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 numerous 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 types. the focal point is on how definite numbers, equivalent to the golden and silver potential, current themselves inside those platforms. half II indicates what number of an analogous numbers and quantity sequences are with regards to the fashionable mathematical examine of numbers, dynamical structures, chaos, and fractals.

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Efficiency being related to the movement dynamic of the Flowforms in question. Naturally, the potentially vigorous lemniscatory or figure-eight movement activity with interposed chaos can be utilized for mixing processes of many kinds. None of this activity can take place without the influence of surface. The Flowform Method also enables research to continue in terms of Adams' original hypothesis regarding the influence of surface and more specifically pathcurve surfaces upon the quality of water.

Flowing movements can also be superimposed on a wave, so that as the wave moves forward, a strong wind can cause the moving current on the back of the wave to move faster than the wave and overshoot the crest and break. In this process, all that is rhythmic in a wave becomes altered. It takes on a spiral form interspersed with hollow spaces in which air is trapped. "Whenever hollow spaces are formed, water is drawn into the hollows in a circular motion, and eddies and vortices arise". This presents us with a new formative principle: The wave folds over and finally curls under to form a circling vortex.

It is of fundamental importance to projective geometry. In fact, projective transformations can be defined to be those transformations of points and lines that preserve cross-ratio. It should be mentioned that this definition of the cross-ratio assumes successive points are in the order A, B, C, D. There are 24 different orderings of these four points, and the cross-ratios in each of these definitions are also preserved. As a matter of fact, we will find the ordering DBAC and its cross-ratio , DB X= BA DC • CA <2-2' most relevant in what follows.

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