Essentially the relations disclosed by moiré patterns are mathematical. A moiré pattern is the graphic solution of an equation. Problems to be solved by the moiré technique must of course be stated in terms of repetitive figures, but they may range in complexity from simple arithmetic to the calculus of vectors. One class of problems that lends itself readily to solution by the moiré technique involves the determination of surface contours. Amateur telescope makers use it frequently for determining when a glass mirror has been polished to the figure of a paraboloid. During the polishing operation the mirror is examined from time to time through a grid of closely spaced lines known as a Ronchi ruling. The rulings interact with their own distorted shadows to generate a moiré pattern. Polishing is stopped when the mirror maker observes the desired result: the pattern generated by the intersection of a grid with its image as reflected by the mirror when it has become paraboloidal.

A similar experiment can be performed with a teaspoon and a piece of window screening. When the bowl of the spoon is viewed through a flat piece of screening in contact with the convex surface of the bowl, a moiré pattern appears as a set of distorted ellipses that depict the contour. A more striking effect can be created by substituting a rubber balloon for the spoon. When the shape of the balloon is altered, even minutely, the moiré pattern presents an exaggerated picture of the change. Gerald Oster of the Polytechnic Institute of Brooklyn, coauthor of the article mentioned above, has suggested this phenomenon as a stratagem for making a contour map of the moon. He would project a grid of known dimensions on the lunar surface and photograph it. The resulting pattern would disclose surface features that escape not only telescopes but also cameras in close lunar orbit. (When I asked Oster what kind of projector he had in mind, he replied: "Having made the proposal, I leave the details to specialists.")

Other moiré patterns can be generated by materials commonly found in the home: two superposed pieces of nylon stocking, for example, or superposed tracings of the grooves in a phonograph record. The most interesting experiments, however, require accurately drawn figures of known dimension. A set of figures of this type has recently been introduced by the Edmund Scientific Co. of Barrington, N.J. This kit includes eight repetitive figures, each presented twice: once on transparent plastic and again on photographic paper. The kit comes with a manual by Oster entitled The Science of Moiré Patterns. One produces moiré patterns by laying a transparency over its opaque counterpart; constant moving of the transparency will produce constantly changing patterns. The eight figures are as follows: a coarse grid of 13 lines per inch; a fine grid of 65 lines per inch; a grid with logarithmic spacing; 144 equally spaced radial lines, concentric circles with a spacing of 65 lines per inch; a "Fresnel zone" plate; a projection of a sphere, and projections of a cylinder [see illustrations on the right].

The moiré patterns that can be generated by this kit are explained most clearly, according to Oster, in terms of projective geometry. Each of the figures can be regarded as the silhouette of a three-dimensional object made of wire. A cone, for example, can be built up by spacing wire circles of decreasing diameter above one another. When the structure is lighted from the top by a point source of light, shadows of the wires appear as a set of concentric circles. Similarly, an inclined ladder of wires casts a shadow in the form of a grid. The spacing between the shadows of the cross wires varies with the angle at which the structure is inclined: it is at a maximum when the ladder is in contact with its shadow and at a minimum when the ladder makes a right angle in relation to the plane of the shadows. When the wire ladder (which represents a plane) is inclined at some angle above the wire cone, the overlapped shadows generate a moiré pattern in the form of either a circle, an ellipse, a parabola or a hyperbola-the form depending on the inclination of the wire grid with respect to the plane of the shadows [see Figure 7 below].

From the point of view of projective geometry, then, a set of concentric circles is equivalent to a cone; a grid, to a plane. When any figure made up of concentric circles is superposed on a grid of about the same spacing, the combination generates a moiré pattern that belongs to the class of curves known as the conic sections. If the spacing between the lines of the overlapping grid equals that between the concentric circles, say 65 lines per inch, the resulting moiré is a parabola, the curve generated when the intersection parallels the side of the cone.

