Some classes of self-similar planar fractals

Dušan Pagon

Faculty of Education
University of Maribor, Maribor, Slovenia

We present some well known classes of planar fractals, based on regular polygons, and introduce a new class of fractals, appearing from the construction of simple branching trees. Most of the studied objects are self-similar in a strong sense. Therefore, the self-similar dimension $d_S={\ln N\over\ln k}$, where $N$ is the total number of congruent sub objects and $k$ is the coefficient of similarity, is introduced and its properties are studied. Besides the dimension $d_S$, various other characteristics of the presented fractals are examined, for instance, when the overlapping occurs, what is the equation of the boundary curve, and what is the density of the embedded object. We also explain the concept of an iterated function system and give the IFS-codes for some of the studied fractals. Most of the results can be generalized to the 3-dimensional space (starting, for instance, with regular solids) and into the higher dimensions.

References
Barnsley M 1993 Fractals Everywhere (London: Academic Press)
Camp D 1995 Int. Rev. Math. Educ. 27 pp143-152
Hutchinson J E 1981 Ind. Univ. Math. Journal 30 pp713-747
Kosi-Ulbl I and Pagon D 2002 Int. J. Math. Educ. Sci. Technol. (London) 33 3
Peitgen H O, Jürgens H, Saupe D 1992 Fractals for the Classroom I, II (New York: Springer Verlag)
Zeitler H 1998 Int. J. Math. Educ. Sci. Technol. (London) 29 1
Zeitler H and Neidhart W 1993 Fractale und Chaos: Eine Einführung (Darmstadt: Wiss. Buchges.)
Zeitler H and Pagon D 2000 Fraktale Geometrie: Eine Einführung (Braunschweig: Vieweg Ver.)