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26 27 curvAture And floW waves and vortices, parabola and ellipses As we have considered symmetry thus far, the emphasis has been on the more static geometries of rotation, reflection, etc. With symmetries of curvature, many of which are implicated in motion and growth, these principles are extended to the dynamic 13. The conic sections 4 were first investigated by Menaechmus in Platos Academy in the 4th century B.C., but it was not until the Renaissance that the importance of their role in physics began to be realised. In 1602 Galileo proved that the trajectory of a thrown object described a parabola. Not long after this Kepler discovered the elliptical nature of planetary motion. Later, it was realised that the hyperbolic curves could represent any relationship in which one quantity varied inversely to another as in Boyles Law. Discoveries of this kind epitomise the way in which a broader understanding of the symmetry principles inherent in mathematics began to uncover the hidden unity of nature. Waveforms also express symmetry, both in their length and period a simple sine curve can be thought of as a projection on a plane of the path of a point moving round a circle at a uniform speed 5. In fact, circular motion is a component of any wavelike event. If this movement is regularly increased or diminished it produces a characteristic sine configuration. 1. Vortices formed by a split airstream in an ogan pipe 2. Wave motion in a liquid medium is essentially circular 3. A train of Karman vortices induced by an obstruction 4. Conic sections and elliptic series. 5. Upper and middle Sine waves. Lower River meanders tend to adopt sine profiles.
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