IN THE GLACIER GARDEN IN Lucerne there is a mirror labyrinth built originally for the Swiss National Exhibition of 1896 in Geneva. I entered this Hall of Mirrors expecting it to be a short maze of the kind I solved easily at state fairs as a child. Almost immediately, however, I became so disoriented that I had to begin feeling my way by holding a brochure in front of me to ascertain when I was getting near a mirror.

The maze was laid out on an array of equilateral triangles marked on the floor. Where there were mirrors they were positioned upright on the borders of the triangles (one or two per triangle) and were outlined with elaborate furnishings. In most directions I saw a jumble of images: sections of the furnishings or scenes intended to imitate the Moorish style of the Alhambra in Granada. In certain directions I saw apparent hallways, each seeming to consist of a series of doorways. The door frames were actually the furnishings around mirrors elsewhere in the maze. At the end of each hallway there was one of the Moorish scenes. I set out along one of the hallways but soon found my way blocked by a mirror. Turning, I followed another apparent hallway and again found my progress blocked by a mirror.

Eventually I began to come on the scenes. One, behind a transparent glass wall at a dead end in the maze, was a flower arrangement between two mirrors forming a 60-degree corner. The mirrors made images of the flowers fill my field of view, as if I were looking into a giant kaleidoscope. Along a different route I passed a sultan staring stony-faced at my wanderings. In due course I came to a door that opened into a small room containing furniture. When I faced back into the maze, I saw the sultan at the end of one hallway and a blank wall at the end of another hallway, even though I was then far from the sultan and even farther from the blank wall. The trip back to the entrance was just as confusing as the trip to the room.

Determined to understand the maze, I explored it again, this time making a map of the mirrors. Mary Golrick, my wife, ventured several triangles ahead of me so that I could determine which hallways led into the maze. Once we had reached the small room we retraced our steps by following the map. We no longer had to feel for the mirrors with the brochure.

I left Glacier Garden with many questions. Does one build an optical maze such as the Hall of Mirrors by planning it on paper? Can the images one sees inside a maze be predicted from a floor plan of the setup? Do different mazes have any common properties? What produces the hallway illusion? What determines which scene lies at the end of a hallway?

I began to study the properties of a typical optical maze by drawing features of it on triangular-coordinate paper made by the Keuffel & Esser Co. The graph paper is ruled with equilateral triangles. I drew mirrors in one color and the paths of light reflecting from them in another color.

I also studied small mazes by building models of them with inexpensive mirrors from Jerryco, Inc. (601 Linden Place, Evanston, Ill. 60202). The mirrors are pushed into a bed of sand for stability and oriented with the aid of a protractor. I adjust the alignment by looking at the reflections. The mirrors, which are eight centimeters along a side, are large enough so that I can peer into the entrance of a maze. In order to examine the view as if I were inside the maze, I hold a small mirror at an angle of about 45 degrees to the horizontal and point it toward whatever I want to look at. The mirror reflects that scene back to me. The mirrors do not give clear images, particularly after many reflections of the light, because the weak reflection from the front of the glass muddles the stronger reflection from the rear surface. Still, the mirrors serve well enough for small mazes.

I first studied a maze with a single opening. I soon found I could predict many of the maze's features by drawing the floor plan. In one four-mirror maze, for example, the mirrors are on the borders of equilateral triangles [see Figure 4]. From my studies I realized the light rays creating a hallway image all are reflected from the mirrors at 60 degrees. Hence they always travel parallel to one side of a triangle. To avoid cluttering the floor plan I drew one line, which I call the light path, to represent the rays creating a single hallway image. I numbered the reflections of the light in sequence. The light enters the maze at the point labeled 0.

Looking into this maze along either of the pathway lines extending from it, you perceive a hallway stretching away from you [see Figure 5]. Suppose the mirrors are framed as the mirrors at Lucerne are. The hallway appears to have four door frames. The floor of the hallway consists of the triangles crossed by the light path. At the end of the hallway is an image of whatever lies at the other end of the light path leaving the maze. If you are standing at point 0, the image at the end of the hallway is your own, angled 60 degrees from the front. The light that forms the image travels from you at point 0, through the reflection sequence 4 through 1 and in the end back to 0, at which point it is intercepted by your eyes.

Another way to depict the maze is to draw a floor plan of the hallway you perceive as you look into the maze [left]. Point O lies at both ends of the hallway. The triangles and the door frames passed by the light path lie between the two ends.

If you stand at point O and look into the maze, you see two hallways: one of them extends from reflection 1 and the other from reflection 4. To see either . of them you must face in the proper direction; all other directions yield a jumble of images. If you stand within the entrance triangle, you will see four hallways by turning around. Two of them consist of a single door frame only. If you are in a triangle farther in the maze and out of direct view of the entrance, six hallways extend from you regardless of whether the triangle is bordered by one mirror or two mirrors. Each hallway is parallel to a side of the triangle. If you look in any other direction, you again see a confusing jumble of images.

