The perplexing patterns consist of bright lines along which the drop focuses the light your eye intercepts. Such a focused display is said to be a caustic-a term derived from the fact that a lens can focus sunlight to burn a hole in a surface. In recent years the patterns have been analyzed by means of what is called catastrophe theory, and so they are now referred to as examples of catastrophe optics.

The bright lines are often outwardly concave and converge to form cusps [Figure 3]. Sometimes the region within a pattern is partly filled with a series of dimmer interference lines that mimic the orientation and curvature of the brighter ones; somewhere a small bright image of the lamp itself is seen.

When a heavy drop hangs pendulously from the glass, the top of the pattern is outwardly convex and there are few cusps, if any, in the vicinity. Just within the border lie tiny, bright "stars." Their visibility is reduced by interference lines, but you can make them dance if you can gently wiggle the glass. If you can rotate the glass around your line of sight, follow one of the cusps. As it approaches the top of the pattern, it shrinks and then enters the interior of the pattern to become a star.

The variety of the patterns may seem endless, but close scrutiny reveals that they are composed of a handful of basic designs, members of the set of "elementary catastrophes." Study of the patterns also demonstrates something more powerful: the catastrophes seen through a drop are actually sections taken through three-dimensional structures, which in turn are sections taken through mathematical structures of higher dimensionality. Sorting out their linkage has been the work of several researchers, notably M. V. Berry and J. F. Nye of the University of Bristol. Here I explore their results and include others from James A. Lock, my colleague at Cleveland State University, and his student James H. Andrews. I shall also describe a simple experiment that produced the most stunning optical displays I have ever seen.

Before I deal with drops on a vertical surface, I should first examine one sitting on a horizontal surface (in the case of which the bright lines are always outwardly concave). Picture a drop on a horizontal microscope slide that is illuminated from below by vertical light rays. Whenever a light ray passes through a tilted surface, it is refracted: its direction of travel is altered. In this case the rays first encounter a tilted surface when they reach the top surface of the drop. To follow the refraction, consider a vertical section of the drop [Figure 4]. Everywhere in the section the surface of the drop is outwardly convex. Pick a point somewhere along the curve, add a tangent to the surface there, and then construct a "normal"-a line perpendicular to the tangent. If the normal leans away from the vertical toward the left, a ray passing through the point is refracted toward the right. If you pick another point with a normal that tilts more to the left, the ray passing through the new point is deflected more to the right.

You can see the results of the refraction with a laser. Arrange for a microscope slide to span the gap between two boxes of

equal height. Place a drop on the slide and position a mirror below it to reflect the laser beam up through the drop. To reveal the refracted light, hold a white card horizontally somewhere above the drop. If the base of the drop is perfectly circular, the rays all go through a "central axis" running vertically through the center of the drop; on the card you are likely to see only a bright point where some of the rays happen to focus. If the slide is soiled or has tiny scratches, minute irregularities break the circular base of the drop-a dust mote might pull out a "finger" of water or indent a "cove" along the perimeter. Either irregularity imposes ridges and gullies along the side of the drop that alter the refraction of light. If there are only one or two irregularities, though, the surface of the drop in any vertical section remains convex even at a ridge or gully, and the light still yields little of interest on the card.

The story picks up when at least three irregularities are present. Their pull distorts the curvature of the water surface in a new way. In a vertical section the surface is no longer entirely outwardly convex but instead has a section that is concave. The point at which the convex and concave curvatures meet is called a point of inflection. The inflections are not isolated but are arranged along a line running around the drop near the base.

When rays pass through the region near an inflection, they are bunched by refraction and focus along a slanted path extending above the drop. If a card intercepts some of the focused light, the bunching at each inflection point gives rise to a bright spot on the card, and the succession of spots forms a bright line that is called a fold-the simplest of the elementary catastrophes. Because the refracted rays are slanted through the central axis, the fold appears on the opposite side of the central axis from the part of the line of inflections that generates it, that is, the pattern is inverted with respect to the drop.

The distance, horizontally, between a segment of the fold and the axis depends on the tilt of the normals in the part of the inflection line that produces the segment. If the normals are barely tilted, so are the rays, and the fold segment is near the axis. When the normals are tilted somewhat more, the fold segment is more distant. The shape of the fold, then, depends on how the tilt of the normals varies along the inflection line encircling the drop.

An overhead view of a drop (with its irregularities greatly exaggerated and with the inflection line indicated) is shown at the left in Figure 6. As the inflection line passes through a and approaches the cove from the left, surface tension forces it to climb up the drop to avoid passing through a short gully at the back of the cove. Along the way the normals in the inflection line become more tilted. The resulting fold is labeled A in the drawing at the right in the illustration, which shows what you would see if you looked down through a translucent card held over the drop. Another segment of the inflection line passes through c and approaches the cove from the right. It too climbs up the drop, producing the fold labeled c The two inflection lines meet at b, and their folds converge at a cusp (B)-another elementary catastrophe.

