Curves and Fractal Dimension
Curves and Fractal Dimension
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DOI：
10.1007/978-1-4612-4170-6
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发表时间：
1994-11
影响因子：
2.7
通讯作者：
C. Tricot
中科院分区：
文献类型：
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作者：
C. Tricot
一个数学家，一个真实的数学家，一个数学对象是抽象的，只存在于他的头脑中或某个遥远的柏拉图宇宙中的数学家，永远不会”看到”曲线。曲线是无限狭窄和不可见的。然而，我们都”看到”直线，圆，抛物线等，当许多年前（对我们中的一些人），我们在学校教小学几何。E.马赫想从物理学中压制一切不能被感知的东西：物理学和形而上学不能共存。许多科学家深受他的哲学影响。克劳德·特里科在他的书中告诉我们，曲线有一个非零宽度。它的宽度是铅笔或钢笔在纸上，或粉笔在黑板上。在这里，我们看不到的、也不真正关心的抽象曲线，是包含它的所有粗曲线的交点，对克劳德·特里科来说，只有粗曲线才是相关的。他详细描述了曲线宽度减小时，曲线上出现的凸起、峰值和不规则性。这不是一个新的观点。事实上，豪斯多夫和Bouligand发起的想法在本世纪初。然而，克劳德特里科设法完善理论广泛和有趣的。他的方法既现实又数学严谨。数学家谁只饲料的抽象以及工程师谁解决有形的问题将享受阅读这本书。
A mathematician, a real one, one for whom mathematical objects are abstract and exist only in his mind or in some remote Platonic universe, never" sees" a curve. A curve is infinitely narrow and invisible. Yet, we all have" seen" straight lines, circles, parabolas, etc. when many years ago (for some of us) we were taught elementary geometry at school. E. Mach wanted to suppress from physics everything that could not be perceived: physics and metaphysics must not exist together. Many a scientist was deeply influenced by his philosophy. In his book Claude Tricot tells us that a curve has a non-vanishing width. Its width is that of the pencil or of the pen on the paper, or of the chalk on the blackboard. The abstract curve which cannot be seen and which does not really concern us here is the intersection of all those thick curves that contain it. For Claude Tricot it is only the thick curves that are pertinent. He describes in detail the way bumps, peaks, and irregularities appear on the curve as its width decreases. This is not a new point of view. Indeed Hausdorff and Bouligand initiated the idea at the beginning of this century. However, Claude Tricot manages to refine the theory extensively and interestingly. His approach is both realistic and mathematically rigorous. Mathematicians who only feed on abstractions as well as engineers who tackle tangible problems will enjoy reading this book.