The Fourth Dimension

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The Fourth Dimension
Field: Geometry
Image Created By: Jason Hise
Website: Wikipedia

The Fourth Dimension

The fourth dimension refers to the concept of a 4-dimensional space, in which four geometric coordinates are necessary to describe any point. In the fourth dimension, our universe is but an infinitesimal slice of the fourth dimension.

Basic Description

What constitutes a dimension? A dimension has an origin and a set of coordinates which each can independently equal any real number. The nth dimension requires n numbers to specify the position of a point in it, and no point in that dimension cannot be described by some combination of the n coordinates. The origin is where every coordinate (however many there are) is zero. Each coordinate represents a distance from the origin in a certain direction, and each of those directions is perpendicular to the rest. If you're reading this, you know already that the first dimension is a line, the second is a plane, and the third is space. What, then, is the fourth dimension?

Even though what four dimensions actually looks like may be beyond us, we know it requires four numbers to describe any possible point in it. The fourth dimension is a hyperspace, since it is of a higher dimension than ordinary space.

A More Mathematical Explanation

Note: understanding of this explanation requires: *Cartesian coordinates, geometry

The Fourth Dimension Explained

Entertaining the notion of hyperspace (i.e., dimensions beyond the [...]

The Fourth Dimension Explained

Entertaining the notion of hyperspace (i.e., dimensions beyond the third) necessitates being able to conceptualize the fourth dimension, and for that, the analogy "4-D is to 3-D as 3-D is to 2-D" is useful. Whenever we try to comprehend an object in 4-space or perhaps a movement of a 4-dimensional object along the fourth dimension that passes through our space, we need only think of a being in 2-space trying to comprehend the corresponding object in 3-space. Since we can understand the latter situation, we can understand the relation, and since we know that the 3-space object relative to the 2-space observer is analogous to the 4-space object relative to us (3-space observers), we can apply the relation to the problem of conceptualizing 4-dimensional things.

So, when thinking of how the third dimension (our space) appears to 4-dimensional beings in the fourth dimension, one need only think of how 2-dimensional spaces appear to us 3-dimensional beings. They look like flat, infinitesimal slices of our space. Likewise, our universe is an infinitesimal slice of the fourth dimension.

The aforementioned analogy, "4-D : 3-D :: 3-D : 2-D," however, is useless without a 4-dimensional body to which to apply it. The tesseract is an object in the fourth dimension. It is the 4-D analogue of the cube, or, to apply the analogy, the tesseract is to the cube as the cube is to the square. Still, what is a tesseract?

Consider a cube where each of the six corners has either 1 or -1 in each of its coordinates (i.e., the corners are (1, 1, 1), (1, 1, -1), (1, -1, 1), (1, -1, -1), (-1, 1, 1), (-1, 1, -1), (-1, -1, 1), and (-1, -1, -1)). Taking 1 of the coordinates, holding it constant at either 1 or -1, and varying the others between 1 and -1 begets a square. Each of the three coordinates can be held constant at 1 and -1, generating the six squares by which the cube is bounded. With a tesseract, there are four coordinates which can be held constant at 1 and -1, and as the other three are varied, cubes are produced. So, the tesseract is bounded by eight cubes.

Possibilities of the Fourth Dimension

The fourth dimension can be treated as a fourth direction, a dimension fundamentally no different from the three we know, except that it is one we humans do not experience. However, a popular and arguably less geometric interpretation of the fourth dimension is time.[1] When it is thought of as time, all four dimensions together are called “spacetime.” To understand how it could be that the fourth dimension is time, think of the fourth dimension that has been discussed thus far. It contains infinitely many infinitely thin 3-spaces like ours. If that fourth dimension is time, then each of those infinitely thin layers of 4-space represents a moment in time, and as you move in the positive direction of the fourth dimension, you go forward chronologically. By this definition of the fourth dimension, if, say, you wanted to touch all of a person, you would not only have to engage them in a total embrace, but you would have to do it for the person’s entire life, since the totality of a person comprises the person’s state of being at every moment in time.[2]

If we take the fourth dimension to be time, then what is a 4-dimensional being? Are we 4-dimensional beings, since we traverse time, the fourth dimension? Or are we not, since we cannot seem to control the speed with which we travel the fourth dimension? Theory on the fourth dimension becomes more complicated as the fourth dimension is thought of as a concept like time instead of simply another direction. If we are not 4-dimensional beings (since we only seem to exist at one cross section of spacetime at any given “time”), and a 4-dimensional being is one that can traverse time at a speed different than time itself, would it have different 3-D cross sections all existing over a range of times?[3] Would it be able to “tell the future”? If it could go further ahead in time than what we call “now,” does that mean that everything that ever will happen in our universe is already determined? Are we just lower-dimensional beings that slowly experience what inevitably will happen in an absolutely deterministic existence?

If time were the fourth dimension, then the fifth dimension would be a collection of infinitely many spacetimes. What, then, is a five-dimensional being? Does it exist in multiple timelines, multiple histories, multiple universes? Does it experience simultaneously different possible outcomes of our universe, or does it experience different universes altogether? Would it be able to travel across universes, entering and exiting their respective histories as it went?

This speculation can continue on indefinitely, though, at a point, it goes from interesting and useful to just interesting, and eventually to completely lacking any utility. Theoretically, there is an indefinitely large number of dimensions (one could keep conceiving of different directions in hyperspace). As far as we know, there do exist dimensions higher than the third, and they may contain beings that comprise more than three dimensions. There could be a set of dimensions in which an entirely different species lived, but perhaps none of their dimensions are the same as any of ours, and so we (probably) would never meet them.[4]

Why It's Interesting

Modern physics and astronomy invite the idea of dimensions beyond the three with which we are innately familiar.

How the Main Image Relates

The animation is a three-dimensional projection of a tesseract rotating in hyperspace. A tesseract is a four-dimensional object which occupies space in the fourth dimension.

Teaching Materials

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  1. Whether or not time is a geometric dimension is not very important. When presented with the possibility that the fourth dimension is time, we do not think of it in the same way as we do the other three dimensions, since it does not seem similar to them in the same way that they are to each other. However, looking back at history, time could be viewed also as a thing of direction, since the positive direction represents going forward in time, it is as though every moment in time takes up an infinitesimal slice of the fourth dimension.
  2. Sider, Theodore. Four-Dimensionalism. New York: Oxford University Press, Inc., 2001. 3. Print.
  3. Talk of an existence over a range of times simultaneously is strange, since it implies a sort of higher time (one that is not traversable by 4-dimensional beings) that exists outside of the defined dimension of time, one in which you can exist at different moments of what we call time at the same “time.”
  4. I use the word “probably” because the notion of cross-dimensional travel in the near future of human technology is inconceivable.

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  [[Description::The fourth dimension refers to the concept of a 4-dimensional space, in which four geometric coordinates are necessary to describe any point. In the fourth dimension, our universe is but an infinitesimal slice of the fourth dimension.|]]