THE REFLECTIONS FROM A SHINY ball, such as might decorate a Christmas tree, form a strangely distorted panorama of you and your surroundings. As though honored, you cannot help but be at the center of the display regardless of how you look at the ball. The mutual reflections of two adjacent balls produce two seemingly endless series of images that shrink toward the point at which the balls touch. A cluster of three adjacent balls is even more arresting, particularly if it is on a dark background. The plentiful images are dazzling, and the region between the balls may appear to be a dark triangle that has encroached on the balls, marring their sphericity. What accounts for these images?

In 1972 Michael V. Berry of the University of Bristol published a study about reflections from such spheres as a Christmas-tree bauble, a steel ball from a bearing and a certain type of lawn decoration. As it happens, Berry is the author of an article-on a different subject-in this issue of Scientfic American [see "The Geometric Phase," I page 46]. Each sphere serves as a convex reflector, yielding miniature and distorted images that seem to lie within the sphere. A larger ball, such as one of the-lawn decorations, works better, because the images it yields are larger and more distinguishable. Within the sphere one sees a world that is more bizarre even than Lewis Carroll's looking-glass world.

If you place a large ball in the center of the lawn and look down on it, the world appears to lie within the ball's circular cross section, which I shall call the image plane. In it your eyes lie at the center and your head is surrounded by the sky. The edge of the lawn (effectively the horizon) forms a circle within the image, and trees that border the lawn stretch toward the center of the image. Tall buildings also stretch toward the center, but they are weirdly distorted with wide bases and narrow tops. Your own image is also warped: your feet seem too large and your body thins grotesquely toward your head like that of a cartoon character. The grass of the lawn rings the edge of the image plane.

The images you see are due, of course, to the reflection of light rays by the ball [see below right]. A ray that originates from h your eyes reflects back to you from the top of the ball; the ray travels down what I call the center line and then back up it. That is why the image of your eyes will always be at the center of the image plane. Rays that originate from other objects reflect and then travel to you at some angle to the center line. Berry pointed out that the extreme case is a ray that barely skims the side of the ball and is deflected negligibly by the reflection. Since it is approximately tangent to the ball, I shall call it a tangent ray. It marks the boundary of what you h can see. Any point on the ground nearer to the ball than the origin of the tangent ray is hidden from view. Everything else in your surroundings has an image in the ball, except of course where your body happens to block rays from reaching the ball.

One of the rays you see originates at what is effectively the horizon, and it approaches the ball along a horizontal path. Rays that reflect higher on the ball than the horizontal ray approach the ball along a path that is angled downward, and they are said to originate above the horizon. Rays that reflect to you from points on the ball lower than the horizontal ray approach the ball from a path that is angled upward, and they are said to originate below the horizon. (This last set of rays is important in the discussion that follows. Notice that these images of the ground lie above the middle of the ball and between the points where the horizon and tangent rays reflect.)

Each reflection obeys a firm rule: The initial ray and the reflected ray must have the same angle with respect to the radial line that runs from the center of the ball to the point where the ray strikes the ball's surface. In the case of the tangent ray the angles between the two rays and the radial line are both 90 degrees; in the case of the ray from your eyes the angles are zero. How the horizon ray reflects depends on the distance between you and the ball. If the distance is large compared with the ball's diameter, the angles for the horizon ray are about 45 degrees and the reflected ray is approximately vertical. If the distance is short, you see another horizon ray that reflects somewhat higher on the ball. The angles involved in the reflection are then larger and the reflected ray slants away from the vertical toward you.

The dark triangle that appears when a cluster of three balls rests on a dark background was noticed by Bob Miller, a San Francisco artist. His display of it can be seen at that city's marvelous playland of science, the Exploratorium. I wondered about the triangle and also about the multiple images of the balls and the sky that are created in the cluster of balls. How exactly do light rays reflect to make the images, and which ball gives rise to any particular image?

Initially I thought the task of identifying the images was virtually impossible. Every time I looked into a cluster of three garden-decoration balls and tried to draw more than a few of the images I became hopelessly confused. I therefore removed one ball and studied the images in the remaining two. In each ball there was then a series of at least a dozen images surrounding the point where the balls touched.

The array is similar to the multiple images you see if you stand between two parallel mirrors. In each mirror you see a series of images of that mirror's frame and the other mirror's frame. The largest image is due to rays that leave the opposite mirror and then reflect to you from the mirror you face. The next image is due to rays that leave the mirror you face, reflect from the opposite mirror and then again from the first mirror. The extra distance traveled by this set of rays makes the image smaller. Each additional image involves one more reflection and so is smaller than the preceding image.

When you look at reflections in adjacent balls, you see an edge of a ball instead of a mirror frame, and the images shrink faster than images in flat mirrors because of the curvature of the reflecting surfaces. Some of the reflections are shown in the illustration below. The top of ball B bears an image of the sky that is due to rays reflecting once. The next-lower image comes from ball A. Its top border is due to a tangent ray from A that reflects to you from B. Just below the border you see an image of the sky that has reflected twice-once from A and once from B.

Somewhat farther down there is an image that comes from B. The top border is due to a tangent ray leaving B, reflecting from A and then from B again. Just below the border you see an image of the sky that has been reflected in the sequence BAB. The rest of the images that you see in B are similar: the images alternate between being reflected first by A or by B, and each time the top border of the image is due to a tangent ray.

