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Symmetry by Dilation Reduction Symmetry 2000, edited by I. Hargittai the title having already been mentioned (at least and T.C. Laurent, I, 2002, 133-141 fleetingly) and italicized, the etymologies of two of those words are worth recalling. In ancient Greek, the scope of summetria went beyond a reference to Symmetry by self-examination in a mirror. A close synonym was in just proportion, and it was mostly used to dilation/reduction, describe a work of art or music as being fractals, and roughness harmonious. In time, most words' meanings multiply and diversify. In developed sciences, groups of invariances-symmetries are abstract and abstruse, but those which underlie fractals are – to Benoit Mandelbrot the contrary – intuitive and highly visual. This paper will show typical ones to be closely related to the simplest mirror symmetry. Roughly speaking, fractals are shapes that look the As to the word fractal, I coined it on some same from close by and far away. Fractal geometry precisely datable evening in the winter of 1975, is famed (or notorious, depending on the source) from a very concrete Latin adjective, fractus, which for the number and apparent diversity of its claims. denoted a stone's shape after it was hit very hard. They range from mathematics, finance, and the Lacking time to evolve, fractal rarely strayed far sciences, all the way to art. from the notion of roughness. Such diversity is always a source of surprise, but this article will argue that it can be phrased so as to become perfectly natural. Indeed, the ubiquity of The Ubiquity of Roughness fractals is intrinsically related to their intimate connection with a phenomenon that is itself The following list combines, in an intentionally ubiquitous. That phenomenon is roughness. It has haphazard fashion, many questions that led to long resisted analysis, and fractals provide the first ideas underlying fractals, and other questions that widely applicable key to some degree of mastery the new ideas found it easy to handle. over some of its mysteries. • How to measure the volatility of the financial A self-explanatory term for looking the same charts, for example to evaluate and compare from close by and far away is self-similar• Of financial risks realistically? course, self-similarity also holds for the straight line • How long is the coast of Britain? and the plane. These familiar shapes are the • How to distinguish proper music (old or new, indispensable starting point in the many sciences good or bad) from plain awful noise? that have perfected a mastery of smoothness. The • How to characterize the course of a river point is that the property of looking the same from untamed by civil engineers? close by and far away happens to extend beyond • How to define wind speed during a storm? the line and the plane. It also holds for diverse • How to measure and compare the surface shapes called fractals. I named them, tamed them structure of ordinary objects, such as a broken into primary models of roughness, and made them stone, a mountain slope, or a piece of rusted iron? multiply. • What is the shape of a cloud, a flame, or a The learned term to denote them is forms welding? invariant under dilation or reduction. In the • What is the density of galaxies in the Universe? professional jargon of some thoroughly developed • How to measure the variation of the load on fields of science, such invariances are called the Internet? symmetries, which would suffice as an excuse for The word, rough, appears in none of those this paper to be included in this book. But there is questions, but the underlying concept appears in another far more direct reason: it will be shown that every one. Irregular is more polite, but rough is more suitable combinations of very ordinary-looking telling. symmetries hold a surprise for the beholder: they An inverse question will provide contrast. produce self-similarity, meaning that they yield • For which shapes do examples of the simplest fractals. smooth shapes of Euclid's geometry provide a The theme having been embedded in the order of sensible approximation? the words in this paper's title, and every word in To Early Man, Nature provided just a few sine/cosine functions. It identifies the fundamental smooth shapes: the path of a stone falling straight and a few harmonics then, in due time, builds a full down, the full Moon or the Sun hidden by a light Fourier series. The latter is periodic, that is, haze, small lakes unperturbed by current or wind. translationally invariant, which, in a broad sense, is In sharp contrast, homo faber keeps adding examples a property of symmetry. Newton's spectral analysis beyond counting. For example, Man works hard at of light is a related example, although the structure eliminating roughness from automobile pistons, of incoherent white light took until the 1930s to be flat walls and tops, Roman-Chinese-American clarified. street