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Some Structures arising from the Farey Fractal

This paper investigates new arithmetic and combinatorial structures derived from the interplay of Farey-type graphs, Fibonacci expansions, and operadic constructions, specifically introducing Fibonadic numbers as an inverse limit under the Zeckendorf shift equipped with a metric, order, and commutative rig structure.

Original authors: Shai Haran

Published 2026-03-24

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Shai Haran

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: A Cosmic Tree of Numbers

Imagine you are looking at a giant, infinite tree growing in a forest. This isn't a tree made of wood and leaves; it's a mathematical tree made of pairs of numbers (like 1/1, 1/2, 2/3, 3/5).

In math, this is called the Farey Tree (or Stern-Brocot tree). It's a way of organizing all possible fractions between 0 and infinity. If you walk down this tree, you are essentially walking through the history of how we build numbers from the simplest building blocks.

The author of this paper, Shai Haran, is asking a fascinating question: What happens if we look at this tree through the lens of the "Golden Ratio" ( $\phi$ )?

The Golden Ratio ( $\phi \approx 1.618$ ) is a special number found in nature (like the spiral of a seashell or the arrangement of sunflower seeds). It has a magical property: $\phi^2 = \phi + 1$ . This means if you have two of them, they make a bigger one.

Haran discovers that this tree has a hidden "DNA" that connects three different worlds:

The Discrete World: Whole numbers and counting (like the Fibonacci sequence: 1, 1, 2, 3, 5, 8...).
The Continuous World: The smooth, unbroken line of real numbers (like 1.5, $\pi$ , $\sqrt{2}$ ).
The Fractal World: A strange, self-repeating structure that bridges the two.

Analogy 1: The "Zeckendorf" Ladder (The Discrete World)

Imagine you want to build a tower of blocks to reach a specific height, say 100 feet.

Normal way: You use 1-foot blocks. You need 100 of them.
Fibonacci way: You have a special set of blocks with heights 1, 2, 3, 5, 8, 13, 21, 34, 55... (the Fibonacci numbers).
The Rule: You can't use two blocks of the same size next to each other (no two 5s in a row).

This is called the Zeckendorf expansion. Haran shows that if you look at the "shape" of the Farey tree, you are essentially seeing this block-building process. Every point on the tree corresponds to a unique way of building a number using these special Fibonacci blocks.

Analogy 2: The "Fibonadic" Numbers (The Strange Middle Ground)

Now, imagine you have a magical machine that lets you use negative block sizes or infinite fractions of blocks.

In the normal world, you stop when you reach your target number.
In Haran's "Fibonadic" world, you can keep going forever. You can have a number that looks like an infinite string of 0s and 1s (like a binary code), but instead of powers of 2, the powers are the Golden Ratio ( $\phi$ ).

Haran calls these "Fibonadic Numbers."

The Twist: These numbers form a "totally disconnected" world. Imagine a cloud of dust where every speck is a number, but there is no smooth path between them. You can jump from one speck to another, but you can't slide.
The Magic: Even though they look like a scattered cloud of dust, if you measure their "weight" using the Golden Ratio, they perfectly fill up the smooth, continuous line of real numbers (from 0 to infinity).

It's like having a digital photo (made of pixels) that, when you zoom out, looks exactly like a smooth painting. The "pixels" are the Fibonadic numbers; the "painting" is the real number line.

Analogy 3: The "Parent and Child" Tree

The paper describes a family tree structure:

The Root: The number 1/1.
The Parents: Every number on the tree has a "Mother" and a "Father" (mathematically defined neighbors).
The Children: A number is created by adding its Mother and Father together (this is called the mediant).

Haran introduces two special tools (maps) to navigate this tree:

$\pi$ (Pi): This is the "Eraser." It takes a number and removes the last digit of its Fibonacci code. It moves you up the tree toward the root.
$j$ (J): This is the "Adder." It adds a zero to the code. It moves you down the tree.

These tools act like a remote control for the tree. If you press "Eraser" enough times, you eventually reach the top. If you press "Adder," you go deeper into the infinite branches.

The "Fractal" Discovery

The most exciting part of the paper is how these tools reveal a fractal pattern.

Imagine you are looking at a map of a coastline.

