Combination Fractals from Different Shape-Unit on Fern Leaves

Toshiyuki Sato¹, Shinichi Matsumoto¹, and Fusayuki Kanda²

¹The Institute of Low Temperature Science, Hokkaido University, Sapporo 060, Japan
²Hokkaido University of Education, Kushiro 085, Japan

(Received October 9, 1991; Accepted November 13, 1991)

Keywords: Combination Fractals, Dryopteris, Ferns, Sakhalin, Shape-Unit

Abstract. Fractal dimensions of leaves of three Dryopteris ferns from southern Sakhalin were calculated by applying different shapes of boxes, equilateral triangle (Dt), square (Ds) and hexagon (Dh). A combination of these dimensions was illustrated as a function of the side length (r-N) and area (S-N) of the shape-unit. Both r-N and S-N produced similar estimates of the three fractal dimensions. The largest r-N values (1.678-1.693) were measured on D.coreano-montana and the lowest values (1.459-1.594) were measured on D.amurensis. The presentation of the three fractal dimensions in combination characterizes the individuality of the sampleshape, i.e., a relative shifting to triangle-unit for D.amurensis and to hexagon-unit for D.austriaca. Such a combinatorial model of fractals derived from variable shapeunits will serve to expand a new science of the quantification of natural diversity and individual shape-character as well as shape-complexity in the wild.