularly sharp and quirky mathematics professors. One claimed to be able to imagine a “tesseract” (a cube in 4D) – he sardonically said so, as if to say, “why can’t you?”. The other claimed that the former was lying. I hoped the latter was right, annoyed about the fact that no, I could not imagine it.

Despite this, we can still say something about both the 4-d world and higher dimensions. We can at least imagine what would happen if our world encountered a world of greater dimensions. To understand this, let us start with something slightly simpler. Suppose we inhabit the 2D world and meet a 3D object. What would we see? Simply the intersection of the 3D object with the 2D surface in which we are confined. Let us imagine a sphere approaching and crossing our world. At first, from a distance, we would not be able to see it because we lack the third dimension. It is outside our world. If the sphere got closer and begun to enter our world, we would see a small circle appear: the intersection of the beginning of the sphere with the plane in which we live. As the sphere continues to cross our plane, the circle would widen, reaching its maximum in the exact moment in which the sphere ought to be precisely halfway across (when it is cut off from the plane at its “equator” so to speak). Consequently, it would then begin to shrink, until it instantly disappears, since it is no longer in contact with our plane.

The fascinating side of this rather trivial example is that something that in a 3D perspective appears as elementary as the movement of a sphere would appear completely irrational to the inhabitants of the 2D world. We could picture them questioning the incomprehensible appearance of a circle and its disappearance into thin air. Consider the scientists of the 2D world faced with the indeterministic appearance and disappearance of objects without any explanation or causality. Their difficulty in imagining a 3D world would be identical to ours in imagining a 4D world.

And in the same way that the appearance of a 3D object would appear 2D in a two-dimensional world, the crossing of a 4D object in our world would look like a 3D object. By moving the object in the 4-dimensional world, we would see it change in size and shape in our world. A bit like in Braque’s extreme cubist paintings, in which the outlines of known objects are perceived, but seem to appear from an indefinite ‘elsewhere’. The world is our own, but it is intersected by forms and entities that spring from other realities. It is as if many worlds were depicted in one, while they interpenetrate in a disjointed, chaotic, and almost disturbing way. There is a video game that according to mathematical laws, reproduces exactly this kind of interaction between the 3D world and 4D objects [1]. The game is not for free, but you can at least watch the explanatory video in which you can get an accurate idea of the phenomenon.

Now, you will say: but nobody on Earth has ever seen three-dimensional objects appear and disappear into thin air or change shape for no reason. This is true… more or less. Physics is full of oddities, mysteries that are still misunderstood and contradictory theories. While scientists like Nobel Prize winner Roger Penrose are sceptical about introducing new dimensions into the description of our universe, there are plenty of scientists who are convinced that this is exactly the way to go. One of the great challenges of contemporary theoretical physics is the unification of the two most famous theories of the 20th century: Einstein’s general relativity (I know, I promised we would overlook him) and quantum mechanics. These two theories, although incredibly precise in predicting many of the phenomena around us, from the movement of large celestial bodies to that of elementary particles, are incompatible with one another under certain aspects. One of the attempts to unify and synthesise these theories is called ‘string theory’. Actually, there are several string theories, which also differ in the number of dimensions they associate with the universe: there are 10-dimensional theories, 11-dimensional, 26-dimensional, *etc*.

I certainly cannot imagine or describe 10D or 26D worlds for you. However, just knowing that there are scientists who devote their careers to understanding the unintelligible, like circles and squares striving to get to the bottom of the existence of spheres and cubes, makes me somewhat proud of humanity and our tireless desire for knowledge.

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