Read the masters! Experience has shown that this is good advice for the serious mathematics student. This book contains a selection of the classical mathematical papers related to fractal geometry. For the convenience of the student or scholar wishing to learn about fractal geometry, nineteen of these papers are collected here in one place. Twelve of the nineteen have been translated into English from German, French, or Russian. In many branches of science, the work of previous generations is of interest only for historical reasons. This is much less so in mathematics.1 Modern-day mathematicians can learn (and even find good ideas) by reading the best of the papers of bygone years. In preparing this volume, I was surprised by many of the ideas that come up.
Table of Contents
Introduction -- J.D.MEMORY -- Blake and Fractals -- 1 KARL WEIERSTRASS -- On Continuous Functions of a Real Argument that do not have a Well-Defined Differential Quotient -- 2 GEORG CANTOR -- On the Power of Perfect Sets of Points -- 3 HELGE VON KOCH -- On a Continuous Curve without Tangent Constructible -- 4 CONSTANTIN CARAMODORY -- On the linear Measure of Point Sets-a Generalization of the Concept of Length -- 5 FEUX HAUSDORFF -- Dimension and Outer Measure -- 6 KARL MENGER -- General Spaces and Cartesian Spaces -- 7 GEORGES BOUUGAND -- Improper Sets and Dimension Numbers (excerpt) -- 8 L. PONTRJAGIN AND L. SCHNIRELMANN -- On a Metric Property of Dimension -- 9 A. S. BESICOVITCH -- On the Sum of Digits of Real Numbers Represented in the Dyadic System (1934) -- 10 A. S. BESICOVITCH -- On Rational Approximations to Real Numbers (1934) -- 11 A. S. BESICOVITCHANDH. D. URSELL -- On Dimensional Numbers of Some Continuous Curves (1937) -- 12 PAULLM -- Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole -- 13 P. A. P. MORAN -- Additive Functions of Intervals and Hausdorff Measure (1946) -- 14 J. M. MARSTRAND -- The Dimension of Cartesian Product Sets (1954) -- 15 A.S. BESICOVITCHANDS.J. TAYLOR -- On the Complementary Intervals of a Linear Gosed Set of Zero Lebesgue Measure (1954) -- 16 GEORGES DE RHAM -- On Some Curves Defined by Functional Equations -- 17 A. N. KOLMOGOROV AND V. M. TIHOMIROV -- e-Entropy and e-Capacity of Sets in Functional Spaces (exerpt) -- 18 KARL KIESSWETTER -- A Simple Example of a Function which is Everywhere Continuous and Nowhere Differentiable -- 19 BENOIT MANDELBROT -- How Long is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension (1967) – Index -- Permissions and Acknowledgments.