
Geometrically, fractals have forms, or features, that repeat at different sizes over ranges of scales. These features can repeat exactly, such as the triangles that repeat with scale on a Koch snowflake or Minkowski sausage (Figure 1). Or, these features might repeat statistically, as on ground or abraded surfaces, where these repeating features create selfsimilar patterns of scratches or over a range of scales. Turned surfaces also show these characteristics inside the feed marks of the surface. This can be seen in Figure 2, illustrated by a turned surface on an aluminum alloy measured by an OLYMPUS LEXT OLS4000 laser scanning confocal microscope.
Mathematical fractal shapes can be generated by recursive algorithms, or recipes, such as the repeating triangles in the Koch snowflake (Figure 1). Conventional geometric shapes are generated by equations. This makes fractals particularly wellsuited for computers. Recursive algorithms can be similar to the phenomena that generate topographies.
The fractal dimension can be used to characterize the intricacy or complexity of geometries. The fractal dimension can be fractional. In Figure 1, the Minkowski sausage has a dimension of 1.5 and is more complex than the snowflake, with a dimension of 1.26. For a profile z = z(x), the fractal dimension is greater than or equal to one and less than two. For a measured surface z = z(x, y), the fractal dimension is greater than or equal to two, and less than three.
The value in characterizing topographies
To be valuable, characterizations of topographies must go beyond simply providing another means of describing topographies; particularly for applications in engineering, technology and science, these descriptions of topographies should advance the understanding of topographic interactions, for example, advance the ability to solve problems related to surface metrology.
There are two fundamental kinds of topographic interactions. The first is manufacturing, creating or modifying topographies. Studies of this kind of interaction seek to understand how topographies are influenced by processing conditions during their creation or modification. The second kind of interaction concerns how topographies influence surface behavior. Studies of this kind of interaction seek to understand the relationships between the topographies and performance. These two kinds of interactions are not mutually exclusive. Some kinds of surface interactions can both modify the behavior and be modified by it simultaneously. The understanding of topographic interactions can help advance science and can solve problems in engineering and technology.
The chaotic nature of roughness and topographies
Many natural surfaces have some kind of fractal appearance over a wide range of easily observable scales. Many manufactured objects, however, only appear to be rough and have this kind of fractal appearance when observed at sufficiently fine scales with special instruments, like confocal microscopes.
At the finest scales, work piece microstructures and manufacturing tools can have chaotic constituents and behaviors that interact to create chaotic topographies. Some of these chaotic topographies can be engineered to be useful. However, to do this engineering effectively and repeatedly, design and manufacturing engineers need to be able to characterize these chaotic geometries in useful ways. Because it is adept at describing chaos, fractal analysis can be valuable for facilitating the design and manufacture of rough surfaces.

In classical, or smooth, geometries, the location of every point in relation to every other point on a surface can be known exactly. The order is perfect, and the entropy is zero. The probability of describing the height of every point on the surface is one; therefore, the information content and the entropy would be zero, indicating that the description of the surface geometry is perfect, and there is no complexity in describing the geometry.
The inherent chaotic nature of measured surface topographies
Surface measurements contain heights z as a function of position in x and y, such that z = z(x, y). Real surface measurements do not produce data sets that are like the surfaces that are defined in smooth, classical geometry. There are two reasons for this:
First, the height of a mathematical point cannot be measured on a real surface. This is because it is infinitesimally small. Sensors that make height measurements must sample some kind of interaction with the surface (over a finite sampling zone). That sampling of the surface is then used to determine a height at that position. The measured height in a topographic data set must represent some function of the heights on the surface in that sampling zone. The actual function depends on the individual sensor and how it interacts with the surface. The function might be something like an arithmetic average, a weighted average, or the maximum.
Second, regardless of how many positions have heights measured on a real surface, the exact heights at other positions that have not been measured can only be predicted within some amount of uncertainty. This is because measured surface topographies tend to have chaotic components. Because of this chaotic nature, statistical descriptions are used to characterize real surface topography measurements.
Fractals and basic geometric properties of surfaces
Fractal analysis can be used to characterize basic geometric properties of chaotic surfaces. It is important to understand that basic geometric properties can vary with the scale of observation, or of calculation, on chaotic surfaces. Basic geometric properties include the area of surfaces, the lengths of profiles on the surfaces, the curvature of surfaces, slopes or inclinations on surfaces, and volumes in surfaces. These properties are important because they are probably many of the basic geometric properties that would be used in modeling interactions that create or modify topographies, or properties that influence topographicdependent behaviors. These geometric properties depend on the specific positioning of the heights on the surface—multiscale characterizations of these geometric properties inherently provide more knowledge about the surface than conventional analyses.
Note: A more comprehensive version of this article previously appeared as a chapter in Areal Surface Topography, ©2013 SpringerVerlag Berlin Heidelberg.