Wondering if the reflections could be catalogued better, I studied the tangent rays from A, starting with one that leaves high on the ball and is reflected by B toward the right [see illustration above]. Imagine what happens if you watch a videotape in which the initial ray from A is slowly angled down along the side of A, always staying tangent to the ball. During the descent the resulting ray reflected by B swings leftward through the vertical and then toward A, and the reflection sequence is AB.

Stop the tape when the final ray, like the initial ray, is tangential to A and note that at that moment the light happens to be reflected along a radial line through B. Such a reflection sends the light back along its previous path, so that it reflects tangentially from A again. When you restart the tape, the light reflects three times in the sequence ABA, and the final ray begins to swing rightward through the vertical and then toward B. When it reaches B, again stop the tape and examine the light path. Note that the lowest segment of the path is horizontal. This feature indicates that the ray's path is symmetrical left and right. Since the initial ray is tangent to A, the final ray is tangent to B.

Continue the tape. There are now four reflections in the sequence of ABAB, and the final ray swings leftward W through the vertical and back to A. Stop the tape when it reaches A. Just then the lowest segment of the light path reflects along a line that is radial to A, and the light is sent back along its previous path and so ends up being tangential to A again.

This much is enough to predict the rest of the taped sequence. An extra reflection is added whenever the final ray becomes tangential to one of the balls. The shape of the light path at that moment can fall into one of three classes that are characterized by the angle of inclination of the lowest segment of the path. In describing the classes I shall call the ball from which the light initially reflects the "original ball" and the other ball the "opposite ball." In class 1 the lowest segment is radial to the opposite ball and the final ray is tangential to the original ball. In class 2 the lowest segment is radial to the original ball. In class 3 the lowest segment is horizontal, which gives left-right symmetry to the path, so that the final ray is tangential to the opposite ball.

If you replay the tape and note the path class when an extra reflection begins, you will find a sequence of 1, 3, 2, 3, 1, 3, 2 and so on. Every other time the lowest segment is horizontal and the path is symmetrical. At intermediate times the lowest segment alternates, slanting first toward one ball and then toward the other. The paths underlying long sequences of reflections can be understood by sketching them in a simplified way that still retains the general shape of each path -see upper illustration at left].

After this analysis I moved the third ball back into place. Where any two balls touched I saw a series of images that resembled the series I had seen in just two balls. The difference was that within any one image of a ball there were now two images, one larger than the other. Inside each of them were two more images, one larger than the other. If you ignore details of the sky in the images and concentrate only on their shape, the array is fractal, reproducing the same pattern on an ever decreasing scale.

In order to identify which ball gives rise to which image, I applied a common technique for multiple reflections. When light rays from an object reflect at a surface such as a flat mirror or a shiny ball, an image is said to be created inside the surface. If the rays then reflect from a second surface, the image inside the first surface serves as the object for the second reflection and an image appears inside the second surface. In a sequence of multiple reflections you can start with the original object and go through the reflections by steps in this way. Each time an image serves as an object for the next reflection, until an observer intercepts the rays and sees the final image.

Figure 6 demonstrates the technique for a sequence in which rays from the sky are first reflected by A and Band then in the order CAB before they are seen on the side of B. The first reflections are not shown, but the combined images of the sky that are produced in A and B are represented by a line joining A and B and having a dot at the B end. In the first part of the illustration the next step in the sequence is indicated. The images in A and B act as objects, and images of them appear in C. The second part of the illustration shows the next step: the images in C act as objects, and new images appear in A. In the third part of the illustration those images act as objects and the final images appear in B, which you see on the side of that ball. Note that with this technique you can determine where the sky image that is first reflected by B ends up. It is to the left of the sky image first reflected by A.

The reflection sequences in the illustration are ACAB and BCAB. After repeating the analysis for other possible sequences, I constructed a map of many of the images seen in B [see upper illustration at left]. Observe that they form levels in the map (and on the ball) and that the ones on a lower level (lower on the ball) are smaller than the ones on a higher level (higher on the ball).

I shall work downward through the shaded ovals in the map until I reach the images I have just described. The top level is labeled B and corresponds to the top of the ball, where light rays from the sky are reflected directly to you. On the left side of the next level there is an image of the sky that is first reflected by A. It surrounds the point where A and B touch and is part of the series seen when only A and B are present. Within that image are two more images. The one on the right is an image in which the rays undergo a reflection sequence CAB. Within that image are two more images that correspond to the sequences BCAB (on the left) and ACAB (on the right). These are the sequences that were shown in the preceding illustration.

You might like to add more layers to the map. The images get smaller and the layers become more crowded. In my

perspective of B the shrinking images of the sky approach a "limit line" that runs from the point where A and B touch to where B and C touch. On the visible part of the ball below the limit line are images of the ground.

What is the limit line? It is a composite of images of the horizon-the far edge of the ground or whatever is the background to the balls [see lower illustration below]. When you view the balls up close, you see that the horizon is curved in all the images; when you are farther away, the curvature is imperceptible and the horizon sections seem to lie along a straight line. The limit line that extends from where A and B touch joins smoothly with the limit line that extends from where B and C touch, and their composite forms one side of the triangle in Miller's demonstration. The other two sides are formed by similar limit lines in the other balls. Below a limit line (in the interior of the triangle) you see only images of the ground that are reflected at least once by the balls. In the center of the triangle you see the ground directly. And so, if the ground is dark then the entire interior of the triangle is dark and that is the source of the dark triangle in Miller's demonstration.

Bibliography

REFLECTIONS ON A CHRISTMAS-TREE BAUBLE. M. V. Berry in Physics Education, Vol. 7, No. 1, pages 1-6; 1972.

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