Fractals: Form, Chance and Dimension
Fractals: Form, Chance and Dimension
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DOI：
10.2307/2286682
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发表时间：
1978-06
期刊：
影响因子：
--
通讯作者：
B. Mandelbrot;M. Aizenman
中科院分区：
文献类型：
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作者：
B. Mandelbrot;M. Aizenman
这是我多年来读过的思想和形式上最美丽的书，而这一点更是因为它是一本有点技术性的数学论文。分形是一个全新的数学领域，它模拟了自然发生形式的最跨学科的抓斗，如海岸线和云，晶体，雪花和宇宙结构。这个新的英文版与其triking格式和插图做正义的特质yncratic天才的作者在某种程度上，吝啬的法国版本没有。曼德尔布罗特曾在经济学、工程学、生理学以及数学界的几位精英中担任过主席，他是IBM的数学专家。分形这个术语和分形领域都是他的发明和宠儿，尽管他也虔诚地讲述了布朗运动中涉及的摆动概念的历史，以及埃德蒙·福尼尔·达尔贝、刘易斯·弗莱·理查德和许多为这一理论做出贡献的中情局数学家的工作。分形的概念扩展到随机和非随机集合，并被定义为Hausdorff-Be icovitch维数严格超过拓扑维数的集合。这两个e维都位于零和其中一个工作的欧几里得空间的维数之间。布朗运动的拓扑维数i总是整数，例如，它是1，而Hausdorff-Besicovitch i的值为2，对于其他分形，它的值不一定是整数。Mandelbrot给出了一个有趣的和简单的illulution的方式，在这种艺术的维数取决于观察对象和观察者的分辨能力之间的相互作用。他建议，考虑一个直径为10厘米的球，缠绕着直径为1毫米的粗线。对于距离为10米的观察者来说，它看起来是一个零维点。在10厘米处，我感觉到一个三维球。在10毫米，它似乎是一个一维混乱的线程。在0.1 mm处，每个螺纹将被视为一个柱，该柱又是一个三维图形。每列0.01 mm
This is the most extraordinarily beautiful book in thought and in form that I have read for many years, and that is all the more peculiar for its being a somewhat technically mathematical treatise. Fractals is a whole new field of mathematics that models the most interdisciplinary grab-bag of naturally occurring forms, such as coastlines and clouds, crystals, snowflakes and cosmological structures. This new English edition with its triking format and illustrations doe justice to the idio yncratic geniu of the author in a way that the parsimonious French ver ion did not. Mandelbrot, who ha had chairs in economics, engineering, physiology, as well as everal of the choice t plums of the world of mathematics, is a jack-of-all-mathematical trades to IBM. Both the term and the field of fractals are his invention and pet, though he is also reverently anecdotal about the' prior history of the concepts of wiggliness involved in Brownian motion and the work of Edmund Fournier D'Albe, Lewis Fry Richard on and the many cia ical mathematicians who hav contributed to this theory. The idea of fractals extend to both random and non-random sets, and is defined as a set for which the Hausdorff-Be icovitch dimension trictly exceeds the topologica] dimen ion. Both of the e dimensionalities lie between zero and the dimension of the Euclidean space in which one works. The topological dimension i always an integer for Brownian motion, for example, it is unity, whereas the Hausdorff-Besicovitch i of value two, and for other fractals its value is not necessarily integral. Mandelbrot gives an intere ting and simple illu tration of the way in which dimensionalities of thi art depend on an interaction between the object observed and the resolving power of the observer. Consider, he suggests, a ball 10 cm in diameter, wound of a thick thread 1mm in diameter. To an observer at a di tance of 10 m it appears as a zerodimensional point. At 10 cm it i perceived a a three-dimensional ball. At 10 mm it seems to be a one-dimensional mess of thread. At 0.1 mm each thread would be seen as a column which is again a three-dimensional figure. At 0.01 mm each column