The kaleidoscope was invented in 1816 by David Brewster, a British physicist who is remembered for his studies of the

polarization of light. The traditional kaleidoscope consists of a tube containing two or three reflecting surfaces such as mirrors or shiny strips of metal that extend along the full length of the interior. The reflectors form either a V or a triangle. You look into one end of the tube, usually through a small aperture, so that your line of sight passes between the reflectors to the opposite end.

The materials at the far end of the tube come in a number of varieties. Bits of brightly colored plastic or glass are common, as are paper clips, pins and other pieces of metal. Part of what you see in the kaleidoscope is a direct view of the material at the far end. In a two-mirror kaleidoscope the direct view is shaped like a slice of pie. In a three-mirror version it is triangular. In both types you also see reflected images, usually of the direct view. I have long puzzled over these images. Why do some kaleidoscopes produce only a few of them whereas others give hundreds? I supposed the difference must be in the quality of the mirrors, but I was wrong.

Many present-day kaleidoscopes incorporate variations on the traditional designs. In some of them the objects at the far end of the tube are suspended in a viscous, clear oil so that the objects move sluggishly when the tube is rotated. Wheels at the far end are also popular. A wheel is mounted on an axis extending from the tube. Sometimes there are two wheels. Wheels can consist of clear plastic containing either butterfly wings or thinly sliced sections of colored minerals. Alternatively a wheel can be made of translucent colored glass or plastic, forming a composite that resembles a stained-glass window. The idea is to turn the wheels as you look into the kaleidoscope. Some kaleidoscopes contain music boxes. When you wind the instrument, the colored array at the far end rotates for several minutes to the accompaniment of a tune.

Another variation relies on the polarization of light. The far end is covered with a polarizing filter. Just inside the end are strips of clear, stretched plastic wrap. Another polarizing filter lies over the viewing aperture. The colorless plastic is made to appear brilliantly colored by the arrangement of filters. The colors depend on the thickness of the strips and the direction in which they are stretched.

In one version of this kaleidoscope the reflections are generated by a sheet .of shiny metal that has been rolled into a loose cylinder, pulled into a spiral at one end and then inserted into the tube. The reflected images are not sharp reproductions of the direct view but smeared, extended stripes of color. I have a kaleidoscope in which the reflecting surface is the highly polished metal interior of the tube, which gives rise to concentric rings around the direct view.

The reflectors in one of my kaleidoscopes are made of light green plastic. They produce reflected images as mirrors do, but the images are swamped by the light leaking through the transparent sides of the kaleidoscope. I cover the instrument with a cloth to improve the visibility.

Many of today's kaleidoscopes are open at the far end, so that what you see is a variously transformed image of what you point the instrument at. Some of them have a converging lens glued to the far end, with the result that more of the scene is squeezed into the view. One of my kaleidoscopes has a hemispherical lens at the far end; its flat side faces the interior. Between the lens and the mirrors is a small spherical plastic ball. Most of the direct view consists of the scene in front of the kaleidoscope, condensed by the hemispherical lens, but part of it consists of the same scene condensed even more by the spherical ball.

I have saved my two favorite kaleidoscopes for last. One is from Kaleidoscopes by Peach of Austin, Tex. One of the partners there, Peach Reynolds, has been a pioneer in the development of modern kaleidoscopes. One version of his instrument is the first "high tech" kaleidoscope: the traditional bits of colored glass and plastic are replaced with an array of light-emitting diodes controlled by a sound-actuated transducer. When the transducer intercepts sound waves, it briefly lights up the colored diodes. To sample different parts of the array the viewer rotates the far end of the tube. I enjoy sitting in front of my stereo system while the diodes pulse to the beat of the music.

My other prized kaleidoscope is produced by Timothy Grannis and Jack Lazarowski of Prism Design in Essex Junction, Vt. A lens at the far end compresses the external scene. The mirrors form equilateral triangles at both ends. The novel feature is that the mirrors flare toward the viewer, so that the triangle at the far end is smaller than the one at the near end.

The resulting array of reflected images is stunning. They are not spread out on a plane passing through the direct view as in other kaleidoscopes. Instead they form a geodesic sphere composed of triangular sections. The illusion that the array is curved in three dimensions is startling. The illusion is enhanced by thin, bright threads that seem to project radially outward from the sphere into a bright, featureless background. When I point the kaleidoscope at a street scene or an active television screen, the sphere bursts into frenzied motion, each section on the sphere showing something different.

