ABSTRACT

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.

chapter Two|14 pages

On the Power of Perfect Sets of Points

ByGeorg Cantor

chapter Four|28 pages

On the Linear Measure of Point Sets—a Generalization of the Concept of Length

ByConstantin Carathéodory

chapter Five|27 pages

Dimension and Outer Measure

ByFelix Hausdorff

chapter Six|15 pages

General Spaces and Cartesian Spaces

ByKarl Menger

chapter Seven|14 pages

Improper Sets and Dimension Numbers (Excerpt)

ByGeorges Bouligand

chapter Eight|11 pages

On a Metric Property of Dimension

ByL. Pontrjagin, L. Schnirelmann

chapter Nine|16 pages

On the Sum of Digits of Real Numbers Represented in the Dyadic System

(On sets of fractional dimensions II.)
ByA.S. Besicovitch

chapter Eleven|10 pages

Sets of Fractional Dimensions (V): On Dimensional Numbers of Some Continuous Curves

ByA. S. Besicovitch, H. D. Ursell

chapter Thirteen|18 pages

Additive Functions of Intervals and Hausdorff Measure

ByP.A.P. Moran

chapter Fourteen|9 pages

The Dimension of Cartesian Product Sets

ByJ. M. Marstrand

chapter Fifteen|15 pages

On the Complementary Intervals of a Linear Closed Set of Zero Lebesgue Measure

ByA. S. Besicovitch, S. J. Taylor

chapter Sixteen|14 pages

On Some Curves Defined by Functional Equations

ByGeorges de Rham

chapter Seventeen|42 pages

ε-Entropy and ε-Capacity of Sets in Functional Spaces (Excerpt)

ByA. N. Kolmogorov, V. M. Tihomirov