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Chapter One

Reflections

In another moment Alice was through the glass, and had jumped lightly down into the Looking-glass room. —Lewis Carroll, Through the Looking Glass

Before we offer the main course, we present four mathematical appetizers. In each of the following four examples, based mainly on the Law of Reflection, the solutions depend on surprising geometric insights that lie just below the surface.

The Color of the Bear

Ada tells Ben about a puzzle that she read in the newspaper, but Ben's solution jumps to a conclusion.

Ada: A hunter leaves his cabin and walks 1 mile south. Then he walks 1 mile east, and, finally, he walks 1 mile north to end up where he started, at his cabin. To his surprise, he finds a bear inside. What is the color of the bear?

Ben: The bear must be a white polar bear because the hunter returns to his initial point after traveling 1 mile south, 1 mile east, and 1 mile north. This is possible only if the hunter starts his journey at the North Pole.

Example 1.1. Ben's last assertion is not entirely correct because the path he describes is one of many possibilities. In addition to the North Pole, find all the other places where such a circuit could start. Hint: At the additional locations, it is more likely that the hunter finds a penguin in his cabin instead of a polar bear.

Solution. Figure 1.1(a) shows an example of such a circuit in the vicinity of the SouthPole. The circle has a circumference of 1 mile with its center at the South Pole. The point Q is an arbitrary point on the circle. The point P is 1 mile north of the point Q. The circuit encircles the South Pole S.

    There are other solutions. For example, the circle in Figure 1.1 (a) can be replaced by a circle with a circumference of 0.5 mile. Then instead of encircling the South Pole S once, we encircle it twice before returning north to point P. Moreover, circuits are possible that encircle the South Pole any number of times. The totality of initial points for the circuit consists of the North Pole and an infinite family of circles centered at the South Pole as shown in Figure 1.1(b).

    In Figure 1.1(a), the hunter follows the shortest path between P and Q. The next example, from the noted British puzzle creator, Henry Dudeney (1857-1930), also deals with a shortest distance.

The Spider and the Fly

Example 1.2. As shown in Figure 1.2(a), in a room measuring 30 feet in length and 12 feet in both width and height, a spider is at the middle of the one of the end walls, 1 foot from the ceiling, and a fly is at the middle of the opposite wall, 1 foot from the floor. The spider observes that he can reach the fly, as shown in Figure 1.2(a), by crawling vertically 11 feet down the wall to the floor, straight across the floor 30 feet to the opposite wall, and vertically 1 foot up the wall to the fly—a total of 42 feet. Assist the spider by showing that there is a path that is 2 feet shorter.

Solution. Imagine that the room is made of cardboard. Figures 1.2(b)-(e) show four ways to cut the box along certain of its edges in order to flatten it. On the flattened box, the shortest path must be a straight line connecting S and F. Figures 1.2(b)-(e) show the lengths of four different line segments corresponding to the four ways of flattening the box. The path in Figure 1.2(b) is the one contemplated by the spider, but the paths in Figure 1.2(d) and (e) are shorter. In fact, the path in Figure 1.2(e) is exactly 40 feet. Surprisingly, as shown in Figure 1.2(f), the shortest path meets five of the six sides of the room.

The Law of Reflection

This search for the shortest path leads to a physical principle called the Law of Reflection. Figure 1.3 depicts the reflection of a light ray emanating from point A, reflected at point C by a mirrored surface l, and arriving at point B. The Law of Reflection states that the angle of incidence [Theta] is equal to the angle of reflection [Phi].

    The Law of Reflection takes a simpler form if we give substance to the "illusion" of reflection. In Figure 1.3, the portion of the figure above the line l shows the physical world in which photons bounce off a mirror. The portion below the line l is a representation of the illusion of reflection—Alice's "Looking-glass House"—in which the half plane above the line l is flipped across that line. After it strikes the mirror, it is useful to represent the further progress of the ray AC by the segment CB', the mirror image of CB. The fact that the angles [Theta] and [Phi] are equal implies that ACB' is a straight line. Since ACB' is the shortest path connecting A and B', it follows that ACB is the shortest path between A and B that meets the line l. The reflected ray always follows the path of minimum length among all paths that meet l and connect the fixed points A and B. This is one of the simplest of the far-reaching minimum principles that play a fundamental role in almost every aspect of physics.

    The Law of Reflection also governs the reflection of sound and the motion of particles under elastic impact, as in the game of billiards.

Billiards

The traditional billiard table has dimensions 73.5 inches by 144 inches. We simplify the game by assuming that the cue ball is a moving point that moves in straight lines and that the ball bounces off the sides of the table, called cushions, according to the Law of Reflection.

Example 1.3. Suppose that the cue ball is placed at a point 46 inches from the west cushion and 20 inches from the south cushion, and suppose that the ball returns to its initial position after striking, in order, the north cushion, the east cushion, and the south cushion. How far does the ball travel to return to its initial position?

Solution. In billiards, a rebound from a cushion is similar to optical reflection except that there is no illusion of a reflected image. Nevertheless, in Figure 1.4, it is useful to show the billiard table and its repeated reflections across the edges of the table. The ball starts at point P, the solid dot; the five hollow dots represent the reflected images of the initial position of the ball. When the ball strikes the north cushion at Q, it bounces along the path QR. Figure 1.4 shows the image QR' of QR in the image of the billiard table reflected across its north edge. The reflected image is useful because, according to the Law of Reflection, PQR' is a straight line. Similarly, R'S' is the double reflection of RS; and S'P' is the triple reflection of SP. The reflected image PP' of the entire path of the ball is a straight line. Thus, the length of the circuit of the ball is equal to the length of the line segment PP', which is equal to 245 inches.

