Abstract:


This paper starts from the Fibonacci sequence to obtain a smooth oscillating Fibonacci function. The function is extended to complex plane via analytical continuation. We find that all the zeros of the function lie on a line. The Fibonacci function map strait lines on the complex plane to golden spirals, which exhibits self-similarity characteristics. We further extend the metal ratios to define metal functions and preliminarily explore their properties.

Key words:  , Fibonacci sequence, Fibonacci function, golden spiral, self-similarity, metal function