The Fibonacci fractal is a new fractal type

摘要

We propose a uniform method for estimating fractal characteristics of systems satisfying some type of scaling principle. This method is based on representing such systems as generating Bethe-Cayley tree graphs. These graphs appear from the formalism of the group bundle of Fibonacci-Penrose inverse semigroups. We consistently consider the standard schemes of Cantor and Koch in the new method. We prove the fractal property of the proper Fibonacci system, which has neither a negative nor a positive redundancy type. We illustrate the Fibonacci fractal by an original procedure and in the coordinate representation. The golden ratio and specific inversion property intrinsic to the Fibonacci system underlie the Fibonacci fractal. This property is reflected in the structure of the Fibonacci generator.