Beyond Measure: A Guided Tour through Nature, Myth, and by Jay Kappraff

By Jay Kappraff

This ebook contains essays that stand on their lonesome yet also are loosely attached. half I files how numbers and geometry come up in different cultural contexts and in nature: the traditional musical scale, percentage in structure, old geometry, megalithic stone circles, the hidden pavements of the Laurentian library, the shapes of the Hebrew letters, and the shapes of organic kinds. the focal point is on how convinced numbers, akin to the golden and silver potential, current themselves inside those structures. half II exhibits what number of an analogous numbers and quantity sequences are relating to the trendy mathematical examine of numbers, dynamical structures, chaos, and fractals.

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G. de Bruin, Kon. Ned. Akad. Wetensch. Proc. Ser. A84 (indaginationes Mathimaticae 43) 27-37. 8 From Chaos, Fractals, and Power Laws by M. Schroeder. H. Freeman. Used by permission. 3 P. Bak and R. Bruinsma, Phys. Rev. Lett. 49, 249-251. Part I Essays in Geometry and Number as They Arise in Nature, Music, Architecture and Design This page is intentionally left blank 1 The Spiral in Nature and Myth The ocean wave... in essence is a kind of ghost Freed from materiality by the dimension of time. Made not of substance but energy.

9, point A transforms to A', which we call B by first projecting to line m through O and then projecting the resulting point back to x through O'. Point B, in turn, transforms to C and C to D, etc. Such a trajectory is called a growth measure. If A begins near the left fixed point X and moves toward Y, then the trajectories start out with small step sizes, which increase in the mid-section between the fixed points and then decrease in size as they approach Y. To reach Y would take an infinite number of steps.

In the formation of vortices, these surfaces are drawn into the whirlpool". The inside of the vortex turns faster than the outside, and corkscrew-like surfaces appear on the surface of the vortex as the result of the disparity of the motion. The vortex is a figure complete in itself with its own forms, rhythms, and movements. 15 Vortex funnel. 16 System of vortices with which Descartes sought to account for the motion of the heavenly bodies consisted of whirlpools of ether. In the case of the solar system the vortex carried the planets around the sun (S).

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