Behavior Research Methods, Instruments, & Computers /99/, 23 (2), /60-/65 SESSION 7 PERCEPTION Moderator-William S. Maki, North Dakota State University Simple geometric fractals DARRELL L. BUTLER Ball State University, Muncie, Indiana Recently, vision scientists have begun to explore fractals. This paper describes a set ofprograms that can be used to create fractal and fractal-like drawings. The programs were implemented on the Apple II series of computers. The programs were primarily designed to create determinis- tic and random fractal-like patterns with fractal dimensionality between 1 and 2. A supplemen- tary program computes the box dimensionality, a measure of dimensionality that does not as- sume an infinite recursive process. The advantages of this measure of dimensionality over the more typical self-similar measure are discussed. The field of fractal geometry is new and is growing at a rapid rate. It concerns shapes that are (or could have been) created by a sequence of operations, often iterations. Mathematicians and computer scientists have proposed that fractal geometry provides a valuable model for com- plex natural shapes (Mandelbrot, 1982; Peitgen & Richter, 1986; Peitgen & Saupe, 1988) because most natural shapes are the result of a sequence of operations. Perhaps be- cause fractal geometry provides methods for quantitatively describing natural shapes, vision scientists in psychology have begun to explore them. Cutting and Garvin (1987) investigated the relation between various fractal variables and judgments of complexity. Marchak (1987, 1989, 1990) explored fractal models in regard to perception of tex- tures and surfaces in nature. Gilden and Schmuckler (1989) studied discrimination of fractal contours. Results of these studies suggest that fractal geometry may have some value, but much more work is needed before an evaluation of the relevance of fractal geometry to human perception can be made. To study the perception of fractals, scientists need tools that can assist in the creation of fractal patterns. Com- puters with appropriate software provide such a tool. The author would like to thank Doug Brown, Kathy Kirkhoff, and Teresa Dinsmore for assisting with the development and debugging of early versions of the software and Danica Radivojevic for aid in debug- ging the most recent version. For reprints, contact the author at the Department of Psychological Sciences, Ball State University, Muncie, IN 47306, or OOdlbutler@BSUvaxl.bitnet. For copies of the program, please send a 5.25-in. diskette and diskette mailer. Some software designed specifically to generate fractals has become available, for example, FractalMagic (Bolme, 1987). One problem with most programs currently avail- able is that they are primarily of value in exploring some specific mathematical fractal sets, such as the Mandelbrot and Julia sets. Although these fractal sets are of particular value to mathematicians, they are not as useful to percep- tion researchers because they do not facilitate the develop- ment of experimental stimuli with well-defined charac- teristics along various dimensions. This paper describes a set of programs that can be used to create fractal and fractal-like stimuli and that create drawings such as those shown in Figures 1 and 2. The programs were first implemented on the Apple n. Conver- sion and enhancement of the programs to run on IBM PCs (in the C language) is in progress. In these programs, fractals and fractal-like patterns are iteratively constructed using two shapes, an initiator (or starting shape) and a generator (or replacement pattern). The initiator is a line, the length and orientation of which are controlled by the user. One way to think about the creation of a fractal is to replace the initiator by a pattern of lines (the generator), then replace each resulting line by a scaled-down version of the pattern, continuing this process iteratively. In this paper, the number of replace- ments is referred to as the number of nestings in a picture or drawing. Figure I illustrates the concepts of initiator, replacement pattern, and nestings using a well-known frac- tal, the Koch snowflake. Technically, fractals are created using an infinite num- ber of nestings, or at least a large number. The term Copyright 1991 Psychonomic Society, Inc. 160