A finger irregularity has similar effects, because there is a gully on each side of it where the base perimeter becomes concave as it begins to curve outward. A finger may create two cusps that overlap or, if it is wide, two distinguishable cusps. Folds and cusps are actually sections of three-dimensional catastrophes. Two such structures are the "swallowtail" (part of it resembles the tail of a swallow) and the "elliptic umbilic," which are illustrated in Figures 8 and 9. Each is shown in perspective with its long axis extending away from you, but when the illustration applies to a waterdrop on a horizontal surface, the long axis is vertical. A horizontal card held over the drop takes a cross-sectional slice through the structure, a slice perpendicular to the long axis. What you see on the card is the part of the structure h that is intercepted by the card.

Picture what happens as you mentally take a series of such slices through the swallowtail, starting at the right side of the illustration and moving toward the left. The first cuts yield an outwardly convex fold that gives no hint of the swallowtail's presence, but eventually the distinctive tail pattern begins to appear and then to grow. The point at which the pattern is about to emerge is said to be the "singularity" of the swallowtail. The mental exercise of shifting a slice along the length of a catastrophe's structure, beginning at its singularity, to disclose the presence of the structure is called "unfolding the catastrophe." Can you actually see an unfolded swallowtail on a card when the drop lies on a horizontal surface? The chances are slim: a small tail pattern is lost in the maze of interference lines, and the curvatures of the water surface are usually too mild to yield any larger tail pattern.

Now try the same mental unfolding with the elliptic umbilic. Here the singularity is a point, and the unfolded pattern is a triangular arrangement of three cusps and three outwardly concave folds. The unfolded pattern would reveal the. catastrophe's presence, but again the irregularities usually fail to generate anything large enough to see. (An unfolded swallowtail or elliptic umbilic may seem elusive, but keep them in mind for what is coming up.)

The third and last of the three-dimensional catastrophes is the hyperbolic umbilic, which is also shown in Figure 7. Its singularity consists of two straight folds that meet at a 60-degree corner. As you unfold the catastrophe from the singularity, the pattern splits into two parts. One is an outwardly convex fold. Inside it is a cusp where two short, outwardly concave folds meet. Either the singularity or the unfolded pattern would signal the catastrophe's presence. You cannot see either of them from a drop on a horizontal surface, however-not because they are too small but because a horizontal drop never has the right shape for creating them.

I now turn to a drop clinging to a vertical glass surface, but first I must explain a point of possible confusion. In the

previous examples I argued that the pattern is inverted with respect to the drop. When you look at a distant lamp through a drop just in front of your eye, you do not see an inversion. For example, the bottom of the pattern comes from the bottom of the drop. The lack of inversion is an illusion. The pattern produced on the retina is actually inverted, as it is on a card, but the brain introduces a second inversion so that the world, including the drop and its pattern, appears right-side up.

When the drop is on a vertical surface, the three-dimensional catastrophes unfold perceptibly because of the slump of the drop and the resulting sharper curvatures on the water surface brought about by surface tension. Recall what is seen through a drop on a spectacle lens. When the drop is small, cusps and outwardly concave folds appear. If the drop is distorted appreciably, they may overlap to yield an unfolded swallowtail When the drop is pendulous, the cusps and outwardly concave folds are restricted to the bottom of the pattern. They result from irregularities along the lower part of the drop's base. The top of the pattern, coming from the top of the drop, is an outwardly convex fold with internal stars.

The fold is a tease, because it is the outside part of an unfolded hyperbolic umbilic. Each star comes from a slightly unfolded elliptic umbilic but only in a masked way. The catastrophe's triangular structure is present (with one cusp pointing down), but its folds produce interference lines that run perpendicularly through the folds. What you perceive is an inverted triangular arrangement of the interference lines, with one cusp pointing up.

Nye was able to unfold the hyperbolic umbilic (including the internal part) and the elliptic umbilic. He did it by attaching several layers of opaque tape to a microscope slide, cutting a hole in the tape, filling the hole with a waterdrop and then mounting the slide vertically or at a slant in a microscope. The hole provided a perch for the drop. When the hole was circular, it also ensured that the base of the drop remained circular in spite of the slide's tilt. Irregularities in the cut of the tape provided the catastrophes. By adjusting the focus of the microscope, he could control just where it effectively took a slice through the light, just in front of the drop. Such a close view is said to be in the "near field." The views in my examples are taken with a card or an eye in the "far field."

I wondered if the idea of a perch might allow me to see unfolded catastrophes in the far field. Instead of using tape, I cut a circle 2.8 millimeters in diameter off of a plastic mechanical-drawing template of circles. I applied glue to one side of my one-circle template, laid that side on a sheet of paper to eliminate excess glue and then pressed the template onto a microscope slide, taking care not to smear glue over the circular aperture.

