Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
In looking at both the sphere and the pseudosphere, we see that they are unlike the plane in that they are both curved surfaces. Furthermore, we saw that "straight" lines (lines of the shortest length, in other words) are not straight at all on these surfaces; rather, they are curves. In order to explore these surfaces, and others that do not obey Euclid's fifth postulate, we need to be able to discuss curves meaningfully. Let's start with a simple curve in a plane:
How can we describe this curve's curviness? We could compare it to a circle, a "perfect curve" in some respects, but it is evident that this curve is not really even close to being a circle. It has regions that seem more tightly curved than others, and it even has regions that curve in opposite directions. When we look at a curve in this way, in the broader context of the plane, we are viewing it extrinsically. By contrast, viewing a curve intrinsically, that is from the point of view of someone on the curve, yields a different perspective and different possible measurements.
So, perhaps a good thing to do would be, instead of talking about the curvature of the whole thing right away, to talk about the curvature at each point along the curve. As theoretical travelers along the curve, we could stop at each point and ask, "What size circle would define the curve in the immediate vicinity of this point?"
First, let's think of the tangent line at this point. This is a line that intersects our curve only at this one point (locally, that is—it's possible that the line might also intersect the curve at other more-distant points). We can also think of the tangent as the one straight line that best approximates our curve at this particular point. Let's then draw a line from this tangent point, perpendicular to the tangent line.
Let this line, called a "normal line" or just a "normal," be of a length that, if it were the radius of a circle, that circle would be the biggest possible circle that still touches the curve in only one place. In other words, the normal line should be the radius of the circle that best approximates the curve at this particular point. Such a circle is called an "osculating circle," which literally means "kissing" circle, because it just barely touches, or "kisses," the curve at this one point.
We can define the curvature of a curve at any particular point to be the reciprocal of the radius of the osculating circle that fits the curve at that point. Let's refer to this curvature as "k" from here on out.
What should we do, though, about the fact that some parts of the curve open upward, whereas other parts of the curve open down? If we designate that the normal always points to the same side of the curve—let's choose upward for our case—then when the normal happens to be on the same side as the osculating circle, we'll call this negative curvature, and when the normal happens to be on the opposite side from the osculating circle, we'll call this positive curvature. At any point where the line is flat (i.e., straight), we don't need an osculating circle, and we'll call this zero curvature. The choice of defining which curvature to consider positive and which to consider negative is completely arbitrary. The method chosen for this example is nice because, if we think of our planar curve as a landscape, then the positively curved areas are the hills and the negatively curved areas are the valleys.
An interesting feature of looking at a curve in this way is that, were we onedimensional beings living on the curve, we would not notice that it is curved at all. This is called an intrinsic view. The only thing that can be measured intrinsically on a curve is its length, and length alone tells us nothing about how curvy a one-dimensional object is.
Remember that the way we quantified the curvature of this curve was to compare it to a circle in the plane. Now, as one-dimensional beings, this requires envisioning one more dimension than would be available to our perception. The curvature becomes apparent only when the curve is viewed by an observer not on the curve itself—that is, one who can see it extrinsically in two dimensions.
Using this system, we can meaningfully talk about any curve in a plane, and we know from previous discussions that once we understand something in a lowerdimensional setting, we can generalize our thinking to a higher-dimensional setting. In this case, instead of talking about plane curves, we will return to our curved surfaces, such as the sphere and the pseudosphere.
Let's take a moment to compare and contrast our plane curve and our curved surface. Our plane curve, though drawn in a two-dimensional plane, is actually only a one-dimensional object. This is because, if you were an ant living on this curve, you would only have the option of traveling forward or backward. Because of this, you wouldn't even really know that your line was curved.
A curved surface, on the other hand, is two-dimensional. If you were an ant living on it, you could move forward, backward, right, or left. As you can see, however, a curved surface requires a third dimension to represent it extrinsically, just as a one-dimensional planar curve requires a second dimension for its extrinsic representation. We said that an ant on a plane curve cannot experience this second dimension and, thus, has no idea that his world is curved. Is the same true for an ant on the surface, however? It can't experience the third dimension, but might it still be able to find out if its world is curved?
To resolve this, we need to find a way to apply our concept of the osculating circle to a curved 2-D surface. Actually, we can begin the same way as before.
Let's pick a point on our surface and define the normal (remember, that's a line that is exactly perpendicular to the surface at this point). If it helps, imagine the plane that is tangent at this one point as a flat meadow, and envision the normal as a tree growing straight up in the middle of the meadow. Now that we have both our point and our normal set, we can look at slices of the surface that contain both the point and the normal.
It is clear that each of these slices through the surface will show a slightly different curve, yet all of them contain our point of interest. So, which of these slices is "the" curvature at this point? We have so many possibilities to choose from!