How are the beautiful displays created in traditional and modern kaleidoscopes? Although I knew reflections were involved, I did not understand how the images fitted so neatly into a geometric order. I was just as puzzled after examining several of my kaleidoscopes. They were too small to be enlightening. Determined to make a much larger kaleidoscope, I bought (in a department store) several flat mirrors 1.3 meters long and .3 meter wide.

Ordinarily such inexpensive mirrors yield poor images. Most of the reflection is from a shiny metallic coating on the back of the glass. The front surface gives less reflection. The two reflections result in blurred edges in the image of an object. Nevertheless, my mirrors (made by Hamilton Glass Products, Inc., 200 Chestnut Street Vincennes, Ind. 47591) gave remarkably clear images.

I stood two of the mirrors upright and taped them together so that they formed a V. The reflecting surfaces were on the inside. The tape hinge enabled me to vary easily the angle between the mirrors.

I put a small box on the floor between the mirrors and then counted the images of the box. The number rose as I decreased the angle. Five images were visible when the angle was 60 degrees. In addition I saw images of the mirror edges, two to each side.

Next I looked down at the box from above the mirrors. Again I saw five images of it as well as the direct view. Now I began to understand how a two-mirror kaleidoscope functions. When the angle between the reflecting surfaces is 60 degrees, the kaleidoscope yields a cluster of six views of the object at the far end. The views lie in sectors of pie-slice shape spread around a hub positioned at the intersection of the mirrors.

One of the views is direct and the others are reflections. All seem to lie in the plane through the direct view, in this case the floor. (I call it the image plane.) In toy kaleidoscopes the reflecting surfaces are poor and the images are dull and blurry compared with the direct view. My mirrors yielded reflected images almost as bright and sharp as the direct view.

When I decreased the angle between the mirrors, the sectors shrank as more images appeared on the side of the hub opposite to the direct view. Usually toy kaleidoscopes are not constructed to yield more than five reflected images because the additional ones are too dull and blurry. With my mirrors I could see sharp images even when the angle was only 22.5 degrees, which produced a direct view and 15 reflected images.

How many images appear at a given angle? You might explore this question with two mirrors. (Some physics textbooks give an incorrect answer. Contrary to what they say, the number of images is not necessarily one less than the ratio of 360 degrees to the angle between the mirrors.)

One easy technique is to build up a sketch of what you see, starting with the direct view. The apex is the hub of the full set of clusters of images. Reflect the direct view around one of its mirrors to form a new adjacent sector. This reflection amounts to rotating the direct view around the mirror until it lies again in the plane of the image field. Make another reflection on the other side of the direct view. Continue to reflect each new sector around its far side as you move both clockwise and counterclockwise around the hut until sectors meet or overlap. The radial boundaries of the sectors, which are truly the reflected images of the bottom edges of the mirrors, are called virtual mirrors because they effective ly reflect one sector into the next.

If the sectors finally meet but dc not overlap, the number of images is easy to calculate: one image per sector. If the sectors overlap, the calculation is more difficult. You should consult the study by An-Ti Chai of Michigan Technological University cited in "Bibliography".

I continued my work by taping a third mirror to the first two to form ar equilateral triangle. I thought the arrangement would yield two addition al sixfold clusters because two more points (A and B) could then serve as hubs. When I peered down into the sys tem, I found the entire image plane covered with sixfold clusters. I was so taken aback by the spectacle that I squeezed my head into the system as far as it would go. At least 20 distinguishable clusters stretched in every direction on the image plane, growing ever smaller with distance from the direct view [see Figure 8]. Farther out I saw dozens more that were too small to be clear.

The image field consisted of triangles outlined by reflections of the bottom edges of the mirrors. Within the triangles were images of the box. Superposed on the field were lines that were images of the vertical, interior intersections of the mirrors. Throughout the image field the clusters reproduced in the clusters around points A, B or C in the direct view. The entire field could be made visible by beginning with the direct view and then repetitively reflecting it around each of its boundaries until the plane was filled.

As yet I did not understand how the light rays were reflected from the three mirrors to fill the image plane. I realized, however, that the optics must be generally related to the formation of images in a similar system of two parallel mirrors [see Figure 9].

Consider the rays leaving the small box at the far end of the system. The ones that go directly to the observer create the direct view. Others leave the box at various angles to the central axis of the system. Consider the ray that reflects at the midpoint on one of the mirrors. When it is intercepted, the observer perceives it to have originated from a point that is on a rearward extrapolation of the ray entering the eye. The point lies at the intersection of the extrapolation and the image plane. The image is perceived to be there.

