In discussing the nature of the Universe, many people refer to it as "finite but unbounded," or even "curved." It is often difficult, particularly for those of with little knowledge of theoretical physics, to understand these concepts, but hopefully this post will help to illuminate their meaning.

When considering dimensionality, it is easiest to first consider the dimensions that we have "access" to - we are "trapped," after all, in the 3rd dimension, and thus cannot physically visualize any higher physical dimensions, but we can certainly think about them using the following analogy. Let's consider a 2-dimensional world, that for the purposes of this explanation we'll call "flatland," in which objects have only length and width, but no height at all. Now, consider what would happen if a 3 dimensional object were to enter this 2-dimensional world; it would exist in flatland in only 2-dimensions, and the residents of flatland would only be able to view the object in two dimensions as it passes through the plane. This must be utterly odd to flatland-ers, as they essentially see an object appear out of nowhere, rapidly change shape, and then disappear again into the nether.

Now, with this analogy in mind, let us consider the other direction of dimensionality - the 4th physical dimension. It is obviously not possible for us to visualize the 4th dimension, but think about it in the following way:

The 0th dimension, we can think about, consists of only a single point. Extend that point into a line, and we have 1 dimension - length. Take that line segment and move it at 90 degree angles to itself, and we have a 2 dimensional square. Take that square and move it at 90 degree angles to itself and we have a 3 dimensional cube, complete with length, width, and height. A cube physically has vertices all of angle 90 degrees, with lines of equal length. When this cube is visualized in 2 dimensions, as on a piece of paper, not all of the angles are conserved, and all of the lines are not of equal length. This is part of the penalty associated with losing a dimension. Now, what would happen if we were to take this 3 dimensional cube and carry it into the 4th physical dimension, in other words, connect each of its vertices at 90 degree angles of the same length? We cannot physically comprehend this object, called a 4-dimensional hypercube, or tesseract, in our 3-dimensional world, but we can visualize it's shadow in a 3-dimensional world, as seen above.

The questions this subsequently raises, are then: what if all 3-dimensional objects that exist in our perceivable universe are only the "shadows," left by 4-dimensional objects? We could never know if this were the case, because our perception is limited to the 3rd dimension. A 4th dimension (or 5th, or 6th, or 7th...) could very well exist, and we may just be blissfully unaware of its existence. The universe may well be curved into a 4th dimension, but we would not know or be able to comprehend this. This theory is fairly well accepted in the scientific community, and has future implications for determining how to "fold" a dimension through another to possibly allow for inter-dimensional travel, something often termed a "wormhole." Our universe may well be curved into other dimensions that we could travel through, but we are still quite far away from fully comprehending this.

Reference: http://www.youtube.com/watch?v=UnURElCzGc0

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