One path toward a solution involves considering the extreme values—in other words, the curve that is most positively curved and the curve that is most negatively curved. We call these the "principal curvatures." If we were then to take the average of these two quantities, we would have a mean curvature for this point.
Would an ant on this surface be able to find, or develop an awareness of, these principle curvatures? To do so, it would have to have some idea of a plane that is perpendicular to the plane of his current existence. The complicating factor here is that the ant has no idea that another perpendicular direction can even exist! It will have a great deal of difficulty trying to figure out curves that can only be seen with the aid of a perspective that it can't have.
All hope is not lost, however. Again, we can turn to Gauss. His Theorem Egregium says that there is a type of curvature that is intrinsic to a surface. That is, it can be perceived by one who lives on the surface. Usually, this curvature, called the Gaussian curvature, is simply defined as the product of the two principal curvatures. For our example here, however, that will not be good enough, because our ant can't even know the principal curvatures!
Instead of trying to find the principal curvatures, the ant can draw a circle on his surface and look at the ratio of the circumference to the diameter. This ratio is often known as pi, and in flat space it is about 3.14159. We usually consider pi to be a universal constant, and it can be, but that depends on which universe we are talking about. In a Euclidean universe, pi is indeed constant. In non- Euclidean universes, however, the value of pi depends on where exactly the circle is drawn—it's not a constant at all!
Consider a circular trampoline. The circumference of this trampoline is fixed, but the webbing in the center is flexible and can be thought of as a surface. When no one is standing on the trampoline, the ratio of its circumference to its diameter is indeed pi. Now, consider what happens when someone stands in the middle of the trampoline: the fabric stretches and the diameter, as measured on the surface, increases.
The circumference, however, remains unchanged as the surface is stretched. This causes the ratio of the circumference to the diameter, pi , to decrease. Our ant could indeed detect such a distortion! This would be positive curvature.
For an example of how an ant could detect the curvature of a surface, consider a globe. If we draw a small circle near the north pole, it will be more or less indistinguishable from circles drawn on a flat plane. Now draw a circle that is a bit bigger, say at the 45^{th} parallel. This line is halfway between the north pole and the equator. Its radius, as measured on the surface, will be considerably longer in proportion to the circumference of the circle than was the case for the small polar circle. Therefore, the ratio of circumference to diameter will be smaller—in other words, larger diameter and smaller circumference.
Now consider the circle represented by the equator. The diameter of this circle, as measured on the surface, will be half the length of the circle! This would mean that for the equator, pi is equal to 2. This is quite a discrepancy from the customary 3.14… value, and it indicates that we must be on a curved surface. Negative curvature can be visualized as a saddle. Such a surface has more circumference for a given radius (and, hence, diameter) than we would expect with either flat or positive curvature.
Gaussian curvature is not as concerned with determining specific values of pi as it is with measuring how pi changes as the radius changes. The more curved a surface is, the faster pi will change for circles of increasing radius.
This idea, that there are certain properties that can be measured regardless of how our curve sits in space, was important in our topology unit and, as we have just seen, it plays a significant role in our discussion of curvature as well. These intrinsic properties of a surface—or the generalization of a surface, a manifold—are definable and measurable without regard to any external frame of reference. The Gaussian curvature is such a property, but the principal curvatures are not.
Recall that to find the principal curvatures, one must take perpendicular slices, which requires that our surface sit in some higher-dimensional space. This is an extrinsic view. The fact that the Gaussian curvature of a surface, as computed by the principal curvatures, yields an intrinsic quantity is quite remarkable. In fact, it is known to this day as "Gauss's Theorema Egregium," meaning "Gauss's Remarkable Theorem."
So, a natural question to ask might be: what kind of surface do we live on? We must have an intrinsic view of whatever space we inhabit—indeed, we have no way to get outside of it! A bit of thought, though, will lead to the realization that, unlike ants, we perceive a third dimension, so whatever this is that we are all living on, it is not a 2-D surface, but rather a 3-D manifold. A manifold can be thought of as a higher-dimensional surface, or it might help to think of it as a collection of points that sits in some larger collection of points.
Furthermore, our everyday experience includes a fourth degree of freedom, time. If we consider time to be part and parcel of our reality, then we are really living in a 4-D manifold called "spacetime." So, is our spacetime the 4-D equivalent of a flat, Euclidean plane, or is our reality curved in some spherical or hyperbolic way? For help in exploring this question, we'll turn to the ideas of a certain former patent clerk whose theories permanently altered the way in which we view our universe. First, however, we need to consider what happens when our surface is not as "nice" as a simple, smooth sphere or pseudosphere.
Next: 8.7 General Relativity
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