Another ray leaves the box and is reflected first from the bottom mirror and then from the top mirror, each at a point a fourth of a mirror length from an end. Again the observer believes the ray originated on the image plane on a rearward extrapolation of the ray that enters the eye. This time the position is farther from the direct view than in the case of a ray reflected only once.

Another ray in the illustration is reflected three times from the mirrors first at the farther one-sixth point on the top mirror, then at the midpoint on the bottom mirror and finally at the nearer one-sixth point on the top mirror. This ray creates an image that is even farther from the true position of the box. Thus images are created upward along the image plane. The greater the distance is between an image and the direct view, the more reflections are required. Ideally there is no limit to the number of images. Eventually, however, the many reflections make the images blurry. An identical set of images extends downward on the image plane. In kaleidoscopes that have two pairs of parallel mirrors (only a few are constructed this way) images also extend to the left and right along the image plane.

Surely a three-mirror system must function in a similar way, but exactly how? I was determined to follow the light rays that entered the system and were reflected from the mirrors to my eyes. I briefly considered placing a laser at the far end and studying how its beam was reflected as I changed its angle of entrance, but I abandoned the notion because I did not want the beam to end up in my eye. Instead, with the room lights off I began to probe the system with a small laser pointed downward.

I knew that the path of a light ray from the box to my eye (even with intermediate reflections) was the same as the path of a ray from my eye to the box. Therefore I pressed the side of the lightweight laser to one temple, closed the opposite eye and moved my head and the laser so that the beam was pointed to an image of the box. The beam continued downward through the mirror system, illuminating the box and thereby adding a red spot to all the images.

The arrangement was not perfect because the beam left the side of my face somewhat displaced from the path of the light rays traveling from a particular image to my eyes. Still, the small angle of error was usually unimportant except when the beam was reflected 10 or more times. I blew smoke into the system to make the path taken by the beam visible.

I first examined a cluster adjacent to the direct view, systematically pointing the laser at each image of the box in the cluster and then noting how the beam crisscrossed between the mirrors to reach the box. After sketching the paths I reversed the arrows that indicate the direction of travel of the light [see Figure 10]. I then had a sketch of how light rays normally leave the box and are reflected from the mirrors to create the sectors in the cluster. Sector J, which is adjacent to the direct view, is created by light rays that are reflected from the far end of the left mirror. Sector K is due to rays that are reflected from the far end of the bottom mirror and then from the left mirror. Sector O is similar except that the first reflection is from the far end of the right mirror.

Sectors L and N require three reflections. For sector L the rays are reflected from the right mirror, then from the bottom mirror and finally from the left mirror. For sector N they are reflected from the bottom mirror, the right mirror and the left mirror. Sector M is more complicated because the rays are reflected four times: bottom mirror, right mirror, bottom mirror again and left mirror.

Once I had made this illustration I began to appreciate why a three-mirror kaleidoscope produces a richer display than a two-mirror version. If the bottom mirror is removed, only sectors J and O remain; they are independent of reflections from that mirror.

I explored more of the image field with the laser beam, counting the number of reflections associated with each sector [see Figure 11]. As I had expected from my study of a parallel-mirror system, the sectors far from the direct view require more reflections. I also noticed a curious sequence in the reflections. Pick any hub and consider the reflections required by the sectors surrounding it. As you would expect, the sector closest to the direct view requires the lowest number of reflections (call it n) for that particular cluster.

Now move around the hub by one sector both clockwise and counterclockwise. Because they are farther from the direct view, these sectors require one more reflection: n + 1. Move one more sector around the hub in opposite directions. These sectors require another reflection: n + 2. Finally move to the sector farthest from the direct view. It requires yet another reflection: n + 3. Every cluster turns out to have such a mathematical sequence around its hub.

When the direct view consists solely of objects lying in the plane across the far ends of the mirrors, every sector in the image field is identical. If the objects are more distant, the content of the sectors must differ. I had noticed this variation when I moved my mirror system outdoors, set it horizontally on a table and pointed it at a busy street. All the sectors differed in content.

This variation results from the perspective of the external scene available at the far end of the mirror system. A simple example is shown in the illustration above. Two objects, represented by a circle and a rectangle, are at the end of the system farthest from the observer. The circle cannot be seen in the direct view because it is too far off the central axis of the system. It can, however, send rays into the system at a sharp angle to the central axis. They are reflected three times before they reach the observer. He perceives the circle as lying in a sector of the image plane distant from the direct view. Similarly, other objects far from the central axis send their rays into the system at other angles and are perceived as being in other sectors.

