Deterministic fractals based on Archimedean solids：确定性分形基于阿基米德多面体.pdfVIP

Scientific Research of the Institute of Mathematics and Computer Science, 1(10) 2011, 93-100 DETERMINISTIC FRACTALS BASED ON ARCHIMEDEAN SOLIDS Andrzej Katunin Department of Fundamentals of Machinery Design Silesian University of Technology, Poland andrzej.katunin@polsl.pl Abstract. In the present work, the construction of fractals based on Archimedean solids was discussed. The methods of 3D fractals construction based on uniform polyhedra were pre- sented. It was shown that the contraction mapping procedure for the construction of fractals with non-overlapping or -disjointed contractions could be applied only for a limited number of the polyhedra. The contraction ratios and the Hausdorff dimensions were determined for the existing fractals with adjacent contractions based on Archimedean solids. Introduction The development of fractal geometry created new ways of solving topical sci- entific problems from quantum physics [1] through structural diagnostics [2] to biology and genetics [3]. They found applications in problems related to the me- chanics of porous media or in modeling the rheology of materials. However, the most intensive development of the application of fractals was noticed in the prob- lems of computer graphics, pattern recognition, image compression and coding [4]. Deterministic fractals are used in testing ray-tracing algorithms during the render- ing of spatial scenes [5]. As is well known, the construction of fractals is based on the self-similarity of the input object. Two general approaches are used in the construction process: random and deterministic ones. The random approach usually uses the well-known Barnsley’s Chaos Game or its combination with the