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Farey graphs and geodesic expansions of complex continued fractions

This paper investigates complex Farey graphs and hyperbolic Farey polygons associated with specific Euclidean imaginary quadratic number fields to recover known Hecke group tessellations, induce Bianchi group tessellations of hyperbolic 3-space, and establish a generalized approach for geodesic complex continued fraction expansions.

Original authors: Hitoshi Nakada, Rie Natsui, Jörg Thuswaldner

Published 2026-03-31

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CC BY 4.0

Original authors: Hitoshi Nakada, Rie Natsui, Jörg Thuswaldner

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

Imagine you are trying to navigate a vast, infinite city where the streets are made of numbers. In the world of regular math (using just whole numbers and fractions), mathematicians have a famous map called the Farey Graph. Think of this map as a giant, intricate web of bridges connecting different rational numbers. If you want to get from one number to another, the "Farey Graph" tells you the most direct, shortest path, avoiding any unnecessary detours. This is called a geodesic path.

This paper takes that familiar map and asks a big question: What happens if we move into a more complex world?

Instead of just using regular numbers, the authors explore a "complex" world involving imaginary numbers (numbers involving $\sqrt{-1}$ ) and specific types of number systems called Euclidean imaginary quadratic fields. You can think of these as different "neighborhoods" of the number city, each with its own unique rules and geometry.

Here is a breakdown of the paper's main ideas using simple analogies:

1. From Flat Maps to 3D Spheres

In the old world (regular fractions), the Farey Graph lives on a flat, 2D surface (like a piece of paper). The connections between numbers are drawn as curves.

The Analogy: Imagine the old map is a flat drawing of a subway system.
The New World: The authors realize that in these complex number neighborhoods, the map doesn't fit on a flat piece of paper. It needs to be built in 3D space (specifically, a hyperbolic space that curves like a saddle).
The Result: Instead of flat bridges, the connections are now spheres floating in 3D space. They call these Ford Spheres. Just as two bridges might touch at a single point in the old map, these 3D spheres touch each other at specific points to form the new, complex map.

2. The "Tessellation" (The Floor Tiles)

When you connect all these points and spheres, they don't just make a messy web; they create a perfect, gapless floor pattern.

The Analogy: Imagine tiling a floor. In the old world, you used triangles. In this new 3D world, the "tiles" are weird, curved 3D shapes (polyhedra) that fit together perfectly to cover the entire infinite space without any gaps or overlaps.
The Discovery: The authors figured out exactly what these 3D tiles look like for five specific number neighborhoods ( $d = 1, 2, 3, 7, 11$ ). Some look like octahedrons (like two pyramids glued together), others like tetrahedrons or more complex shapes. These shapes are the "standard cells" that build the whole universe of these numbers.

3. The "Shortest Path" Problem (Geodesic Expansions)

The main goal of the paper is to solve a navigation problem: How do you write a complex number as a "continued fraction" (a specific type of math recipe) that represents the absolute shortest path on this 3D map?

The Problem: There are many ways to write a number as a continued fraction. Some are long and winding; some are short and direct. A "geodesic" expansion is the shortest, most efficient one.
The Solution: The authors looked at specific algorithms (mathematical recipes) that people have used for a long time (like the "Nearest Integer" method). They proved that for certain number neighborhoods, these old recipes always produce the shortest path. They are "geodesic" by nature.
The Catch: They also found that for some neighborhoods (like the one with $d=7$ ), the old recipes can sometimes take a detour. They found a specific "trap" where the path looks short but isn't actually the shortest, much like a GPS that takes you down a scenic route that turns out to be longer than the highway.

4. Why Does This Matter?

You might wonder, "Who cares about 3D number maps?"

The Big Picture: This isn't just about drawing pretty pictures. Understanding these shortest paths helps mathematicians understand the "entropy" (or chaos) of these number systems. It's like understanding the traffic flow in a city.
The Connection: The paper connects the geometry of these 3D shapes to the algebra of number recipes. It shows that if you understand the shape of the "floor tiles" (the tessellation), you automatically know the best way to navigate the numbers.

Summary in One Sentence

The authors built a 3D map of complex numbers using floating spheres and curved tiles, proving that certain mathematical recipes always find the shortest route through this strange, curved universe, while warning us about specific traps where the route might look short but isn't.

The "Aha!" Moment:
Just as a flat map of the world distorts distances near the poles, trying to map complex numbers on a flat surface distorts the "shortest path." By lifting the map into 3D hyperbolic space, the authors revealed the true, natural geometry of these numbers, showing us the most efficient way to travel through them.

1. Problem Statement

The paper addresses the generalization of classical Farey graphs, Farey tessellations, and geodesic continued fraction expansions from the rational numbers $\mathbb{Q}$ to Euclidean imaginary quadratic number fields $\mathbb{Q}(\sqrt{-d})$ , where $d \in \{1, 2, 3, 7, 11\}$ .

While the relationship between Farey graphs and continued fractions is well-established in the real case (via the modular group $PSL(2, \mathbb{Z})$ acting on the hyperbolic plane $\mathbb{H}^2$ ), the complex case involves the Bianchi groups $PSL(2, \mathcal{O}_d)$ acting on 3-dimensional hyperbolic space $\mathbb{H}^3$ . Previous work by Hockman covered the Gaussian field ( $d=1$ ), but a unified approach for all Euclidean imaginary quadratic fields was lacking. The authors aim to:

Define complex Farey graphs and tessellations in $\mathbb{H}^3$ .
Establish the geometric structure of these tessellations using hyperbolic polyhedra.
Determine which complex continued fraction algorithms produce geodesic expansions (shortest paths in the Farey graph).

2. Methodology

The authors employ a geometric approach rooted in hyperbolic geometry and the theory of discrete groups.

