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/vol02/jeh2frac/ | © Copyright 1995 | |||
| Volume 02 | Received: Accepted: |
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Fractals: A Mathematical Framework
John E. Hutchinson |
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| Abstract | |
| We survey some of the mathematical aspects of deterministic and non-deterministic (random) fractals that have been useful in applications. Sets and measures (or grey-scales) are included. Some new results on random fractals are presented. Directions that may be worth further exploration in image compression are marked with a *. Some of the underlying mathematics is explained in more detail, but still at an informal level, in [8]. Other general references at an elementary level are [4] and [12]. Examples of sets with scaling properties, and whose dimension is not an integer, have been known to mathematicians for a long time. However, it was Mandelbrot who introduced the term fractal and who developed the connections between these ideas and a range of phenomena in the physical, biological and engineering sciences (see [10]). One point that sometimes causes confusion is the following. A "mathematical" fractal in a certain precise sense looks the same at all scales; when examined under a microscope at no matter what the magnification it will appear similar to the original object. On the other hand, a "physical" fractal will only display this "self-similarity" for a range of magnifications or scales. The mathematical object will be an accurate model only within this particular range. | |
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