If you zoom out, you see a smooth curve.
If you zoom in, you see jagged rocks.
If you zoom in further, you see even smaller jagged rocks that look just like the big ones.

Haran shows that the Farey tree works the same way.

He defines "Layers" of the tree based on how big the numbers are.
He discovers that the shape of the tree at a specific "depth" looks exactly like the shape of the tree at a different depth, just scaled up or down.
The "DNA" of the tree is encoded in the Golden Ratio. The way the tree branches is dictated by the equation $\phi^2 = \phi + 1$ .

The "Rig" (A Number System Without Negatives)

The paper introduces a new kind of math system called a Rig (a Ring without negatives).

Think of it like a bank account where you can only deposit money, never withdraw.
In this system, you can add and multiply these "Fibonadic" numbers.
The result is a system that behaves like the real numbers (you can measure lengths, areas, etc.) but is built entirely out of discrete, countable steps (like counting blocks).

Why Does This Matter?

This paper is a bridge.

For Mathematicians: It connects the study of prime numbers, the structure of trees, and the properties of the Golden Ratio in a way that wasn't fully understood before.
For Everyone: It shows that the universe of numbers is more flexible than we thought. You can build a smooth, continuous world (like the real numbers) out of a scattered, disconnected world (like the Fibonacci numbers) if you use the right "lens" (the Golden Ratio).

In a nutshell:
Shai Haran found a secret code hidden in the way fractions are arranged. This code uses the Golden Ratio to turn a jagged, pixelated world of counting numbers into a smooth, continuous world of real numbers. It's like discovering that a digital video game is actually a window into a real, physical world, provided you know how to look at it through the Golden Ratio.

1. Problem Statement

The paper addresses the structural analysis of Farey sequences and the Stern–Brocot tree, specifically focusing on the geometry of the set of coprime pairs $X = \{(x, y) \in \mathbb{N}_+ \times \mathbb{N}_+ \mid \gcd(x, y) = 1\}$ .

While the shape of these sequences is known to relate to the Zeckendorf expansion (representing integers as sums of non-consecutive Fibonacci numbers), the author seeks to:

Formalize the relationship between the discrete structure of these trees and the continuous properties of the Golden Ratio ( $\phi$ ).
Construct a new algebraic structure, the "Fibonadic numbers" ( $\mathcal{F}$ ), which serves as a bridge between the discrete world of integers and the continuum of non-negative real numbers.
Analyze the asymptotic behavior of the tree layers using $\phi$ -norms to understand the fractal nature of the Farey graph.

2. Methodology

The author employs a synthesis of number theory, combinatorics, and algebraic geometry (specifically rig theory and inverse limits).

Zeckendorf Expansion & Operators: The paper utilizes the Zeckendorf representation $R = \sum z_n f_n$ $R = \sum z_{n} f_{n}$ (where $f_n$ $f_{n}$ are Fibonacci numbers). It defines two key operators:
- $\pi$ : Erases the last digit of the expansion (shifts right).
- $j$ : Adds a zero to the expansion (shifts left).
- These operators are shown to have coboundaries in $\{-1, 0, 1\}$ , indicating a near-homomorphic behavior regarding addition.
Inverse Limits (Fibonadic Numbers): The set $\mathcal{F}$ is defined as the inverse limit of $\mathbb{N}$ under the map $\pi$ . Elements of $\mathcal{F}$ are infinite sequences of 0s and 1s (with no consecutive 1s) representing "Fibonadic numbers."
Laurent Series & Rig Homomorphisms: The author embeds these sequences into the rig of Laurent series $\mathbb{N}((t))$ $N ((t))$ . A valuation map $V_\phi$ $V_{ϕ}$ is defined by substituting $t = \phi$ $t = ϕ$ .
- $V_\phi(z) = \sum z_n \phi^n$ .
- A canonical projection $P: B \to \mathcal{F}$ is constructed to normalize series into valid Fibonadic expansions using specific reduction rules (moves A and B) that preserve the $\phi$ -value.
Tree Geometry & Norms: The paper defines a $\phi$ -norm on the tree vertices $v$ :
$|v|_\phi = |M(v)| + \phi^{-1}|F(v)|$
where $M(v)$ and $F(v)$ are the "Mother" and "Father" nodes in the Stern–Brocot tree. This norm allows for the partitioning of the tree into layers based on $\phi$ -powers.