grids, and – last but not least – from most More generally, the harmonies that Kepler saw parts of mathematics (contributing to the in the planets' motions have largely been widespread view that geometry is “cold and dry.”) discredited, yet it remains very broadly the case Forgetting this last question, the others set a that the starting point of every science is to identify pattern one may extend forever. The simple reason harmonies in a raw mess of evidence. is that roughness is ubiquitous in Nature. In the works of Man, it may not be welcome, but is not Scale Invariance Perceived as Playing for always avoided, and may sometimes be Roughness the Role That Harmonious unavoidable. Examples are found in some parts of mathematics, where they were at one time Sound Played in Acoustics described as pathological or monstrous, and, once again, in the above list of questions. By contrast to acoustics, the study of roughness Needless to say, each of these questions belongs could not, until very recently, even begin tackling to some specific part of knowledge (science or the elementary questions this paper listed earlier. engineering) and the practical attitude is not to My contribution to science can be viewed as waste time studying roughness but instead to get centered on the notion that, like acoustics, the study rid of it. Fractal geometry, to the contrary, has of roughness could not seriously become a science embraced roughness in all its forms and studies it without beginning by the following step: it had to for its intrinsic interest. first identify a basic invariance/symmetry, that is, a deep source of harmony common to many Roughness Lagged Far Behind Other structures one can call rough. Until the day before yesterday, roughness and Senses, for Example, Sounds, in Being harmony seemed antithetic. An ordering of deep Mastered by a Science human concerns, from exalted to base, would have surely placed them at opposite ends. But change By customary count, a human's number of sense has a way of scrambling up all rankings of this sort. receptors is five. This may well be true but the As candidates for the role of harmonious actual number of distinct sense messages is roughness, I proposed the shapes whose roughness certainly much higher. is invariant under dilation/reduction. Take sound. Even today, concert hall acoustics In the most glaring irony of my scientific life, this are mired in controversy, recording of speech or first-ever systematic approach to roughness arose song is comfortable with vowels but not from a thoroughly unexpected source in extreme consonants, and drums are filled with mysteries. mathematical esoterica. This may account for the Altogether, the science of sound remains delay science experienced in mastering roughness. incomplete. Nevertheless, it boasts great achievements. Meek Symmetries Let us learn some lessons from its success. Typical of every science, it went far by exerting In the form known to everyone, the concept of healthy opportunism. Side-stepping the hard symmetry tends to provoke love or loathing. My questions, it first identified the idealized sound of own feelings depend on the context. I dislike string instruments as an icon that is at the same symmetric faces and rooms but devote all my time reasonably realistic and mathematically scientific life to fractals. Let us now move manageable even simple. It clarifies even facts it indirectly, step by step, towards examples of does not characterize or explain. It builds on the fractals. harmonic analysis of pendular motions and the Among symmetries, the most widely known and of Henri Poincaré. But it did not develop in any best understood involves the relation between an way that mattered outside of esoteric pure object and its reflection in a perfectly flat mirror. To mathematics. Outside mathematics, it remained obtain an object that is invariant – unchanged – dormant until the advent of computer graphics and under reflection, it suffices to “symmetrize” a my work, as exemplified in what follows. Because completely arbitrary object by combining it with its of fractal geometry, Poincaré's idea now matters mirror reflection. As a result, there is an infinity of broadly and will provide us with a nice transition such objects. from the simplest symmetry to self-similar A fairgrounds carousel helps explain symmetry roughness. In the next section, the meaning of with respect to a vertical axis. Symmetry with roughness is a bit stretched; in the section after the respect to a point is not much harder. next, almost natural. Now, replace the symmetry with respect to one mirror by symmetries with respect to two parallel The Self-Similar "Thrice-Invariant Dust mirrors. Any object can be symmetrized D," With Three Part Generator Symmetry “dynamically,” by being reflected in the first mirror then the second, then again the first, again the second, and so on ad infinitum.
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