Fractal Surfaces

Frontmatter

1. Introduction

Abstract

It is impossible to begin a book on fractals without first acknowledging the debt owed to Benoit Mandelbrot. Fractal surfaces, and the many other natural manifestations of fractal behavior (only a few of which will be described here), existed long before Mandelbrot described them, but his 1982 book The Fractal Geometry of Nature collected together many ideas in one place, brought them with the aid of striking graphics to a general audience, and placed the made-up word “fractal” firmly in the modern vocabulary. His popularization of the concepts of fractal geometry has spurred many other workers to explore various aspects of this newly revealed subject.

John C. Russ

2. Measuring the Fractal Dimension of Boundary Lines

Abstract

One of the earliest observations of fractal behavior of boundary lines was carried out by Richardson, although it was published obscurely (Richardson 1961). Driven by an interest in finding relationships between the economic and military conflict between nations and the length of their borders, Richardson tried to measure these lengths from maps. But he quickly discovered that the perimeter lengths varied markedly depending on the scale of the map, regardless of the care he took in performing the measurement.

John C. Russ

3. The Relationship between Boundary Lines and Surfaces

Abstract

A fractal surface can be understood by direct analogy to the fractal boundary lines or profiles already discussed. It is possible to construct a regular fractal surface using the same approach as the Koch island, as shown in Figure 1. In this case, the surface is initially a Euclidean plane. Adding and removing blocks in a regular pattern produces an increase in surface area without changing the enclosed volume. Repeating this operation with ever smaller blocks allows the surface area to increase without limit. The fractal dimension describes the rate at which the surface area increases. For the particular structuring element shown in Figure 1, this is 2.792.

John C. Russ

4. Hurst and Fourier Analysis

Abstract

Although the primary focus of this text is on fractal surfaces, there are some tools which can be best introduced using other data types. Fourier analysis is most often applied to time-varying signals. Likewise, Hurst, or rescaled range analysis of fractals was initially performed on time-based historical data (Hurst, Black et al. 1965). Consider a record of some temporal phenomenon, such as the annual flood of the Nile river, or daily temperature readings, or the Dow-Jones industrial average. These examples and many others exhibit the same kind of self-similarity we have been discussing in earlier chapters.

John C. Russ

5. Light Reflection and Scattering

Abstract

There has been interest in characterizing the quality of surface finish for machined or otherwise man-made objects, and the intentional roughness of surfaces such as catalysts, since long before fractal geometry came along. The same measurement tools which are used for these purposes can also provide the raw data for a fractal interpretation of the surfaces. These include the scattering of light or photons of other energies (from X rays to radio waves), as well as various forms of microscopy. Basically, a surface that is rough will show a pattern of scattered intensity, or of measured brightness in an image, that has variations. Analysis of these brightness variations can be used to describe the roughness.

John C. Russ

6. Modeling Fractal Profiles and Surfaces

Abstract

In order to test and understand the characterization tools, and to compare real surfaces, it is important to be able to generate fractal profiles and surfaces using mathematical models. The description in Chapter 1 of a patterned Koch profile shows one way to do this for a line profile. A similar effect for surfaces can be achieved by adding and removing blocks; as smaller and smaller blocks are added and removed from the surface, the volume beneath it is not changed but the surface area increases without bound. The pattern of blocks added and removed may either be strict (as in the classic Koch fractals) or randomized.

John C. Russ

7. Mixed Fractals

Abstract

From a purely mathematical point of view, a fractal curve or surface is defined as having a statistically self-similar form at any dimensional scale. In other words, the surface area or line length should increase without limit as the image magnification is increased. Some purely mathematical fractal objects, such as the now well-known Mandelbrot set, can be enlarged without limit to show ever more richness of fine detail, as illustrated in Figure 1. The only limitation is the finite numerical precision in the computer, and even with that constraint it is possible to find a tiny representation of the entire M-set magnified more than 20,000 times. Compare this to the magnification of the fern in Chapter 1, Figure 23.

John C. Russ

8. Examples of Fractal Surfaces

Abstract

Discussion of some of the situations in which real surface fractal measurements have been made can be organized in two different ways. The preceding chapters have discussed the various measurement tools, including the kinds of microscopes (light, electron, scanned probe) used to acquire surface range or elevation images and the mathematical procedures applied to obtain the various fractal dimensions. Some mention of the findings from these studies and the use of these tools and simulations for real-world samples has been essential to explain the various approaches, but the emphasis has been on measurement techniques.

John C. Russ

Backmatter