Design for Manufacturing of IFS Fractals from the Perspective of Barnsley's Fern-leaf

ABSTRACT

This study deals with the design for manufacturing (DFM) of fractals created by a random walk called iterated function system (IFS). In particular, the DFM of an IFS-created fractal called Barnsley's fern-leaf is considered. The IFS dedicated for creating virtual models of a fern-leaf uses a set of four strictly-contracting affine mappings in the onto manner. The interactions among these mappings are studied in detail in order to identify some data structures. Based on the identified data structures, a DFM procedure is proposed. In the proposed DFM procedure, three out of the four mappings are employed in both the onto and one-to-one manner. The proposed DFM procedure is applied to the redesign of the shape (fern-leaf). Physical models of the redesigned fern-leaf are manufactured using both additive and subtractive manufacturing technologies (3-D printing and milling). The factors affecting accuracy of the physical models are also described. Although this study is limited to the shape of the fern-leaf, other IFS-created shapes can be redesigned using the proposed DFM procedure. Nevertheless, this study sheds some light on our understanding of how to develop more accurate physical models of IFS-created fractals.

GRAPHICAL ABSTRACT

Keywords:

• iterated function system
• fractals
• design for manufacturing
• CAD/CAM
• additive manufacturing

ACKNOWLEDGMENTS

The first author acknowledges Mr. Ryuta Omori with whom the author initiated the work described in this article. The authors thank the reviewers for their constructive suggestions.