On the fibbinary numbers and the Wythoffarray

TL;DR

This paper explores the structure of fibbinary numbers, establishing a bijection with natural numbers via Zeckendorf representation, and reveals a connection between the fibbinary array and the Wythoff array, providing new insights into their fractal properties.

Contribution

It introduces the set of fibbinary numbers, constructs a bijection with natural numbers, and demonstrates that the fibbinary array corresponds to the Wythoff array, linking these structures.

Findings

Fibbinary array is the image of the Wythoff array under a specific bijection.

The fibbinary table offers visual insight into the Wythoff array's fractal structure.

The Wythoff table simplifies understanding of the associated fractal, comparable to Steinhaus tree.

Abstract

This paper defines the set fib of fibbinary numbers and displays its structure in the form of a table of a specialised type, and in array form. It uses the Zeckendorf representation $n \in \mathbf{N}$ to define a bijection $\mathcal{Z}$ between $\mathbf{N}$ and fib. It is proved that the fibbinary array is the image under $\mathcal{Z}$ of the famous Wythoff array. The fibbinary table proves useful pictorial insight into the fractal defined by the Wythoff array. The Wythoff table, obtained as the image under the inverse of $\mathcal{Z}$ of the fibbinary table, leads to a simpler view of the fractal, and may be compared with the (1938) Steinhaus tree.