Hyperbolic Setting: They work in the upper half-space model $\mathbb{H}^3 = \{x_1 + x_2i + x_3j : x_3 > 0\}$ . The boundary is the complex plane $\mathbb{C} \cup \{\infty\}$ .
Generalized Farey Graphs ( $E_d$ ): They define the Farey graph $E_d$ for $\mathbb{Q}(\sqrt{-d})$ as a graph with vertices in $\mathbb{Q}(\sqrt{-d}) \cup \{\infty\}$ . An edge exists between $\alpha$ and $\beta$ if they are images of $\infty$ and $0$ under a Möbius transformation in the Bianchi group $PSL(2, \mathcal{O}_d)$ . A key property (Proposition 1) is that an edge exists iff $|p_1q_2 - p_2q_1| = 1$ for the corresponding fractions.
Ford Spheres: They generalize Ford circles to Ford spheres in $\mathbb{H}^3$ . These are Euclidean spheres (or planes) tangent to the boundary $\mathbb{C}$ . The tangency of Ford spheres corresponds to the existence of edges in the Farey graph.
Farey Polygons and Tessellations: Inspired by A. Schmidt's work, the authors define Farey triangles, quadrangles, and hexagons in $\mathbb{H}^3$ . These are hyperbolic polyhedra bounded by geodesic arcs. They construct a "standard cell" $C_d$ for each $d$ and show that the action of the Bianchi group on these cells generates a tessellation of $\mathbb{H}^3$ .
Connection to Hecke Groups: For specific walls in the tessellation, the authors identify subgraphs of $E_d$ that are isomorphic to Farey graphs associated with Hecke groups $H_4$ and $H_6$ . This links the complex case to real continued fraction algorithms involving $\sqrt{2}$ and $\sqrt{3}$ .
Geodesic Analysis: They analyze the "nearest integer" continued fraction algorithms (generalizing Hurwitz and Lakein) to determine if the resulting paths in the Farey graph are geodesics (shortest paths).

3. Key Contributions and Results

A. Geometric Structure of Complex Farey Graphs

Tessellation of $\mathbb{H}^3$ : The paper proves that the Bianchi group action generates a tessellation of $\mathbb{H}^3$ $H^{3}$ by hyperbolic polyhedra (Theorem 4).
- For $d=1, 2$ , the fundamental domain is a single polyhedron $C_d$ .
- For $d=3, 7, 11$ , the fundamental domain requires two polyhedra ( $C_d$ and its reflection $C'_d$ ).
Classification of Standard Cells (Proposition 7):
- $d=1$ : $C_1$ is an octahedron (8 faces).
- $d=2$ : $C_2$ is a tetradecahedron (14 faces: triangles and quadrangles).
- $d=3$ : $C_3$ is a tetrahedron (4 faces).
- $d=7$ : $C_7$ is a pentahedron (5 faces).
- $d=11$ : $C_{11}$ is an octahedron (8 faces: triangles and hexagons).
Disjointness: The authors prove that distinct Farey polygons intersect only at common cusps or common edges, ensuring the tessellation is well-defined (Proposition 4).

B. Connection to Hecke Groups

The paper establishes that the restriction of the Farey graph $E_d$ to specific "walls" (vertical planes) in $\mathbb{H}^3$ is isomorphic to the Farey graphs of Hecke groups $H_4$ (for $d=2, 7$ ) and $H_6$ (for $d=11$ ). This provides a bridge between complex continued fractions and real continued fractions with specific irrational bases (Theorem 3).

C. Geodesic Continued Fraction Expansions

Nearest Integer Algorithms: The authors prove that the nearest integer continued fraction algorithms (defined for $d=1, 2, 3$ ) always produce geodesic expansions (Theorem 5). This means the sequence of convergents corresponds to the shortest path in the Farey graph from $\infty$ to the target number.
Counter-example for $d=7$ : While the nearest integer algorithm works for $d=1, 2, 3$ , the paper provides a subtle counter-example for $d=7$ (Example 2). A specific path in the Farey graph for $d=7$ satisfies local optimality conditions (Propositions 8 and 9) but is not a global geodesic. This highlights the increased complexity of the $d=7$ and $d=11$ cases compared to $d=1, 2, 3$ .
Large Partial Quotients: Theorem 6 establishes that if all partial quotients $a_k$ in a continued fraction satisfy $|a_k| \geq 4.17209$ , the expansion is guaranteed to be the unique geodesic expansion. This generalizes results from the real case to the complex setting.

4. Significance

Unification: The paper provides a unified geometric framework for studying continued fractions across all Euclidean imaginary quadratic fields, moving beyond the Gaussian case ( $d=1$ ).
Geometric Insight: By visualizing continued fractions as paths in 3D hyperbolic space bounded by polyhedra, the authors offer a powerful geometric intuition for the convergence and optimality of these algorithms.
Algorithmic Implications: The results clarify which algorithms are optimal (geodesic). The finding that nearest integer algorithms are geodesic for $d=1, 2, 3$ validates their use in computational number theory for these fields.
Complexity of Higher Fields: The counter-example for $d=7$ demonstrates that properties holding for small discriminants do not automatically extend to larger ones, suggesting that the arithmetic of $d=7$ and $d=11$ requires more nuanced analysis (potentially involving non-standard algorithms or different fundamental domains).
Future Applications: The authors note that these results on geodesic expansions have potential applications in dynamical systems, specifically regarding the entropy of $\alpha$ -continued fraction maps in the complex domain.

In summary, this work significantly advances the understanding of complex continued fractions by rigorously defining their underlying hyperbolic geometry and characterizing the optimality of standard algorithms through the lens of geodesic paths in Bianchi group tessellations.