The dyadic monoid can be visualized as an infinite binary tree; more generally, if the set S has p elements, then the monoid may be represented as a p-adic tree. The automorphisms of the dyadic monoid is the modular group; the automorphisms can be pictured as hyperbolic rotations of the binary tree. A more general notion than self-similarity is Self-affinity. Examples The Mandelbrot set is also self-similar around Misiurewicz points. Self-similarity has important consequences for the design of computer networks, as typical network traffic has self-similar properties. For example, in teletraffic engineering, packet switched data traffic patterns seem to be statistically self-similar. This property means that simple models using a Poisson distribution are inaccurate, and networks designed without taking self-similarity into account are likely to function in unexpected ways. Similarly, stock market movements are described as displaying self-affinity, i.e. they appear self-similar when transformed via an appropriate affine transformation for the level of detail being shown. Andrew Lo describes stock market log return self-similarity in econometrics. Finite subdivision rules are a powerful technique for building self-similar sets, including the Cantor set and the Sierpinski triangle. In cybernetics The viable system model of Stafford Beer is an organizational model with an affine self-similar hierarchy, where a given viable system is one element of the System One of a viable system one recursive level higher up, and for whom the elements of its System One are viable systems one recursive level lower down. In nature Self-similarity can be found in nature, as well. To the right is a mathematically generated, perfectly self-similar image of a fern, which bears a marked resemblance to natural ferns. Other plants, such as Romanesco broccoli, exhibit strong self-similarity. In music Strict canons display various types and amounts of self-similarity, as do sections of fugues. A Shepard tone is self-similar in the frequency or wavelength domains. The Danish composer Per Nørgård has made use of a self-similar integer sequence named the 'infinity series' in much of his music. In the research field of music information retrieval, self-similarity commonly refers to the fact that music often consists of parts that are repeated in time. In other words, music is self-similar under temporal translation, rather than (or in addition to) under scaling. See also References External links "Copperplate Chevrons" — a self-similar fractal zoom movie "Self-Similarity" — New articles about Self-Similarity. Waltz Algorithm Self-affinity Fractals Scaling symmetries Homeomorphisms Self-reference