I soon realized I would never discover the common properties of mazes by drawing them without a plan. I needed to

construct them systematically from some basic starting design and under specified rules. The starting design had to be the simplest of all the mazes that produce a hallway image: a single triangle bordered by two mirrors. The light path for the hallway images in this maze involves one reflection from each mirror. If you stand at the entrance of the maze, you see two hallways, each with two door frames.

From the simplest maze I built more complex mazes by an operation I call a foldout. It entails replacing a mirror on the outer boundary of a triangle in the maze with two new mirrors on an adjacent triangle outside the maze. The maze grows larger because the foldout adds a triangle to it. Sometimes a foldout produces a double-sided mirror that projects into the maze from the perimeter.

The advantage of building a maze by repeated foldouts is that each foldout leaves unchanged the light path in the rest of the maze. After you add a foldout the light path passes through the new triangle with one reflection from each new mirror and then continues through the rest of the maze as before. This fact allows for some generalizations about a maze built from the simplest maze by repeated foldouts. No matter how large or complex the maze becomes, the light path enters the maze, is reflected once from each mirror and then leaves. Hence the hallway you see when you look into the maze at point O must have a door frame for each mirror and an image of you at the other end. Within the hallway you see the entire interior of the maze. When you are inside the maze, each of the six hallways that extend from you has its entrance at its far end

I constructed a fairly complex maze from the simplest one by repeated foldouts [Figure 8]. Note that the procedure left several double-sided mirrors in the maze. The reflections of the light path are numbered in sequence on the basis of an arbitrary choice of direction for the path. (The sequence could be reversed by choosing the opposite direction.) Even without tracing the light path through the maze, I can guarantee that the path is reflected once from each mirror because the maze was built by foldouts.

The illustration also includes a way of representing the sequence of reflections as a loop. The line for the light path enters the loop at point O and runs past all the numbers representing reflections until it leaves the loop at 0. All mazes built by foldouts have similar loops, each loop with a single line running past all the numbers.

Your position within a maze can be represented by a point on the loop. The sections of the loop between your point and the entrance point O represent two of the hallways you can see. Since from any triangle deep within a maze you can see three pairs of hallways, your position can be represented by three points on the loop. If the triangle has no mirrors, the points are well separated on the loop. If the triangle has a single mirror on its perimeter, two of the points are adjacent. If the triangle consists of a 60-degree corner formed by two mirrors, all three points must be adjacent.

Suppose one of the mirrors on the perimeter of the maze is removed to provide a second opening to the maze. For example,

if you mentally remove the mirror associated with reflection 17, the loop is altered. Under this arrangement a line enters at 0, passes 1 through 16 and leaves the loop at 17. Another line enters the loop at 17, passes 18 through 23 and leaves at 0. In brief, the removal of a perimeter mirror splits the loop and produces within the maze two light paths that are independent.

At entrance point O you will still see tw6 hallways, but they are now unrelated. In one direction you see a hallway with door frames corresponding to reflections 1 through 16. In another direction the hallway has door frames corresponding to the reflections from 23 through 18. At the end of each hallway you see whatever lies outside the maze at the new opening. When you are within the maze, some of the hallways extending from you have the original opening at their ends, whereas other hallways have the new opening at their ends. (You can alter the maze in a similar way if instead of making a new opening you insert a nonreflecting scene at position 17.)

What happens to the light path if, the mirror having been returned to position 17, a double-sided mirror is removed from the maze? I removed the mirror for reflections 1 and 20, the one for 3 and 16 and the one for 4 and 11. These moves break the light path into four parts [see Figure 9]. If you stand at the entrance of the maze, you see a hallway consisting of only three door frames (reflections 21, 22 and 23). At the end of it there is an image of yourself.

You can see deeper into the maze only by walking into it as far as the second triangle. There you intercept another light

path consisting of reflections 2, 17, 18 and 19. Since this light path is closed (unbroken by an entrance or a nonreflecting scene), the hallways it generates are in principle infinite in extent. For example, if you face toward reflection 2, you see a hallway that stretches away from you with an infinite number of door frames. Your own image appears periodically along it. (With real mirrors the distant door frames and images of you become too muddled to resolve.)

Much of the maze remains hidden even when you move several more triangles deeper into it. Imagine someone hiding in the maze at the position of the black dot just in front of the mirror that makes reflection 7. When will you first see him in a hallway? He will appear only when you enter the closed light path in which he is standing. You first reach that light path at the colored dot after rounding the corner of the mirror that makes reflections 5 and 22. Before then you might catch tantalizing glimpses of him in the jumble of hallway images, but they give no clue to his location. Even at the colored dot you can be misled because the hallways he is in extend toward reflections 5 and 10. If you follow the hallway toward reflection 10, rounding the Gorner of the mirror that makes reflections 9 and 18, you will finally see the person directly.

