The Fibonacci Space-Filling Curve

TL;DR

This paper introduces a novel construction of a space-filling curve using Fibonacci substitution, providing a new proof of the curve's ability to fill a square completely, differing from traditional methods.

Contribution

It presents a new proof of the space-filling property of the Fibonacci-based curve using Cartesian product, offering a different approach from existing constructions.

Findings

Successfully constructs a Fibonacci-based space-filling curve

Provides a new proof of the curve's space-filling property

Differentiates from traditional space-filling curve constructions

Abstract

Can you stretch and reform a curve such that it fills a square completely? This question dates back to 18th century, the origin of space-filling curves. It was proved affirmatively by many great mathematicians. In this document, we reconsider the problem and present a different proof using Cartesian product of Fibonacci substitution with itself. Our construction differs from other curves by design.