Lissajous's apparatus used light to trace the patterns. To a pair of tuning forks he attached small mirrors that vibrated at different frequencies in separate planes. A beam of light reflected by the mirrors fell on a nearby screen, where the vibrations generated a slowly changing pattern of interlacing curves. The demonstration so fascinated audiences that the design and operation of harmonographs mushroomed into a widespread pastime.

The pastime still has devotees. Last year, for instance, Katherine 0. Reed of Concord, Mass., joined with her father, Thomas B. Reed, a chemist at the Lincoln Laboratory of the Massachusetts Institute of Technology, to construct a harmonograph as a project for her high school science fair. By the time the fair had ended the senior member of the team found himself deeply immersed in the century-old avocation. As a result the Reeds have since turned out a succession of harmonographs for creating ever prettier doodles, and they are still at it. Their latest machine is a complicated affair that weighs more than 50 pounds.

The Reeds write as follows: "Harmonograms can be generated by a number of devices, all of which have certain elements in common. A 'writing' apparatus of some kind is essential; it can be pen and paper, a beam of light that writes on photographic paper, a feather that traces marks on soot-coated glass, an electric spark that darkens conductive paper or a beam of electrons that writes on the phosphor of an oscilloscope. The 'pen,' whatever form it takes, must be moved across the paper by an oscillating mechanism.

"In general, mechanical harmonographs use either pendulums or cranks for oscillators. The movement of a horizontal crank when viewed from the side is the same as that of a pendulum when viewed from the bottom. Both are examples of harmonic motion Pendulum machines are easier to construct and have the added charm of generating figures that diminish in size as the amplitude of the swing dies away.

"One simple version of the pendulum machine consists of a flashlight suspended upside down by a slender wire. The bulb is masked so that light emerges through an aperture the size of a pinhole. When the flashlight is made to swing in an ellipse and is photographed from the bottom by time exposure in a darkened room, the light pattern takes the form of a spiraling ellipse. The axis of the ellipse rotates because of unavoidable irregularities in the suspension of the flashlight.

"A more sophisticated pendulum machine that generates figures like those made by Lissajous is credited to the British physicist Hugh Blackburn. This pendulum is made by hanging a weight from two slender wires of equal length attached a short distance apart to the bar or beam used for suspension. At a selected point the wires are bound together by a snugly fitting ring; they then form a single strand that can be lengthened or shortened by sliding the ring up or down. Pendulums of this type, often exhibited at science museums, are equipped with a bob in the form of a conical- cup that has a small hole in the bottom for releasing a trail of sand that records the motion.

"A simple version of a machine of the crank type is easily constructed of wood. One made recently by Tom 13arnard of Tucson, Ariz., rests on a board about 3/4 inch thick and 18 by 24 inches in its other dimensions. The board supports three pulleys linked together by a common belt [see Figure 1]. The upper side of the largest pulley, which is about 10 inches in diameter, serves as a turntable for rotating a sheet of paper taped in place. An adjustable crank arm that carries the pen is driven by a smaller pulley. The machine makes its most interesting designs when the diameter of this pulley is not a multiple of the diameter of the largest pulley. The crank arm engages any one of a series of radially spaced holes in the driving pulley. This scheme enables the operator to increase or decrease the throw of the crank as he wishes. The distant end of the arm slides on a notched board at the rear of the machine; the particular notch in which the arm is placed determines the radius of the harmonogram. A conventional ball-point pen is attached to the arm by a sheet metal bracket and a rubber band. A third pulley, turned by hand, powers the device.

"A rather more sophisticated but less portable machine is the twin-pendulum harmonograph [see illustration on the left]. Two pendulums, linked independently to the pen, swing at right angles to each other. A small table supports the paper in contact with the pen. The rate of vibration of each pendulum can be separately adjusted by raising or lowering the appropriate bob. We made our pendulums of 1/2-inch wooden dowel rods a yard long. The rods pivot on knife-edges about 10 inches from the top end. The bobs are tin cans containing approximately six pounds of gravel.

"We found that the pen writes most reliably when it is weighted with two ounces of iron washers. This arrangement results in figures with rather widely spaced lines because the drag of the pen slows the machine quickly. Patterns of closely spaced lines were made by substituting a Leroy lettering pen for the ball-point pen. The Leroy pen exerts only a few grams of pressure. We enlarged its ink capacity by pushing the upper part of the pen into a 1/2-inch length of tubing with an inside diameter of 3/16 inch. We used the No. 0 and No. 1 pen sizes.

"Our most recent machine is a version of the elegantly simple double-elliptic pendulum first described in 1907 by British experimenters. It consists of a small wooden platform weighted with 50 pounds of iron [see lower illustration at left]. The platform is suspended through a yoke and a length of iron pipe by ball-bearing gimbals fixed to the ceiling. The figure is drawn on paper that is taped to the platform; the drawing instrument is a pen attached to one end of a counterweighted lever arm. This simple arrangement generates only circles, ellipses and straight lines. The interest of these figures can be enhanced by making identical drawings and superposing one on the other with a slight displacement; moiré patterns then appear.

