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According to Benoit Mandelbrot, "A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension." [1] Presented here is a list of fractals, ordered by increasing Hausdorff dimension, to illustrate what it means for a fractal to have a low or a high dimension.
Escape-time fractals – use a formula or recurrence relation at each point in a space (such as the complex plane); usually quasi-self-similar; also known as "orbit" fractals; e.g., the Mandelbrot set, Julia set, Burning Ship fractal, Nova fractal and Lyapunov fractal. The 2d vector fields that are generated by one or two iterations of escape ...
Fractals are self-similar geometric objects with both aesthetical and scientific uses. Subcategories. This category has the following 6 subcategories, out of 6 total. ...
Music theory analyzes the pitch, timing, and structure of music. It uses mathematics to study elements of music such as tempo, chord progression, form, and meter. The attempt to structure and communicate new ways of composing and hearing music has led to musical applications of set theory, abstract algebra and number theory.
The complement of the Cantor ternary set is an example of a fractal string. In arithmetical terms, the Cantor set consists of all real numbers of the unit interval [ 0 , 1 ] {\displaystyle [0,1]} that do not require the digit 1 in order to be expressed as a ternary (base 3) fraction.
Chaotic maps and iterated functions often generate fractals. Some fractals are studied as objects themselves, as sets rather than in terms of the maps that generate them. This is often because there are several different iterative procedures that generate the same fractal. See also Universality (dynamical systems).
For n > 3, the result is a 3-dimensional bulb-like structure with fractal surface detail and a number of "lobes" depending on n. Many of their graphic renderings use n = 8. However, the equations can be simplified into rational polynomials when n is odd. For example, in the case n = 3, the third power can be simplified into the more elegant form:
This notion of fractal dimension can be generalized to that of complex dimension, which may be used to deduce geometrical information regarding the local oscillations in the geometry of the fractal. For example, the complex dimensions of a fractal string (such as the Cantor string) may be used to write an explicit tube formula for the volume of ...