Chapter 3

Section 2: Exploring “Dimensions”

What exactly is a dimension? The colloquial use of this word references such a variety of meanings that it is easy to become confused when trying to decipher what physicists mean by ‘dimension.’ In general settings, a dimension can be used to reference a thought, a quality, an experience, or the scope or importance of a given topic. But in physics there is a very specific meaning given to the word ‘dimension.’

Dimensions are independent parameters that map a framework. Spatial dimensions each provide independent information about position and temporal dimensions give us independent information about time. Let’s explore this a little. We are most familiar with spatial dimensions, namely x, y, and z, or length, width and height, but why do we call these parameters dimensions?

To grasp this fundamental concept, let’s imagine the x-y plane as a flat sheet that extends to infinity in both directions. Let’s pick an origin and label it (0,0) — zero in the arbitrarily chosen x‑direction and zero in the perpendicular y-direction (Figure 3-1). We can talk about the x‑coordinate of any point on that plane, which is the distance the point is from the origin (the x = 0 line) in the x-direction. Similarly, we can label the y‑coordinate of any point. However, by using only the variables x and y, we are completely incapable of illuminating anything about any distance above, or below the x-y plane. That is why z is another dimension—because it cannot be expressed through terms of x or y. These dimensions are called spatial because they orthogonally (perpendicularly and independently) map space. Each dimension provides unique and independent information about the map, and this independence is expressed in their perpendicular geometric configuration. [1]

 

Figure 3-1 Two-dimensional Cartesian plane.

 

Our common experience suggests that four dimensions are sufficient to completely express where and when any event occurs. We think a specific event is perfectly nailed down once we have described where it occurs in terms of x, y, z and when it occurs in terms of t. But, and this is an important point, physical reality might be made up of more than four dimensions. Events may require more than x, y, z, t information to perfectly nail down.

Physicists have been discovering hints that suggest Nature’s most simplistic and complete description may actually exist in a realm of higher dimensionality. If this turns out to be true it would mean that more than four independent parameters (dimensions) are required to completely identify precisely where and when events occur in Nature.

This possibility should make us quite interested in the fact that the laws of quantum physics are fundamentally governed by the complex number system, because the complex numbers can be seen as encoding a higher dimensional setting. In the complex number system each familiar spatial dimension (x, y, z) is morphed into a complex plane through the addition of an ‘imaginary’ dimension (i, j, k). The resultant planes allow us to mathematically describe a six-dimensional manifold, and each plane can be separately graphed as in Figure 3-2.

 

[FIGURE PLACEHOLDER]

Figure 3-2 A complex plane.

 

Instead of following the rules dictated by the combination of three space dimensions (x, y, z,) Nature’s microscopic realm seems to follow rules that suggest it is composed of a union between three planes which are each like the one shown here.

 

When attempting to map the microscopic realm, these so called ‘imaginary dimensions’ are absolutely necessary in order to construct a map that can account for observation. But what does this mean? Do these imaginary dimensions have physical existence? Or are they just some kind of mathematical trick?

While we let that question hang in the air, let’s note that we have no physical theory that dictates that there should be only three dimensions of space and one dimension of time. What we do have is an ever-growing collection of observations (most significantly from the fields of cosmology and quantum mechanics) that stubbornly resist explanation within a four-dimensional framework.

Some of these observations can be found very close to home. For example, Polytetrafluoroethylene, the material commonly known as Teflon (which most of us can find in our kitchens as the nonstick coatings on our pans), possesses a lattice structure that is not allowed within three-dimensional space. It belongs to a class of structures called quasicrystals, which are structures that can only form their specific geometric lattice patterns in higher dimensional spaces. (Randall 2005, 4-5)

All crystals are symmetric lattices of repeating basic elements of atoms or molecules. The number of dimensions those elements have access to determines the type of patterns available for their arrangements. Three-dimensional frameworks allow more geometric arrangements than two-dimensional frameworks do, but these arrangements are still limited.

What is interesting about quasicrystals is that they do not conform to any of the lattice structures allowed in three-dimensional frameworks — yet they exist. In other words, it’s only in higher dimensional frameworks that quasicrystals can be described as ordered repetitive structures (the definition of a crystal), and therefore it’s only with the presence of additional dimen­sions that their formation can be explained. The very presence of quasicrystals suggests that our current view is laced with dimensional tenuity. Even the nonstick surfaces on our frying pans contain echoes of additional dimensions.

 

 

Figure 3-3 Penrose Tiling

An example of what a higher-dimensional repetitive crystalline lattice looks like when we suppress it to fit our familiar dimensional mold. These structures look like they make use of two geometric shapes, but when we rotate these shapes through another dimension we see that the two shapes are really the same shape with different orientations.

 



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