Picturing Spherical Lissajous Figures

quotient of integers, such as 1,12) so that the angular frequencies O>x and O>yare commensurable, then the curve is closed and the motion repeats itself at regular intervals of time [1]. Fascinating and beautiful graphical representations of three-dimensional spherical Lissajous curves can be generated from these equations: Fig. 4. Clifford Pickover, spherical Lissajous figure created using a graphics supercomputer. Its shape was generated with 6/lp = 2. x= rsin(9t)cos(ljlt) :1:= rcos(9t) y = rsin(9t )sin(ljl t) (3) (4) (5) PICTURING SPHERICAL LISSAJOUS FIGURES where A is the amplitude and a and 0 are the phases of the sinusoids. These equations describe oscillations at right angles and were first demonstrated byJules A. Lissajous in 1857. If O>x/O>y is a rational number (a number that can be expressed as a Clifford A. Pickover (computer scientist), Visualization Systems Group, IBM ThomasJ.Watson Research Center, Yorktown Heights, NY 10598, U.SA Received 24 April 1989. Acceptedfor pub-lication by RogerF. Malina. The classic Lissajous figures in physics can be obtained by the equations References and Notes l. H. Miyahara, K. Endou, A. Domae and T. Satao, "Arrhythmia Diagnosis by the IBM Electrocardiogram Analysis Program", journal ofElectrocardiology 13 (1980) pp. 17-24. See also H. Miyahara, A. Domae and T. Satao, "The Reproducibility of Interpretation of 10 Computer ECG Systems by Means of a Microprocessor-Based ECG Signal Generator", Computers and Biomedical Research 17 (1984) pp. 311-325. 2. C. Pickover, "On the Use of ComputerGenerated Symmetrized Dot Patterns for the Visual Characterization ofSpeech Waveforms and Other Sampled Data", journal oftheAcoustical Society ofAmerica80, No.3, 955-960 (1988). See also C. Pickover, "The Use of Random Dot Displaysin the Study of Biomolecular Conformation", journal ofMolecularGraphics 2, No. 34 (1984); L Peterson, "Picture This", Science News131, No. 25, pp. 392-395 (1987) (and cover picture). 3. B.Julez, ''Experiments in the Visual Perception of Texture," ScientificAmerican232 (1975) pp.34-40. 4. For a discussion of the theory behind the SDP and a precise computational recipe for implementation , see C. Pickover, Computers, Pattern,Chaos and Beauty (New York: St. Martin's Press, 1990). • x =Ax cos( O>xt + 0) y =Aycos(O>yt + a) (1) (2) These curves lie on the surface of a sphere of radius r and exhibit closure when 9/ljl is rational. If9/ljl is not rational, then after a long time the curve will have passed through every point lying on the surface of the sphere. Color Plate A No.3 and Fig. 4 show curves drawn using a graphics supercomputer to render the hidden surfaces and shading. For each point on the curve a sphere is plotted, and the interpenetrating spheres trace out the final object. For artistic purposes , three light sources illuminate the object. Color Plate A No.3 was created with 9/ljl = 0.5. The alternating stripes were generated using black and white spheres. Figure 4 was generated using 9/ljl =2. Although high-powered computers are useful in displaying these figures in a beautiful rendition, readers with access to simple computer graphics routines can explore the unusual properties of the curves simply by plotting points as a function of time. Readers can experiment further by adding a phase term to Eqs. 3-5. Reference l. D. Halliday and R. Resnick, Physics, Pts. 1-2 (New York:Wiley, 1966). Abstracts 361 ...