Newsgroups: sci.fractals Path: unixg.ubc.ca!cs.ubc.ca!nntp.cs.ubc.ca!destroyer!sol.ctr.columbia.edu!math.ohio-state.edu!cs.utexas.edu!uunet!pipex!uknet!gdt!mapimhn From: mapimhn@gdr.bath.ac.uk (I M H Nadiadi) Subject: Re: What are Fractals? Message-ID: <1993Aug7.043741.14556@gdr.bath.ac.uk> Organization: School of Mathematical Sciences, University of Bath, UK References: <23r9sdINNk6r@uwm.edu> <23tej2$2qa@zaphod.axion.bt.co.uk> <23tht0$jt2@math.mps.ohio-state.edu> Date: Sat, 7 Aug 1993 04:37:41 GMT Lines: 236 In the referenced article, edgar@math.ohio-state.edu (Gerald Edgar) writes: > >In <23tej2$2qa@zaphod.axion.bt.co.uk> pwhite@axion.bt.co.uk (Peter White) wrote: >>Hang on, doesn't Man(d)lebrot define a fractal as an object whose >>Hausdorff dimension is greater than its Euclidean dimension ? >> > >In his book THE FRACTAL GEOMETRY OF NATURE, Mandelbrot did, indeed, >define a fractal as an object whose >Hausdorff dimension is greater than its topological dimension. >However, in the second printing there is an appendix added at the end >in which he retracts that definition. > > Gerald A. Edgar In the referenced article, price@convex.csd.uwm.edu (Gregory Neil Price) writes: >Dear Readers of Sci.fractals; > >....... From what I understand, fractals >or fractal analysis was invented by an economist (Benoit Mandelbrot). Also >I would appreciate suggestions on good introductory texts ....... > >G. Price > Fractal geometry provides a means by which to express and simulate both simple and complex, natural geometry (Mandelbrot \cite{Mandelbrot:82}, Feder \cite{Feder:88}, \etal{Peitgen} \cite{PeitgenSaupe:88}, Barnsley \cite{Barnsley:88}). The word ``fractal'' was coined by Benoit Mandelbrot in the early 1970's. It was derived from the Latin word ``fractus'' which aptly means ``broken'', i.e., fragmented or irregular. Mandelbrot \cite{Mandelbrot:82} observed that certain natural geometries, e.g., coastlines, terrain and clouds, exhibited a simplifying invariance under scale, i.e., their geometries possessed similarities that were invariant to changes in magnification or resolution. He discovered that this invariance to scale existed in a large variety of artificial and natural phenomena. This invariance to scale, i.e, ``self-similarity'', is central to ``Fractal Geometry''. A wide class of natural geometries appear to possess this underlying fractal character within a range of scale. Although, Mandelbrot dates the origin of ``Fractal Geometry'' from 1975 \cite{MandelbrotInPeitgenSaupe:88}, he acknowledges that many mathematical objects and artistic drawings whose geometries are now formally perceived as ``fractal'' existed long before that decade (Jones \cite{Jones:91}). ``I have coined `self-affine' and `self-similar' in 1964 (the latter is so accepted now, that its age has become hard to believe), but `affine' goes back to Euler'' -- Mandelbrot \cite{Mandelbrot:86ICTP1}. The formal mathematical definition of a fractal set, as defined by Mandelbrot, is one whose ``Hausdorff dimension is not an integer'' (Peitgen and Richter \cite{PeitgenRichter:86}). Dimensions form the main mathematical tools for the study of fractal sets; ``Dimensions contain much information about the geometry of fractals such as properties of their projections, intersections or local structure'', Falconer \cite{Falconer:91}. They are non-integer indices that can be measured approximately by experiment. They quantify the static geometry of an object, an objective means by which to compare fractal sets and quantify the ``fractal properties'' of objects observed in the natural world. The Hausdorff-Besicovitch dimension (D_{HB}) is a pure, rigorous measure-theoretic definition of a fractal dimension, theoretically applicable to all fractal sets, but, in practice, it is not a viable measure of image ``fractality'' and is impractical to compute. ``Recall that I once defined a fractal, as a set for which D_{HB} > D_{T} (= topological dimension). This `tentative' definition has looked less and less attractive, and I have long abandoned it (my book's \cite{Mandelbrot:82} second and later printings, p. 458). An alternative was to view as fractal those sets for which the dimensions in a certain list (of definitions for fractal dimension) coincide. This alternative is no longer promising'' -- Mandelbrot \cite{Mandelbrot:86ICTP1}. In addition, due to certain anomalies \cite{Mandelbrot:86ICTP3} ``I now fear that the Hausdorff Besicovich definition