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Unit 5

Other Dimensions

5.2 Degrees of Freedom

FUNDAMENTAL NOTIONS

The concept of dimension is, in its most basic and intuitive form, the concept of measuring certain aspects of an object independently from all of its other aspects. This idea of dimension is also known as "degrees of freedom." If an object has three degrees of freedom—height, width, and length, let's say—that means that it is able to "change" in any one of those three ways, and a change in one has no effect on the other two. So, if we are navigating the streets of a city laid out on a grid system, for instance, we are free to change our east-west position or our north-south position, depending on whether we're moving along an avenue or a street. These are our two degrees of freedom. In a city whose grid system is perfectly oriented to the four cardinal directions, going north on an avenue does not affect your east-west position.

In order to examine the basic nature of spaces of different dimensions, we will look at how many numbers it takes to specify the location of a point. For our purposes, a point is an object with no other properties other than its location. A point, by itself, has no degrees of freedom—it is effectively a space of zero dimensions.

We consider a space to be a collection of locations. The zero-dimensional space has only one location and, thus, allows for only one point. A space with more than one possible location allows for at least one degree of freedom for a point in that space. It also allows for the existence of multiple points, which then can be grouped to form line segments, polygons, solids, and so on, depending on the exact dimension of the space.

All spaces are not created equal. Their differences can be characterized in various ways, such as how one defines distance, whether or not angles exist, and how many degrees of freedom are afforded the objects in that space. We will concern ourselves only with the last of these properties. To help you get a handle on this concept of degrees of freedom, here's another way to look at it: a space of locations in which a point has only one degree of freedom is a space in which points can differ from one another in only one way. A number line is a model of this type of space.

number line

Furthermore, two points in this space of one degree of freedom can never have anything in common. If they did, they would be the same point!

A space of two degrees of freedom allows for points to differ from one another in more than one way. For instance, (0, 1) is different from (0, 2), even though both have a zero in common. The points (1, 0) and (2, 0) are distinct from both each other and from (0, 1) and (0, 2), even though all four points incorporate a zero value somewhere. A space of two degrees of freedom, thus, allows for a greater variety of locations than are possible with only one degree of freedom.

one degree of freedom

In this section we will look at a few familiar spaces in terms of their dimension. We will also give passing consideration to other properties, such as distance and area, but our primary concern will be with dimensionality and its consequences.

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LINELAND

Let us first start by examining a one-dimensional space with which we are all familiar, the number line.

number line

Life in a one-dimensional (1-D) space is, well, just not that interesting. If you were a point in 1-D space, all that we would need to pin down your exact position is one number. That number would simply be how far you were, in whatever units we're using, from some agreed-upon reference point. The units could be whatever we choose, as long as they are uniform. For our present discussion, we'll simply use the term "units." The reference point is assigned the value of zero and is more commonly known as "the origin."

If we take two points in 1-D space and connect them, we form a line segment. This line segment has a property that no single point has, length. The length of a line segment in 1-D space can be found from the positions of the two endpoints via subtraction.

number line

That's about all the "news" from one-dimensional space. Forwards or backwards, this side of the origin or that side, long or short line segments—these are pretty much the only things we could possibly care about if our world were one-dimensional. So, let's move on to explore a significantly more interesting place, two-dimensional space.

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FLATLAND

quadrants

In a two-dimensional (2-D) world, we have an added degree of freedom over a one-dimensional world. One number is no longer enough to specify a unique location. For instance, on the Cartesian plane a "3" on the horizontal direction, or axis, can be paired with many different vertical values, and each pairing defines a different, unique location in the space. Due to the fact that the horizontal and vertical directions are "measured" completely independently of each other, we need two numbers to pin down a location in 2-D space.

2-D space

Also, the origin now is not only the reference point for the horizontal axis, as with the number line, but also for the vertical axis. It, too, requires two numbers to define its location, so we define the origin as the point (0, 0). Notice now that the question of direction is much more interesting than in 1-D space. In one dimension, you can only go back and forth, but in two dimensions you can go back and forth, up and down, or any combination of these.

Imagine that we have a line segment that starts at the origin and goes to (3, 4).

2-D space

It's obvious that this line segment has neither a strictly vertical nor a strictly horizontal orientation, but rather some hybrid of the two directions. Furthermore, finding the length of this segment is now not a simple subtraction problem, as before. We can, however, still determine a length by examining the line segment's directional components.

line segments

The components of the line segment can be thought of as its "shadows" on the horizontal and vertical axes. This idea of finding a shadow will help us in understanding how objects with components in multiple independent dimensions can be visualized, but we'll get to that a little later.