In the case of a wire model the spacings of the grid can be altered by changing either the inclination of the ladder or its distance from the light. Moving the ladder closer to the light causes the shadows of the grid to spread apart, just as though it were inclined at a smaller angle. At this "inclination" the pattern becomes an ellipse, as can be demonstrated by lifting the transparent grid of the Edmund set while viewing the concentric circles. Two sets of ellipses will be seen because the grid represents either or both of two planes, one inclined to the left and the other to the right. Interchanging the figures-viewing the grid through the circles-decreases the spacing of the grid in relation to that of the circles (increases the angle of intersection). The figures now generate sets of hyperbolas.

The logarithmic grid, as interpreted by projective geometry, is a curved surface: the shadow pattern that would be recast by a ladder bent into a logarithmic curve. Cross wires in the steepest part of the curve cast narrowly spaced shadows. The spacing increases smoothly as the curve flattens. Superposing the concentric circles on the logarithmic grid is equivalent to cutting a cone with a curved knife: the "intersection" generates the whole family of conic sections. Again, the experimenter must expect some ambiguity; a set of concentric circles can be interpreted as the projections of abutting cones. Similarly, the "curve" of the logarithmic grid can be interpreted as bending upward or downward [see illustrations at left].

By the same reasoning a pattern of radial lines can be interpreted as a helix seen by viewing a circular staircase from the top or bottom. The radial lines give no clue, however, to the direction in which the helix rotates. For this reason a pattern of radial lines superposed on a repetitive structure of any kind displays two sets of moiré patterns, and, if it is used as an analogue computer, it generates two solutions. The solution of interest must be distinguished by taking the conditions of the problem into account.

The Fresnel-zone plate is the projection of a paraboloid. The intersection of a pair of closely spaced paraboloids that point in the same direction generates a plane. When a pair of Fresnelzone plates are overlapped at a short center-to-center distance, the resulting moiré pattern consists of straight lines that represent one edge of the planes of intersection. When one parabola (Fresnel-zone plate) of the pair is reversed, the intersection becomes circular. This can be demonstrated by overlapping the figures at a wider center-to-center spacing. Because only about half of each figure is now involved, the ambiguity is reduced in favor of paraboloids that intersect while facing in opposite directions. The moiré pattern takes the form of circles.

The projection of a sphere resembles that of the paraboloid in that each is an array of concentric circles. In the case of the Fresnel-zone plate, however, all lines and all spacings occupy equal areas, whereas in the case of the spherical projection the equivalent areas diminish from the center to the edge. When a grid is superposed on the spherical projection, the moiré patterns depict the intersection of a sphere by a plane, the angle of intersection being determined by the period of the grid spacing with respect to that of the spherical projection. When the logarithmic grid is superposed on the spherical projection, the resulting moiré pattern indicates two points of intersection as though the sphere were cut simultaneously at two points by a curved knife. One cut appears closer to the equator than the other because the curvature of the "knife" constantly increases. It can also be demonstrated that the cylindrical projection is characterized by similar properties.

Many natural phenomena can be simulated by the eight repetitive figures. A pair of piano strings that are tuned to the same frequency and vibrate in step, for example, emit a set of sound waves that in some respects is analogous to a pair of grids overlapped so that the lines match. The effect of detuning one piano string can be demonstrated by lifting one grid slightly above its matching print while keeping the lines parallel. The separation has the optical effect of changing the relative period of the grids. Moiré patterns appear in the form of widely spaced fringes, the counterpart of the beat note emitted as the vibrations of the strings alternately fall into and out of step. The spacing between the fringes of the moiré pattern can be interpreted as an amplified copy of the difference in the spacing of the lines constituting the grids. Therefore it can be used as a sensitive measure of the period of either grid if the spacing of one grid is known. If the fringe spacing is designated d and the periods of the grids are respectively a and b, then d = ab/a - b.

Straight fringes also appear when the grids overlap at a slight angle. This effect can be employed as a sensitive measure of angular differences. The distance d between the fringes is equal to the quotient of the distance a between the lines of the grating, divided by two times the trigonometric sine of half of the angle at which the gratings overlap [d = a/2 sin (/2)]. In the case of the coarse Edmund gratings the line separation, or repeat length of the figure, is .0683 inch. If the transparency overlaps its matching print at an angle of 45 degrees, the fringe spacing is therefore .0685/2 x .3827, or .089 inch. (The sine of 45/2 degrees is .3827.) For angles smaller than five degrees the equation can be approximated to d a/, in which the fringe spacing d is approximately equal to the repeat length of the figure a divided by the angle, expressed in radians. (One radian is about 57 degrees.)

