A rational source of plane fractals and its application to fragmentation analysis of thin plates

It is demonstrated in this paper that the Euler equation which corresponds to the subdivision of a torus into a polyhedron is a rational source of plane fractals. The vertex orders and polygon edges within a plane fractal are an invariant of the fractal. The process of forming a fractal is intimately related to the topological and infinitely divisible properties of the solutions for the Euler equation. Based on this equation it is shown that only four linear plane fractals are available and the boundary of a plane fractal is a fractal curve, such as the Koch curve. Certain relations between a plane fractal and a fractal formed by the Julia set are also discussed with the aid of a new plane fractal pattern given in this work. As an application of the present analysis, brittle fragmentation of a thin plate is considered. Present results provide a new expression for estimating the average size of a fragment based on an energy balance principle and an idea about quasi-identical plane fractals. Present theoretical analysis is in agreement with experimental results and previous investigations.