Complexity of the Fibonacci snowflake

TL;DR

This paper investigates a Fibonacci-related fractal curve on the square lattice, calculating its fractal dimension and complexity, contributing to understanding geometric properties of Fibonacci-based fractals.

Contribution

It introduces a new Fibonacci snowflake curve, computes its fractal dimension, and provides a complexity measure, expanding knowledge of Fibonacci-related fractal geometries.

Findings

Fractal dimension of the Fibonacci snowflake is computed.

A complexity measure for the Fibonacci snowflake is established.

The curve's interior tiles the plane via translation.

Abstract

The object under study is a particular closed curve on the square lattice $\Z^2$ related with the Fibonacci sequence $F_n$. It belongs to a class of curves whose length is $4F_{3n+1}$, and whose interiors by translation tile the plane. The limit object, when conveniently normalized, is a fractal line for which we compute first the fractal dimension, and then give a complexity measure.