has lost its earlier `special standing' ''. The Hausdorff-Besicovitch dimension provides the framework by which to derive a variety of other, often related, definitions of fractal dimension and algorithms for their estimation, e.g., Minkowski-Bouligand, mass-box, gap, compass (divider), packing, index-\alpha (spectral exponent) (Falconer \cite{Falconer:90} and Mandelbrot \cite{Mandelbrot:86ICTP1,Mandelbrot:86ICTP2,Mandelbrot:82}). In general, they are practical attempts to measure and quantify our perception of how densely a fractal set occupies the space in which it exists (Voss \cite{VossInPeitgenSaupe:88} and Barnsley \cite{Barnsley:88}). References ---------- @INCOLLECTION{Jones:91, Author = "Jones, H.", Year = "1991", Title = "Fractals before Mandelbrot", Chapter = "1", Pages = "7--34", Booktitle = "Fractals and Chaos", Editor = "Crilly, A.J. and Earnshaw, R.A. and Jones, H.", Publisher = "Springer-Verlag"} @INPROCEEDINGS{Falconer:91, Author = "Falconer, K.J.", Year = "1991", Title = "Dimensions -- their determination and properties", Pages = "211--254", Booktitle = "Fractal Geometry and Analysis", Editor = "B\'{e}lair, J. and Dubuc, S.", Address = "Montr\'{e}al, Canada", Month = "3--21 July", Organization = "NATO Advanced Study Institute and S\'{e}minaire de math\'{e}matiques sup\'{e}rieures", Publisher = "Kluwer Academic Publishers, Netherlands"} @INPROCEEDINGS{Mandelbrot:86ICTP1, Author = "Mandelbrot, B.B.", Year = "1986", Title = "{Self-affine fractal sets, I: The basic fractal dimensions}", Pages = "3--16", Booktitle = "Fractals in Physics", Editor = "Pietronero, L. and Tosatti, E.", Series = "Proceedings of the Sixth Trieste International Symposium", Address = "Trieste, Italy", Month = "9--12 July", Organization = "International Center for Theoretical Physics (ICTP)", Publisher = "Elsevier Science Publishers, B.V. (North-Holland)"} @INPROCEEDINGS{Mandelbrot:86ICTP2, Author = "Mandelbrot, B.B.", Year = "1986", Title = "{Self-affine fractal sets, II: Length and surface dimensions}", Pages = "17--20", Booktitle = "Fractals in Physics", Editor = "Pietronero, L. and Tosatti, E."} @INPROCEEDINGS{Mandelbrot:86ICTP3, Author = "Mandelbrot, B.B.", Year = "1986", Title = "{Self-affine fractal sets, III: Hausdorff dimension anomalies and their implications}", Pages = "21--28", Booktitle = "Fractals in Physics", Editor = "Pietronero, L. and Tosatti, E."} Some Books On Fractals ---------------------- @BOOK{PeitgenJurgensSaupe:92, Author = "Peitgen, H.O. and Jurgens, H. and Saupe, D.", Year = "1992", Title = "Chaos and Fractals: New Frontiers of Science", Publisher = "Springer-Verlag"} @BOOK{CrillyEarnshawJones:91, Editor = "Crilly, A.J. and Earnshaw, R.A. and Jones, H.", Year = "1991", Title = "Fractals and Chaos", Publisher = "Springer-Verlag"} @BOOK{Falconer:90, Author = "Falconer, K.", Year = "1990", Title = "Fractal Geometry: Mathematical Foundations and Applications", Publisher = "John Wiley \& Sons"} @BOOK{Edgar:90, Author = "Edgar, G.A.", Year = "1990", Title = "Measure, Topology and Fractal Geometry", Publisher = "Springer-Verlag"} @BOOK{Barnsley:88, Author = "Barnsley, M.F.", Year = "1988", Title = "Fractals Everywhere", Publisher = "Academic Press"} @BOOK{PeitgenSaupe:88, Editor = "Peitgen, H. and Saupe, D.", Year = "1988", Title = "The Science of Fractal Images", Publisher = "Springer-Verlag"} @BOOK{Feder:88, Author = "Feder, J.", Year = "1988", Title = "Fractals", Publisher = "Plenum Press"} @BOOK{PeitgenRichter:86, Editor = "Peitgen, H.-O. and Richter, P.H.", Year = "1986", Title = "The Beauty of Fractals", Publisher = "Springer-Verlag"} @BOOK{StanleyOstrosky:86, Author = "Stanley, H.E. and Ostrosky, N.", Year = "1986", Title = "On Growth and Form: Fractals and Non-Fractal Patterns in Physics", Publisher = "Nijhoff, Boston, Mass."} @BOOK{Falconer:85, Author = "Falconer, K.J.", Year = "1985", Title = "The geometry of fractal sets", Publisher = "Cambridge University Press"} @BOOK{Mandelbrot:82, Author = "Mandelbrot, B.B.", Year = "1982", Title = "The Fractal Geometry of Nature", Publisher = "W.H. Freeman and Company"} @BOOK{Mandelbrot:77, Author = "Mandelbrot, B.B.", Year = "1977", Title = "Fractals: Form, Chance, and Dimension", Publisher = "W.H. Freeman and Company"} \\ Iqbal M. Nadiadi Internet: mm902p@uk.ac.cranfield \\ // Applied Mathematics & Computing Group (mapimhn@uk.ac.bath.gdr)// \\ The Cranfield Institute of Technology (imn@uk.ac.bath.maths) \\ // Bedfordshire, UK, MK43 0AL. //