3. Key Contributions

A. The Fibonadic Rig ( $\mathcal{F}$ )

The paper constructs $\mathcal{F}$ as a commutative rig (a ring without negatives).

Structure: Elements are sequences $z = (z_n)$ with $z_n \in \{0, 1\}$ and no consecutive 1s.
Operations: Addition and multiplication are defined via the projection $P$ of the standard polynomial operations in the Laurent series ring.
Isomorphism: There is a surjective rig homomorphism $V_\phi: \mathcal{F} \to \mathbb{R}_{\ge 0}$ . The kernel of this map (modulo an equivalence relation $\sim$ generated by $z \sim z^-$ ) identifies $\mathcal{F}/\sim$ with the non-negative reals.
Significance: This provides a discrete, "pro-finite" arithmetic system that naturally converges to the real numbers via the golden ratio.

B. The Farey Fractal and Tree Layers

The author decomposes the Stern–Brocot tree $X$ into layers $\ell(n, z)$ based on a Fibonadic number $z$ .

Level Definition: A vertex $v$ belongs to layer $n$ if its $\phi$ -norm satisfies:
$\phi^{n-1} \cdot V_\phi(z) \le |v|_\phi < \phi^n \cdot V_\phi(z)$
Base Sets: The set of vertices up to layer $n$ , denoted $b(n, z)$ , forms a finite subtree. The collection of these bases $\{b(n, z)\}$ forms a new planar rooted tree $B$ .
Bijection: There is a bijection between the set of "Principal Units" $D$ (Fibonadic numbers starting with $1, 0, \dots$ ) and the set of infinite paths in the configuration tree $B$ .

C. The $\phi$ -Norm and Asymptotics

The paper establishes that the Zeckendorf expansion's asymptotic behavior is governed by the $\phi$ -norm.

The condition for a node $v$ to be relevant to the $n$ -th layer is purely determined by $|v|_\phi \in [\phi^{n-1}, \phi^n]$ .
This connects the discrete combinatorial structure of the tree to the continuous scaling properties of the golden ratio, effectively describing the "Farey Fractal."

4. Results

Metric Space: $\mathcal{F}$ is a complete metric space with metric $d(z, z') = \phi^{\delta(z, z')}$ , where $\delta$ is the index of the first differing digit.
Ordering: A total order is defined on $\mathcal{F}$ which is compatible with the $\phi$ -value map.
Discontinuity: The operations on $\mathcal{F}$ are not continuous with respect to the metric topology (e.g., limits of sums do not equal sums of limits), highlighting the subtle difference between the discrete Fibonadic structure and the continuum $\mathbb{R}$ .
Fundamental Domain: The set $D$ acts as a fundamental domain for the action of powers of $\Phi$ (the image of $t$ ) on $\mathcal{F} \setminus \{0\}$ . The map $V_\phi$ restricted to $D$ maps onto the interval $[1, \phi]$ .
Tree of Configurations: The paper explicitly constructs the tree of finite subtrees $B(n)$ , showing how the structure branches and how the "configurations" (finite approximations of the infinite tree) refine as $n \to \infty$ .

5. Significance

Unification of Discrete and Continuous: The work provides a rigorous algebraic framework (the Fibonadic rig) that unifies the discrete Zeckendorf expansion with the continuous real numbers via the golden ratio. It explains why the shape of Farey sequences relates to Fibonacci numbers.
New Arithmetic System: It introduces a non-standard arithmetic system where addition and multiplication are defined via normalization of Laurent series, offering a new perspective on "pro-finite" number systems.
Geometric Insight: By defining the $\phi$ -norm, the paper offers a precise geometric interpretation of the Stern–Brocot tree, characterizing its fractal layers through scaling by $\phi$ . This deepens the understanding of the "Farey Fractal" as a self-similar structure governed by the golden ratio.
Operadic Structure: The paper hints at an operad structure on the finite subtrees, suggesting deep connections to higher algebraic structures and category theory.

In summary, Shai Haran's paper reinterprets the classical Farey sequence and Stern–Brocot tree through the lens of Fibonadic numbers, creating a novel algebraic rig that bridges integer arithmetic and real analysis, and revealing the fractal geometry inherent in the distribution of coprime pairs.