    The fourth and final example is an application of the Law of Reflection to a problem of geophysical exploration.

An Ear to the Ground

A seismic reflection survey is a method for mapping structures deep within the earth. Multiple energy sources and listening devices are employed. The energy source can be an earthquake, an explosion, or a device called a vibroseis truck, a vehicle weighing more than 10 tons that raises itself off the ground on a hydraulic plunger and vibrates the ground in the subsonic range of 10 to 56 cycles per second. Typically, four vibroseis trucks are used. The listening devices are called geophones and are used in an array of hundreds. This technique is used to map structures as deep as 30 miles below the surface of the earth.

    The following example utilizes much simpler equipment—one seismic pulse and one listening device.

Example 1.4. Assume that there is a plane reflective layer of rock at an unknown depth below the plane surface of the earth and that the earth above the rock layer is homogeneous so that sound travels only in straight lines. Our trained elephant stamps his foot once and hears exactly two echoes from within the earth. What can be said about the angle between the plane of the rock layer and the plane of the earth's surface?

Solution. Sound is propagated in rays in every possible direction. These rays are reflected by the rock layer and possibly re-reflected by the earth's surface according to the Law of Reflection. When a reflected ray returns to the earth's surface at the elephant's location, then he hears an echo.

    Multiple echoes are caused by multiple reflections. If the rock layer is parallel to the earth's surface, then multiple echoes resemble the multiple images from parallel mirrors as is sometimes seen in a barber shop. I recall as a child sitting in a barber chair for the first time and watching an infinity of images of my small self. If the rock layer is parallel to the earth's surface, then an indefinite number of reflections is possible; the number of echoes heard is limited only by the elephant's ability to hear faint sounds. However, if the rock layer and the earth's surface are not parallel, then the number of echoes is limited regardless of the keenness of the elephant's hearing. For example, if the rock layer is a vertical wall, then no echo whatever is possible because no reflected ray can return to the earth's surface.

    Figure 1.5(a) shows how it is possible for the elephant to hear two echoes from two reflected rays in case the rock layer is at an angle of 30° with the earth's surface. Point E represents the elephant's location. The first echo is reflected from the rock layer at A directly back to E, but the second echo is reflected at B by the the rock layer, then at C by the surface of the earth, and, finally, again at B by the rock layer. In both cases, the ray retraces its path to return to E.

    In Figure 1.5(b), the lower half-plane is dissected into six wedge-shaped regions. The first region is bounded by the rock layer, tilted at 30°, and the surface of the earth. As we proceed counterclockwise, each region is the reflection of its predecessor across their common boundary. The points [E.sub.1], [E.sub.2], and [E.sub.3] are the reflected images of the elephant's location E.

First echo. In Figure 1.5(b), the line segment EA is the downward path of the first echo, and line segment [AE.sub.1], the reflected image of AE in Figure 1.5(a), represents the reflection from the rock layer back to the elephant. It follows from the Law of Reflection that the segments EA and [AE.sub.1] fit together to form the straight line segment [EE.sub.1].

Second echo. The line EB is the downward path of the second echo; [BC.sub.1] is the reflection of the upward path BC in Figure 1.5(a); [C.sub.1][B.sub.1] is the image of the downward path CB; and, finally, [B.sub.1][E.sub.2] is the image of the upward path BE in Figure 1.5(b). Again, it follows from the Law of Reflection that, in Figure 1.5(b), the four segments EB, [BC.sub.1], [C.sub.1][B.sub.1], and [B.sub.1][E.sub.2] fit together to form the straight line segment [EE.sub.2].

    The line segments [EE.sub.1] and [EE.sub.2] represent two echoes, but the line segment [EE.sub.3] does not represent an echo. In fact, a line from E to a reflected image of E represents an echo only if that line proceeds downward from E. For example, the line segment [EE.sub.2] does not represent an echo because it does not proceed downward from E. However, if the angle between the rock layer and the earth's surface were less than 30°, then [E.sub.3] would be below the surface of the earth and [EE.sub.3] would represent a third echo. We conclude, if at most two echoes are heard, then the angle must be 30° or greater.

    Figure 1.5(c) shows a rock layer that meets the surface of the earth in an angle of 45°. The figure shows multiple reflections of the wedge-shaped region between the rock layer and the earth's surface; the first reflection is across the rock layer, the second reflection is across the vertical image of the earth's surface, and so on. The point E represents the position of the elephant—the point at which the echoes are heard. The points [E.sub.1] and [E.sub.2] are the images of E in the reflected regions. The straight line segment [EE.sub.1] represents an echo that is reflected from the rock layer directly to the earth's surface, but [EE.sub.2] does not represent an echo because it does not proceed downward from E. However, if the angle were less than 45°, then the point [E.sub.2] would be below the surface of the earth and the line [EE.sub.2] would represent a second echo. We conclude, if two or more echoes are heard, then the angle is less than 45°.

    From the above two paragraphs, it follows that if the elephant hears exactly two echoes, the angle must be at least 30° but less than 45°.

    In this chapter, we have seen some mathematical insights without the use of mathematical formulas. Nevertheless, formulas, though not an end in themselves, are a useful mathematical supplement to ordinary language. There are many beautiful things that we can see more readily with just a bit of algebra. For example, in the next chapter, an equation helps to solve a problem that would otherwise be difficult.