Encyclopedia of Computer Science and Technology

functional languages 203Page, Clive. “Clive Page’s List of Fortran Resources.” Availableonline. URL: http://www.star.le.ac.uk/~cgp/fortran.html.Accessed August 4, 2007.Reid, John. “The Future of Fortran.” Available online. URL: http://www.ieeexplore.ieee.org/iel5/5992/27213/01208645.pdf.Accessed August 4, 2007.fractals in computingFractals and the related idea of chaos have profoundlychanged the way scientists think about and model theworld. Around 1960, Benoit Mandelbrot noticed that supposedlyrandom economic fluctuations were not distributedevenly but tended to form “clumps.” As he investigated othersources of data, he found that many other things exhibitedthis odd behavior. He also discovered that the patterns ofdistribution were “self-similar”—that is, if you magnified aportion of the pattern it looked like a miniature copy of thewhole. Mandelbrot coined the term fractal (meaning fractured,or broken up) to describe such patterns. Eventually,a number of simple mathematical functions were found toexhibit such behavior in generating values.Fractals offered a way to model many phenomena innature that could not be handled by more conventionalgeometry. For example, a coastline that might be measuredas 1,600 miles on a map might be many thousands of mileswhen measured on local maps, as the tiny inlets at everybay and beach are measured. Fractal functions could replicatethis sort of endless generation of detail in nature.Fractals showed that seemingly random or chaotic datacould form a web of patterns. At the same time, Mandelbrotand others had discovered that the pattern radicallydepended on the precise starting conditions: A very slightdifference at the start could generate completely differentpatterns. This “sensitive dependence on initial conditions”helped explain why many phenomena such as weather (asopposed to overall climate) resisted predictability.Computing ApplicationsMany computer users are familiar with the colorful fractalpatterns generated by some screen savers. There are hundredsof “families” of fractals (beginning with the famousMandelbrot set) that can be color-coded and displayed inendless detail. But there are a number of more significantapplications. Because of their ability to generate realistictextures at every level of detail, many computer games andsimulations use fractals to generate terrain interactively.Fractals can also be used to compress large digital imagesinto a much smaller equivalent by creating a mathematicaltransformation that preserves (and can be used to recreate)the essential characteristics of the image. Militaryexperts can use fractal analysis either to distinguish artificialobjects from surrounding terrain or camouflage, or togenerate more realistic camouflage. Fractals and chaos theoryare likely to produce many surprising discoveries in thefuture, in areas ranging from signal analysis and encryptionto economic forecasting.Further ReadingFractal Resources & Links. Available online. URL: http://home.att.net/~Novak.S/resources.htm. Accessed August 4, 2007.Gleick, J. Chaos: The Making of a New Science. New York: Viking,1987.Mandelbrot, Benoit. The Fractal Geometry of Nature. New York:W.H. Freeman, 1982.Peitgen, Heinz-Otto, Hartmunt Jüurgens, and Dietmar Saupe.Chaos and Fractals. 2nd ed. New York: Springer, 2004.A Mandelbrot fractal generated using Adobe PhotoShop and theKPT (Kai’s Power Tools) Fraxplorer filter. (Lisa Yount)functional languagesMost commonly used computer languages such as C++and FORTRAN are imperative languages. This means thata statement is like a “sentence” in which the value of anexpression or the result of a function is used in some way,such as assigning it to another variable or printing it. Forexample:A = cube(3)passes the parameter 3 to the cube function, which returnsthe value 27, which is then assigned to the variable A.In a functional language, the values of functions are notassigned to variables (or stored in intermediate locations asfunctions are evaluated). Instead, the functions are manipulateddirectly, together with data items (atoms) arrangedin lists. The earliest (and still best-known) functional languageis LISP (see lisp). Programming is accomplished by