"The most astonishing increase in the versatility of the machine can be made, however, by suspending a second, less massive pendulum from

the bottom of the platform, again using a gimbal arrangement. When it vibrates at its own natural period, the shorter pendulum perturbs the motion of the swinging platform in an infinite number of ways. The handsomest patterns are generated when the frequency of the upper pendulum bears a whole-number ratio to that of the lower pendulum-a ratio such as 3:2 or 2:1. The adjustable factors that describe any given figure are then the two amplitudes of the upper pendulum, the phase angle between them, the phase between the upper and lower pendulum, the phase and amplitudes of the lower pendulum and the ratios of the frequencies of the upper and lower pendulums. We have scarcely begun to investigate the variety of patterns that can be generated with the double-elliptic pendulum machine.

"Equally interesting are crank machines, particularly those that generate hypotrochoids and epitrochoids. A hypotrochoid results when one circle is made to roll inside another; an epitrochoid, when one circle rolls outside another. Machines of this type can be adjusted for generating the family of curves that includes the cardioid, limacon, deltoid, astroid, trifolium, quadrifolium, ellipse, circle and even the straight line.

"A simple crank machine has been constructed by Everett Clement of Peterborough, N.H. He undertook the project as a means of investigating the patterns that appear on the face of a cathode-ray oscilloscope when two electrical signals of appropriate wave form and phase relation are applied to the input terminals. His device consists of a motor-driven arm that carries at its outer end a second arm rotated by means of a planetary sprocket and a ladder chain that engages a 'sun,' or fixed, sprocket [see upper illustration above]. A flashlight carried by the second arm is photographed by time exposure in a darkened room. The motion of the first arm describes a circle, whereas the flashlight, which marks a point on the second circle, generates hypotrochoids. The radius of the figure can be adjusted by altering the position of the flashlight on its supporting arm.

"A more complex and versatile machine of this type has been made by H. A. Cata of Geneva, Switzerland. He calls his machine a cyclograph [left]. It generates both hypotrochoids and epitrochoids. In addition to all the figures that can be made with Clement's machine the cyclograph also demonstrates that the cardioid, limagon and nephroid are merely special cases of the epitrochoid. Spirals, cubics, quartics and other curves of higher order are beyond its present capabilities, however.

"All machines except the simple pendulum types can be made to generate stereoscopic harmonograms, which are patterns that appear in three dimensions when viewed through a stereoscope or with the aid of a prism. With practice some people find it possible to see the patterns in three dimensions by crossing their eyes in such a way that the left eye views only the right figure and the right eye only the left one. To make stereoscopic pairs with a double-pendulum machine, such as the one shown in the upper illustration in Figure 4, the pendulums are first released simultaneously. When the resulting figure is finished, its stereoscopic mate is made by releasing one pendulum slightly before the other; thus it has a head start of about five degrees of swing. The resulting figures are then photographically reduced in size so that they do not overlap when they are mounted side by side at a center-to-center distance of 25 inches. When mounted, the figures must be oriented so that they would be in register if they were displaced and superposed. Stereoscopic harmonograms can be made by machines of the gear type merely by increasing the radius of the pen arm about 5 percent when generating the second figure.

"Although several versions of easily constructed harmonographs have been presented along with examples of the figures they generate, the subject would not be complete without at least passing mention of the basic physical and mathematical principles that underlie the production of the designs. Simple harmonic motion has rather a special meaning in the field of physics: the projection on any diameter of a point-moving in a circle at uniform velocity. As rigorously described, the position of the projected point at any instant is equal to the product of the angular velocity of the point, the time in seconds that it has moved and the trigonometric sine of the angle through which it has moved. This type of motion-rapid near the center and slowing down gradually to a stop at one end, then accelerating toward the center only to slow down and stop again at the opposite end-was doubtless as familiar to the caveman dangling from a vine as it is to the modern child on a swing. Other examples include the motion of a weight bobbing at the end of a rubber band and various kinds of vibration-of a violin string, of air compressions in wind instruments of the pipe family and of subatomic particles responsible for the emission of electromagnetic waves. In short, the mode by harmonograms is found at the root of all natural phenomena."

The subatomic particles that made the trails in the unanalyzed nuclear interaction that appeared in this department last month were: trail a, p; b, ⁰; c, K₁⁰; d, ⁺; e, p; f, ^-; g, ^-; h, ⁺.

Bibliography

HARMONIC CURVES. William F. Rigge, S.J. The Creighton University, 1926.

HARMONIC VIBRATIONS AND VIBRATION FIGURES. Joseph Goold, Charles E. Benham, Richard Kerr and L. R. Wilberforce, edited by Herbert C. Newton. Newton and Co., 1909.

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