The slanted mirrors in the kaleidoscope made by Grannis and Lazarowski distort the image field into what appears to be a three-dimensional geodesic sphere. To examine this distortion I stood in front of one of my mirrors while I tilted its top edge away from me. The image of the floor tilted away as if the bottom edge of the mirror were on the edge of a precipice.

Similarly, each sector in the kaleidoscope with flared mirrors seems to be tilted away from me because the shape of the sectors appears as an equilateral triangle viewed at a slant. As I move my view to one side of the direct view the sectors seem to slant more, creating the illusion that they are or; the side of the geodesic sphere. The rays extending radially outward from the sphere are actually the images of the internal edges of the mirrors where they intersect one another. The bright, featureless background surrounding the sphere is the reflection by the mirrors of one another.

I also investigated image fields produced by mirrors in an isosceles triangle. (The mirrors were not flared.) Two mirrors formed the legs of the triangle and a third mirror served as the base. As I looked into the system I decreased the angle between the legs. More images appeared in the primary cluster around the intersection of the legs; the rest of the image field became cluttered with incomplete reflections of the scene.

When the angle was 22.5 degrees, the primary cluster consisted of the direct view and 15 copies of it (either exact or reflected copies). The rest of the image field consisted of only parts of that cluster [right]. When I looked at the base mirror, I saw the center of the primary cluster. As I changed my angle of view I changed the part of the primary cluster I could see.

The angle of view also determined what I saw in other parts of the field. One way to explain the rest of the field is to employ the concept of virtual mirrors. Note that the base mirror is reproduced in every reflection sector in the primary cluster. Each reproduction of the base mirror serves as a virtual mirror, capable of reflecting the primary cluster. (This reflection amounts to rotating the primary cluster about the virtual mirror until it is again in the image plane.)

When I looked directly into a virtual mirror (as though it were real), I could see the center of the primary cluster, but adjacent virtual mirrors eliminated the rest of the reflection from that mirror. Where the rest of the cluster should have been I saw other perspectives of the cluster reflected by the adjacent virtual mirrors. (This dependence on perspective is not indicated in the illustration.) My most expensive kaleidoscope produces such a display. Initially I believed it must have 16 mirrors. (I dared not open such an expensive instrument to check.) Now I realize it has only three highly reflecting mirrors.

This type of image field is fundamentally different from the field formed by an equilateral triangle. The field is ambiguous, in the sense that the content at a particular place in the field depends on the angle of view into the system. A kaleidoscope with an equilateral triangle produces an unambiguous image field because the content at a particular place does not change when the angle of view is varied.

Ambiguous image fields are not as pleasing to me in spite of the beautiful symmetry in the primary cluster. Are there more configurations of three mirrors that give unambiguous fields? Can they be produced with systems containing more than three mirrors? I shall leave these questions for you with a simplifying question: Is there a geometric figure that can be reflected repetitively about each of its sides until a plane is filled with the figure without overlap? Next month I shall reveal the general solution I have discovered.

You might also enjoy studying systems of mirrors formed as a regular polygon with an even number of sides. Parts of the image fields from them should be similar to the scene generated by two parallel mirrors. Are there any differences? What happens when the polygon has an odd number of sides? What do you see when you look into a kaleidoscope in which the interior is a reflecting cylinder and thus is in effect a polygon with an infinite number of parallel sides? Would you expect a kaleidoscope with curved mirrors to yield unambiguous fields?

I bought my kaleidoscopes at High , Tide/Rock Bottom, 1814 Coventry, Road, Cleveland Heights, Ohio 44118. I Many types of kaleidoscopes are also available from the Light Opera Gallery, Ghirardelli Square, No. 102, San Francisco, Calif. 94109. Kaleidoscopes by Peach is at 10731 Manchaca Road, Austin, Tex. 78748. Prism Design is at 87 Upper Main Street, Essex Junction, Vt. 05452. If you discover any unusual designs or invent some of your own, please let me know.

Bibliography

THE NUMBER OF IMAGES OF AN OBJECT BETWEEN TWO PLANE MIRRORS. An-Ti Chai in American Journal of Physics, Vol. 39, No.11, pages 1390-1391; November, 1971.

MULTIPLE IMAGES IN PLANE MIRRORS. Thomas B. Greenslade, Jr., in The Physics Teacher, Vol. 20, pages 29-33; January, 1982.

REFLECTIONS IN A POLISHED TUBE. Laurence A. Marschall and Emma Beth Marschall in The Physics Teacher, Vol. 21, page 105; February, 1983.

THROUGH THE KALEIDOSCOPE. Cozy Baker. Beechcliff Books, 100 Severn Avenue, Suite 605, Annapolis, Md. 21403; 1985.

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