What is the Fibonacci word?

It's a specific infinite binary sequence built the same way the Fibonacci numbers are built, but by concatenating strings instead of adding numbers: starting from "0" and "01", each next string is the previous two concatenated in order (previous-plus-the-one-before-it), so the sequence grows 0, 01, 010, 01001, 01001010, and so on, with each string's length following the Fibonacci numbers.

How does the turtle turn into a fractal curve?

A virtual turtle starts facing one direction and draws one unit-length segment for every character in the word, moving forward each time. At every '1' character (and only at '1's, '0's just move straight ahead), the turtle turns 90 degrees, alternating between turning left and turning right each time it encounters a '1'. That alternating-turn rule is what turns a flat sequence of 0s and 1s into a self-similar, branching-looking curve.

What does the Order setting control?

Order (1 to 22) selects how many string-concatenation steps to run before drawing, which determines the Fibonacci word's length and therefore how many segments the turtle draws. Higher orders produce Fibonacci-numbered word lengths and correspondingly longer, more detailed curves.

Who popularized this construction?

The turtle-graphics interpretation of the Fibonacci word as a plane fractal curve was popularized by Alexis Monnerot-Dumaine in a 2009 paper, and it's now a standard, documented example of a substitution-sequence-driven fractal curve.

Fibonacci Word Fractal Generator

Overview

Introduction

The Fibonacci sequence shows up everywhere in mathematics, but the Fibonacci word fractal is one of its more surprising appearances: turn the Fibonacci word's string of 0s and 1s into turtle-graphics turning instructions, and a genuinely fractal-looking curve emerges.

This tool builds that curve directly: choose an order, and a virtual turtle draws one segment per character of the corresponding Fibonacci word, turning at every '1' to trace out the resulting shape.

What Is Generate a Fibonacci Word Fractal?

A generator for the Fibonacci word fractal, the documented turtle-graphics construction popularized by Monnerot-Dumaine, built by interpreting the Fibonacci word's characters as move-and-turn instructions.

There is no text input or output here, just a numeric Order control; the tool builds the underlying Fibonacci word internally and renders the resulting turtle path as an SVG curve.

How Generate a Fibonacci Word Fractal Works

The Fibonacci word itself is generated first, by starting with the strings "0" and "01" and repeatedly setting the next string equal to the current one concatenated after the previous one, for as many steps as the chosen order.

A virtual turtle then walks through the word character by character, drawing one unit-length line segment forward for every character, whether it's a '0' or a '1'.

At every '1' character, and only at '1's, the turtle turns 90 degrees before drawing its next segment, alternating between a left turn and a right turn each time a '1' is encountered. That alternating rule is what produces the curve's distinctive self-similar, comb-like structure rather than a simple zigzag.

When To Use Generate a Fibonacci Word Fractal

Use it to explore how a purely combinatorial sequence (the Fibonacci word) can encode genuine geometric, fractal-like structure once interpreted as turtle-graphics instructions.

It's a good source for a visually distinctive fractal curve illustration, especially for material connecting number theory, substitution sequences, and geometry.

Often used alongside Generate a Z-order Curve and Generate an H-fractal.

Features

Advantages

Demonstrates a direct, well-documented bridge between a combinatorial sequence and a geometric fractal curve.
Order range extends to 22, covering a wide span from a short, simple path to a long, richly detailed curve.
Exports as clean vector SVG or a flattened PNG for use in illustrations or teaching material.

Limitations

Order is capped at 22 for renderability, since the Fibonacci word's length grows exponentially (following the Fibonacci numbers) with each additional order.
The curve is deterministic and fully specified by the order alone, there's no way to adjust the turning angle or turn rule independently of the underlying Fibonacci word.

Examples

Order 1

Input

Order = 1

Output

A very short path with only a couple of segments and at most one turn.

At low orders, the Fibonacci word itself is short ("01" at order 1), so the turtle only draws a couple of segments.

Order 14

Input

Order = 14

Output

A long, richly self-similar curve with a comb-like, branching visual texture built from hundreds of segments.

At order 14 the underlying Fibonacci word is hundreds of characters long, and the alternating 90-degree turns at every '1' build up the curve's characteristic fractal texture.

Best Practices & Notes

Best Practices

Start around order 8 to 12 to see the curve's characteristic self-similar texture clearly before pushing toward the order-22 maximum, where the path becomes very dense.
Download as SVG if you want to trace or recolor sections of the curve afterward in a vector editor.

Developer Notes

The Fibonacci word is built iteratively (`b = b + a; a = previous b`), avoiding recursion so long orders don't risk a stack overflow, then a single pass over the resulting string drives the turtle: each character advances position by one unit at the current heading, and every '1' toggles an alternating left/right 90-degree turn before the next segment is drawn. The path is normalized into the shared 1000x1000 SVG viewBox afterward.

Generate a Fibonacci Word Fractal Use Cases

Illustrating how substitution sequences (like the Fibonacci word) can encode geometric, fractal-like structure
Generating a visually distinctive turtle-graphics curve for a math or generative-art illustration
Teaching the connection between combinatorics on words and fractal geometry

Common Mistakes

Expecting every character to trigger a turn, only '1' characters do, '0' characters just move the turtle straight ahead.
Expecting the turn direction to be fixed (always left, or always right), it alternates: the first '1' turns one way, the next '1' turns the other way, and so on.

Tips

Try the Z-order Curve Generator next to compare a substitution-sequence-driven curve against a coordinate-interleaving-driven one.
Increase the order gradually and watch how the curve's self-similar sub-patterns repeat at larger scales as the word grows.

Generate a Fibonacci Word Fractal