Fractal - Characteristics


One often cited description that Mandelbrot published to describe geometric fractals is "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole"; this is generally helpful but limited. Authorities disagree on the exact definition of fractal, but most usually elaborate on the basic ideas of self-similarity and an unusual relationship with the space a fractal is embedded in. One point agreed on is that fractal patterns are characterized by fractal dimensions, but whereas these numbers quantify complexity (i.e., changing detail with changing scale), they neither uniquely describe nor specify details of how to construct particular fractal patterns. In 1975 when Mandelbrot coined the word "fractal", he did so to denote an object whose Hausdorff–Besicovitch dimension is greater than its topological dimension. It has been noted that this dimensional requirement is not met by fractal space-filling curves such as the Hilbert curve.

According to Falconer, rather than being strictly defined, fractals should, in addition to being differentiable and able to have a fractal dimension, be generally characterized by a gestalt of the following features:

  • Exact self-similarity: identical at all scales; e.g. Koch snowflake
  • Quasi self-similarity: approximates the same pattern at different scales; may contain small copies of the entire fractal in distorted and degenerate forms; e.g., the Mandelbrot set's satellites are approximations of the entire set, but not exact copies, as shown in Figure 1
  • Statistical self-similarity: repeats a pattern stochastically so numerical or statistical measures are preserved across scales; e.g., randomly generated fractals; the well-known example of the coastline of Britain, for which one would not expect to find a segment scaled and repeated as neatly as the repeated unit that defines, for example, the Koch snowflake
  • Qualitative self-similarity: as in a time series
  • Multifractal scaling: characterized by more than one fractal dimension or scaling rule
  • Fine or detailed structure at arbitrarily small scales. A consequence of this structure is fractals may have emergent properties (related to the next criterion in this list).
  • Irregularity locally and globally that is not easily described in traditional Euclidean geometric language. For images of fractal patterns, this has been expressed by phrases such as "smoothly piling up surfaces" and "swirls upon swirls".
  • Simple and "perhaps recursive" definitions see Common techniques for generating fractals

As a group, these criteria form guidelines for excluding certain cases, such as those that may be self-similar without having other typically fractal features. A straight line, for instance, is self-similar but not fractal because it lacks detail, is easily described in Euclidean language, has the same Hausdorff dimension as topological dimension, and is fully defined without a need for recursion.

Read more about this topic:  Fractal

Other articles related to "characteristics":

Democratic Party – - History - Characteristics
... In 2001, these frictions - combined with the prospect of a devastating defeat in the upcoming election - led to an exodus of conservative and liberal conservative members around Tusk who joined former members of the UW's senior coalition partner, the conservative Solidarity Electoral Action, to form the new party Civic Platform. ...
Gary Becker - Discrimination
... Arrow is "the valuation in the market place of personal characteristics of the worker that are unrelated to worker productivity." Personal characteristics can be physical ...
History Of Mexico - Prehistory and Pre-Columbian Civilizations - Religion
... They frequently took on different characteristics and even names in different areas, but in effect they transcended cultures and time ... of new gods to the collection of existing gods may have been one of the shaping characteristics for the success during the Christianization of Mesoamerica ... with existing ones that seemed to share similar characteristics or responsibilities ...
Orion (constellation) - Characteristics
... There are alternative ways to visualise Orion ... From the Southern Hemisphere, Orion is oriented south-upward, and the belt and sword are sometimes called the saucepan or pot in Australia and New Zealand ...