When the glue had cured, I cleaned the back of the slide and blew compressed air from a can into the aperture to clean it. Next I mounted the slide vertically in a laser beam by sticking one end into a mound of modeling clay. The beam was expanded by a lens so that it illuminated slightly more than the aperture, and then it continued on several meters to a white wall. I filled the aperture with tap water by simply touching it with a drop of water from the end of a syringe (an eyedropper would serve as well) and soaked up stray water around the aperture with tissue paper. The entire preparation took only 15 minutes, but when I switched off the room lights, I was prepared for a long search for unfolded catastrophes.

Instead something astonishing appeared on the first try. The top of the meter-wide display on the wall (which came from the bottom of the drop, which is to say from the bottom of the circle) consisted of the routine cusps and outwardly concave folds. Surprisingly, the bottom of the display was similar to the top, unlike what I expected from my experience with a drop on a spectacle lens.

Entranced, I watched as the drop began to evaporate and its curvature changed. All the cusps along the bottom of the display shrank, "pierced" the fold and were transformed into unfolded elliptic umbilics that were masked as stars [see Figure 10]. The lowest cusp and its associated star led the way. The abandoned fold at the bottom straightened, and both it and the stars shifted upward. I felt I was watching things that were alive.

Soon the cusps of the hidden elliptic umbilic of the lead star began to appear. Two of them curved upward like the horns of a bull; the lowest one grew downward. By then the fold had became outwardly convex. Within minutes the emerging elliptic umbilic reorganized to become the internal structure of an unfolded hyperbolic umbilic-it consisted of a downward-pointing corner formed by two straightening folds. The structure marched downward while the fold at the bottom of the pattern advanced upward. When they met, they coincided neatly to form a single corner with straight sides at roughly 60 degrees: the singularity of a hyperbolic umbilic. Then the pattern split, one part becoming an outwardly convex fold and the other becoming an interior corner with curved sides. I had seen the complete unfolding of the hyperbolic umbilic from end to end and also its transformation from an unfolded elliptic umbilic!

I moved the slide and its mount to a window, refilled the aperture and then sighted through the drop at a distant streetlamp in the otherwise dark outdoors. This time the performance was tiny, inverted from what had been on the wall and in white light instead of the laser's red light, but it was no less intriguing. Other small circles cut from the original template worked just as well.

When Lock and I watched a similar performance later, we found I had seen only the first act of the play. The theatrical transformations of the act were dominated by inflection lines at the top of the drop. (When a card in the laser beam shadowed the top of the drop, the bottom of the wall display vanished.) The second act was a reversal of the first act, and its theatrics were dominated by inflection lines at the bottom of the drop. In the first act the wall pattern had been inverted from the drop, but in the second act evaporation had allowed the water surface to sink into the aperture, and the new curvature righted the pattern.

Here is a run-through of the entire play. The action starts when the bottom cusps retreat through the bottom fold and become stars. Recall that a star masks a triangular arrangement of folds and cusps, with one cusp pointing downward. When the first star begins to transform into a corner, the folds on the left and right sides of the triangle emerge from hiding and begin to straighten to form the corner.

Meanwhile, the top fold of the triangle moves upward on the wall, comes out of hiding and sprouts an upward cusp. The cusp is a clone of the one that created the star but is inverted in orientation. For example, if the original downward cusp was on the left side of the wall pattern, its upward clone is on the right side. As another star undergoes the transformation, its corner aligns with the first corner and another cloned cusp sprouts. As more stars are transformed, the corner pattern on the wall brightens and the region of sprouted cusps becomes crowded. The internal structure of the wall pattern comes to resemble an ice-cream cone with spikes sticking out of the top.

In the meantime, the external structure of the pattern on the wall shrinks and takes on the same cone shape. The singularity of the hyperbolic umbilic is reached when the bottom parts of the cones are completely straight and overlapping. The roles of the two structures are then reversed: what had been the external structure continues to shrink while what had been the internal structure continues to expand. The new internal structure begins to release a flock of new stars, each of which migrates downward on the wall, passes through the bottom fold of the new external structure and becomes a cusp. The final pattern on the wall is largely an inversion of the h initial one.

Bibliography

OPTICAL CAUSTICS IN THE NEAR FIELD FROM LIQUID DROPS. J. E. Nye in Proceedings of the Royal Society of London, Series A: Mathematical and Physical Sciences, Vol. 361, No. 1704, pages 21-41; May 3, 1978.

CATASTROPHE OPTICS: MORPHOLOGIES OF CAUSTICS AND THEIR DIFFRACTION PATTERNS. M. V. Berry and C. Upstill in Progress in Optics, Vol. 18. Edited by Emil Wolf. Elsevier North-Holland, 1980.

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