Notice that the line segment forms the hypotenuse of a right triangle whose legs are the horizontal and vertical components. This means that we can determine the length of the line segment—or, in other words, the distance from the origin to (3, 4)—by using the Pythagorean Theorem.

(horizontal component)2 + (vertical component)2 = (hypotenuse)2

If we rewrite this, taking the square root of both sides, we get:

hypotenuse = hypotenuse

So, plugging in the horizontal value, 3, and the vertical value, 4, we get the familiar 5 for the length of our line segment.

The fact that we can use the Pythagorean Theorem to calculate the distance between two points means that the version of 2-D space that we have been studying is Euclidean. There are other ways to define distance, and this turns out to be a good way to distinguish between spaces that, although they have the same dimension, exhibit different behaviors.

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SPACELAND

We have seen that in the 2-D world, horizontal and vertical directions are independent dimensions. To think about a 3-D world, we need one more direction that can change independently of horizontal and vertical changes. We know this direction as movement "toward" or "away." For simplicity's sake, from here on out we will follow convention and represent horizontal distance by the letter x, vertical distance by the letter y, and distance toward (the "positive" direction) or away ("negative") by the letter z.

line segments

Notice that using just two numbers won't uniquely specify a point in this space. For instance, the designation (3, 4) pins down a location only in the xy-plane—it tells us nothing about location in the z-direction, or in other words, how near to us or how far from us the point is. In fact, in 3-D space (3, 4) defines a line, one that is parallel to the z-axis. In other words, because no z value is specified, the assumption is that z can take on any value, from positive infinity to negative infinity. By contrast, (3, 4, 12) does indeed designate a uniquely defined point in three dimensions.

In the 2-D world, we saw that we could use the Pythagorean Theorem to find the distance from one point to another. Does it also work in the 3-D world?

Let's see.

To find the distance from the origin to (3, 4, 5), we can imagine two right triangles like so:

triangle

The first triangle is formed in the xy-plane, with its hypotenuse being the line segment that extends from the origin to (3, 4). We saw earlier that the length of this hypotenuse can be calculated directly from the Pythagorean Theorem:

32 + 42 = 52

Thus, the hypotenuse of the first triangle measures 5 units. This line segment now becomes a base of the second triangle, with vertices at the origin, (3, 4, 0), and (3, 4, 12):

Again, we can use the Pythagorean Theorem to find the length of the hypotenuse.

triangle

52 + 122 = 132

So, the length of the line segment from the origin to (3, 4, 12) is 13 units. Notice that if we plug in the expression for the square of the first hypotenuse into the expression for the second hypotenuse, we get:

32 + 42 + 122 = 132

More generally:

(the x distance)2 + (the y distance)2 + (the z distance)2 = (total distance)2

This shows us that the Pythagorean Theorem generalizes quite nicely from the 2-D world to a 3-D world. In fact, we could continue this development into 4-D, as we will soon see.

As we stated earlier, the addition of each new dimension to a space introduces a new property that lower-dimensional spaces don't have. For instance, in 2-D space we can have not only line segments but also planar shapes, such as squares and discs, which exhibit the new property of "area." Similarly, 3-D space introduces the property of volume. Shapes with the property of volume, called solids, are not possible in any space with fewer than three dimensions.

Also, note that we have been referring to dimension primarily as a spatial measure, but it doesn't have to be. Any quantity that can be measured independently of others qualifies as a dimension. So, imagine that we have a particle at a particular location in 3-D space. We might be concerned with other properties of this particle besides its three spatial coordinates, such as its mass, charge, or color. If we included each of these three independent measures as basic attributes in our description of the particle, we would have a six-dimensional object—that is, it would be uniquely determined in a space of six dimensions. Such a space is not very easy to visualize, but it presents no problems mathematically. We simply realize that it is the space that contains all sets of six numbers. Only three of those numbers are spatial coordinates, but we don't necessarily need to limit ourselves to these. We have seen that ideas from lower-dimensional spaces generalize quite nicely as we step up to higher-dimensional realms. We can use this idea to leverage our intuitive understanding of lower-dimensional spaces to spaces of four dimensions
and higher.

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Next: 5.3 Journey into the Fourth Dimension


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