A relatively simple but astonishingly sensitive instrument for measuring small changes in length employs a pair of grids as the amplifying element. According to C. Harvey Palmer of Johns Hopkins University the apparatus can measure changes in rotary motion as small as a billionth of a degree and displacements of a trillionth of a centimeter-less than the diameter of an atom! In an apparatus of simple design light rays from an illuminated grid would pass through a lens and be reflected by a pivoted mirror back through the same lens to focus on a second grid. A condensing lens beyond the second grid would concentrate the light on a photocell. A moiré pattern would appear when the image of the first grid overlapped the second grid. The amount of light reaching the photocell would be determined by the moiré pattern, which in turn would depend on the position of the pivoted mirror, as shown in Figure 12.

This oversimplified apparatus would be sensitive to small changes in the intensity of the lamp, instabilities in the photoelectric circuit and aberrations in the optical system. Such sources of error are balanced out in practical instruments by splitting the light into a pair of compensated beams and illuminating two grids mounted 180 degrees out of phase that energize a pair of photoelectric cells constituting two arms of a Wheatstone bridge. The beams are focused by a well-corrected lens and adjusted for proper phase by a pair of glass plates that can be rotated to after the effective lengths of the optical paths.

In one experiment made with an easily constructed version of the instrument, Palmer supported the mirror by two wires, one of iron and the other of brass. The wires were surrounded by a small coil of copper wire. When the coil was energized by a flashlight battery, the resulting magnetic field altered the length of the iron wire and rotated the mirror only a minute amount, but enough to send the pointer of the meter off scale!

Amateurs on the lookout for an inexpensive method of measuring the absolute amplitude of microseisms, minute changes in the pressure of a gas, the magnetostriction of solids and so on might well investigate this application of the moiré technique. A practical apparatus of the type, specially designed for amateur construction, is described in the book Optics: Experiments and Demonstrations, by C. Harvey Palmer (Johns Hopkins Press, 1962).

Numerous phenomena can be investigated merely by overlapping the repetitive figures. The set of concentric circles, for example, in addition to representing a cone, can be interpreted as a set of concentric waves either radiating from or converging on the center. Because waves of all kinds behave in much the same way the analogue is valid for sound waves, water waves, electromagnetic waves and so forth. Wave interference is easy to demonstrate by over lapping the transparent pattern of concentric circles on its print. Points of destructive interference, where the crest of one wave falls in step with the trough of another, are represented by moiré patterns in the form of hyperbolas. Compare the moiré pattern generated by the pair of concentric-circle figures with the photograph of wave interference in a ripple tank that appears in Figure 11. The same general equations that predict the behavior of waves in a ripple tank apply when the concentric-circle patterns are used as an analogue computer for investigating wave interference in other media, a subject discussed in this department for October, 1962.

Conversely, the properties of some materials can be investigated quantitatively by their effect on repetitive figures. The mineral calcite, for example, has the property of splitting light into two polarized components that are physically displaced. The effect can be demonstrated, and the displacement measured, simply by observing a single figure of concentric circles through the mineral. The resulting moiré pattern is a measure of the optical displacement. Spherical waves, such as those emitted in phase by a pair of radio antennas, are also simulated by the patterns. The effect can be studied by overlapping the transparent figure on its print so that the center-to-center distance is less than one interline spacing. The resulting moiré pattern displays two intense beams 180 degrees apart, demonstrating the directional property of paired antennas. When the center-to-center distance is increased, side lobes appear, again as in the case of radio antennas.

The figure composed of radial lines can also simulate a broad class of natural phenomena characterized by potential fields in two dimensions such as the electrostatic force between elements of a vacuum tube or the motion of fluids from an inlet or toward a sink. In the case of a fluid emerging from a port and spreading uniformly over a surface, the radial lines depict the streamlines. Velocity of flow is represented by the spacing between the radial lines; the higher the velocity, the narrower the spacing. The direction of the flow cannot be determined, however, by examining a single pattern, whether the fluid moves toward or away from the center.