The removal of the double-sided mirrors can be studied with a loop representation of the maze. The loss of a double-sided mirror creates a "bridge" in the loop. For example, the loss of the mirror that makes reflections 1 and 20 creates a bridge between those positions in the loop. Similarly, the loss of the 3-16 and 4-11 mirrors creates two more bridges. The line in the loop that enters at 0 and heads toward reflection 1 is immediately redirected by a bridge to reflection 21. Then it passes 22 and 23 and leaves through 0. Another line passes from reflection 2 through reflections 17, 18 and 19, being trapped between two bridges. The third line, which is also trapped between a pair of bridges, consists of reflections 12 through 15. The light path crosses a bridge from 15 toward reflections 3 and 4 but is turned back by another bridge. The final line, which is bounded by the bridge between 4 and 11, consists of reflections 5 through 10.

In this way you can study how the loss of a double-sided mirror alters the light paths in a maze without building or even redrawing the maze. Begin with the loop of the maze, add a bridge between the numbers corresponding to the reflections the mirror had and then draw lines within the loop. The first line begins at 0. When you come to a bridge, take it across the loop. Sometimes you may find another bridge immediately on the other side of the loop.

In one experiment I removed all the double-sided mirrors from the maze [below right]. The loop representation is complex but consists of only two lines, indicating that the maze has only two light paths One line enters at 0, follows a bridge to 21, takes another bridge to 6, 7 and 8 and then passes to 19 by another bridge. From there it takes bridges to 2, 17 and 10. Next it follows a bridge toward 4 but is immediately turned back onto another bridge to 23. Finally it passes from 23 through 0 and out of the maze.

The other line is simpler; it corresponds to a closed light path. It consists of reflections 12 through 15 and is constrained by two bridges. This maze would not be fun for a game of hide-and-seek even though the second light path is far from the entrance to the maze. When you enter the maze, you can see most of the interior. The layout needs no hallways.

The action of removing a double-sided mirror can be generalized with the aid of a loop representation of the maze. Imagine a loop already sectioned by bridges from previous losses of mirrors. If the new removal adds a bridge within a section, that section is split into two sections. The removal therefore splits a light path within the maze into two light paths. If instead the removal adds a bridge between two sections of the loop, the sections are joined. This removal joins two light paths within the maze.

Another way to modify a maze is to add double-sided mirrors. I studied how this step would modify the empty maze I have described [left]. For convenience I renumbered the reflections and redrew the loops as two separated sections. The larger section consists of point 0 and reflections 1 through 9. The smaller section is closed and consists of reflections 1 through 4.

In the example there are two types of insert. On the left side of the maze a mirror is inserted at a point where the two independent light paths cross. The new mirror connects them and hence connects their loop sections. I designated the reflection on one side of the new mirror la (because it is intermediate between 1 and 2 in the initially longer light path) and the reflection on the other side 4a (because it is intermediate between reflections 4 and 1 in the initially shorter light path). These reflections lie on bridges joining the two loop sections.

On the right side of the maze the inserted mirror is at a place where one of the light paths was crossing itself. I labeled the new reflections 4a and 7a. In the loop section the 4a reflection is on a bridge between 4 and 8 and 7a is on a bridge between 5 and 7. This type of insert splits a light path into two paths As with the removal of a double-sided mirror, the addition of such a mirror can either split a loop section and a light path or it can connect two sections and two light paths.

In principle you can construct any maze by beginning with the simple two-mirror version. Expand it with foldouts. So far

there is one light path, and you can see the entire interior by looking along a hallway at the entrance. The hallway exhibits one door frame for each mirror; your image appears at the end of the hallway. When you are in the maze, all six hallways extending from you show an image of the entrance at their end.

If you change the number of double-sided mirrors, make a new opening or replace any mirror with a nonreflecting scene, you split the light path. Your view along a hallway from the entrance or anywhere within the maze is then limited. As you walk through the maze you pass into different light paths and find new hallways with different scenes at their end or ones that extend indefinitely. You might enjoy tracing the light paths and making the loops for the Hall of Mirrors. Imagine walking into that maze. Where will you first see each scene? Where can a person hide from you?

Suppose you see in an unclosed hallway an image of someone who is not in your direct view. The number of images you see depends on the arrangement of mirrors near him and his position in the hallway. Suppose the person stands near one end of a mirror, outside the mirror's triangle [see above]. The light path along which you view him extends toward the lower right. Suppose the person faces in that direction. Then you see a single image of his front that is due to light rays moving directly along the light path.

If he moves into the mirror's triangle, you see an additional image of him that is due to light rays starting from his right rear, reflected by the mirror and traveling down the hallway to you. If he is in a 60-degree corner, you see a third image of him that is due to rays starting from his left rear, reflected once by each mirror and traveling down the hallway.

Can you see multiple images of the person with other arrangements of the mirrors? What is the maximum number of images you can see of that person in a single hallway that is not closed? What is the maximum number of images of yourself that you can see in such a hallway?

Bibliography

MULTIPLE IMAGES IN PLANE MIRRORS. Thomas B. Greenslade, Jr., in The Physics Teacher, Vol. 20, No. 1, pages 29-33; January, 1982.

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