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Fringes M. Ft. Cullen
Mike Cullen is Professor of Mathematics and formerchairman at Loyola MarymountUniversity Los Angeles, and has been a in faculty member at Grinnell College, L.S.U., and New Mexico Highlands University.He received his Ph.D. in mathematics at of age 24 fromthe University Iowa and has done post-graduate work in biomathematics at the Universityof Washington. His publications includeresearch papers in classical complex analy? sis and two undergraduate books in mathematical biology. He takes special delight in uncovering nonstandard applications of undergraduate mathematics.
two geometric patterns are superimposed, additional geometric patterns (known as moire patterns) may become visible. In the "Op Art" posters of the 1960's two superimposed screens create additionalimages that appear to change rapidly with the slightest movement of the viewer [2, pp. 239-252], , . To create a moire or "watered" silk, two layers of ribbed silk are steam pressed and a third pattern emerges which resembles reflections on the surface of together a pool of water . The scientific applications of the mathematical theory of overlapping geometric patterns range from waveinterference phenomena in physics to the detection of stress patterns in metals and of aberrations in lenses , , . Two objects can be positioned precisely by arranging for moire patterns to appear with any small displacement in alignment . Particular examples will be given later in the paper. Figure 1 shows a family of concentric circles that has been superimposed on a grid of horizontallines. A third family of oval-shaped curves, the moire pattern of the two original families of curves, can be clearly seen near the vertical bisector of the figure. Notice that each individual oval (or moire fringe) appears as the eye follows successive intersections of lines and circles. The moire fringes arising from
Figure 1 370 THE COLLEGE MATHEMATICS JOURNAL
such families of linesand circles are indeed conic sections, as we will show. We first explain exactly how these fringes are formed. General Moire Fringes
Let f(x, y) = c denote the family of level curves of a function of two variables. When the collection of values that c assumes is discrete, we call the family a grating and write fix, y) = cm, where m ranges over a set of consecutive integers and ck 0, and we...