What would be the effect of introducing a sink at some point in the field? This rather difficult mathematical problem is easily solved by overlapping the radial print with its transparent mate so that the centers are relatively close. The case of two inlets and no sink can be demonstrated by sliding the centers of the figures apart. At a certain distance the pattern displays a "stagnation point," the region of opposing flow from the two inlets.

The parallel straight lines of a grid represent the uniform motion of a fluid, as in the case of an ideal river in which the flow is unperturbed, whereas the logarithmic grid represents uniformly accelerated flow. By overlapping the radial figure with the logarithmic grid the interaction of radial flow and linear accelerated flow can be demonstrated.

At points of high linear velocity the stagnation point, indicated by the moiré pattern, moves close to the source of radial flow. The coaxial moiré circles generated by superposing the radial figure on its copy similarly represent, according to Oster, the stress lines in a solid in which the centers of the figures coincide with the locations of applied stress. The patterns also simulate magnetic-field lines between the poles of a horseshoe magnet, just as such lines are conventionally mapped with iron filings.

Why does one see moiré patterns? In investigating this question Oster devised several experiments that provide clues, if not a definitive answer, to the mystery. "A hint as to what compels us to choose only one type of intersection between lines, the type that we recognize as the moiré pattern," he writes, "can be gained by drawing two straight lines that cross at some small angle. Note the conspicuousness of the intersection, where the lines appear pinched as though two wires had been twisted around each other. The effect is enhanced when the intersection is viewed in red light, diminished in blue light. Evidently the eye is unable to resolve the intersection. When many parallel lines cross, as in the case of two overlapped grids, the eye unconsciously searches the field and ties together these preferred points of intersection. The effect is greatest for lines that cross at small angles.

"To examine the effect of intersections in another way, lay a hair on a grid of 65 lines per inch. (Preferably this should be a red hair; it is finer than a blond or brunet one.) Neither the hair nor the rulings can be resolved if the combination is examined at arm's length. Yet the points of intersection between the hair and the rulings will stand out clearly as a line of dots. Incidentally, the dots constitute a direct measure of the first derivative, or slope, of the curve described by the hair. Conversely, as readers familiar with the calculus will recognize, for a dot distribution that follows a certain function the curve of the hair is the solution of the first-order differential equation.

"Many effects can be seen by close observation of single patterns, preferably figures printed on a white background. When the print of the radial figure is examined by one eye at a distance of less than 10 inches, a gray, blurred pattern is observed. The blur arises from the inability of the eye to focus properly at short range on the moiré pattern that develops when the radial figure and its afterimage register on the retina. Other interesting effects appear when the eye is fixed on a pattern that moves. For example, if the coarse-line figure is held by diagonally opposite corners and rocked, a dark bar will link the grasped corners, flanked on each side by curved bars in a lighter tone. Again, this moiré arises from the intersection at a small angle of the immediate image of the grid on the retina and its afterimage. Observe any of the centrosymmetrical figures while you hold it steadily in your hand and move your hand in a circle. Moiré patterns will be seen that arise from the interaction of the immediate images and afterimages and will appear to rotate in step with the movement of the figure.

"It is even possible to see colors by staring at the black and white prints, particularly when attention is fixed on the strongly illuminated center of the radial figure. Various individuals report different colors. I see mostly gold and pink. In addition to appearing colored, the central portion of the circular pattern appears to be in rapid rotation. Children see the effects readily and many adults do not. These effects are intimately tied up with the granularity of the retinal receptors, with the cones of the fovea and also with the rapid involuntary movements of the eyeball-the saccadic movements and tremors. It is conceivable that the fovea acts as an almost unperceived circular figure superposed on our vision, but we are so accustomed to its presence that we do not notice it."

Bibliography

REPRESENTATION AND SOLUTION OF OPTICAL PROBLEMS BY MOIRÉ PATTERNS. Gerald Oster in Proceedings of the Symposium on Quasi-Optics. New York Polytechnic Press, in press.

The Society for Amateur Scientists (SAS) is a nonprofit research and educational organization dedicated to helping people enrich their lives by following their passion to take part in scientific adventures of all kinds.