Ford Circles as Horocycles
In hyperbolic geometry, the Poincaré half-plane model represents
hyperbolic space as the upper half of the complex plane {z : Im(z) > 0}, with the
metric ds² = (dx² + dy²) / y². The x-axis is the boundary at infinity.
A horocycle is a curve of constant geodesic curvature 1 — the
limit of circles as the center moves to infinity. In the half-plane, horocycles
are horizontal lines or circles tangent to the x-axis.
Here's the key insight: Ford circles are exactly the horocycles tangent
to the x-axis at rational points. The Ford circle for p/q has radius 1/(2q²)
and center at (p/q, 1/(2q²)). This isn't just analogy — it's identity. The geometry
of rationals is hyperbolic geometry.
The Modular Group Tessellation
The modular group PSL(2,ℤ) consists of Möbius transformations
z ↦ (az + b)/(cz + d) where a,b,c,d ∈ ℤ and ad - bc = 1. These are exactly the
orientation-preserving isometries of the hyperbolic plane that map rationals to rationals.
The fundamental domain is the region {z : |z| > 1, |Re(z)| < 1/2}.
Every point in the half-plane can be mapped to this region by some element of PSL(2,ℤ).
The group is generated by T: z ↦ z + 1 (translation) and S: z ↦ -1/z (inversion).
The tessellation you see is formed by repeatedly applying T and S to the fundamental
domain. Each tile is hyperbolically congruent — they all have the same hyperbolic area
π/3, even though they look different in Euclidean terms.
Geodesics in the Half-Plane
Geodesics — the shortest paths between points — are vertical lines
and semicircles perpendicular to the x-axis. The boundary points (on the x-axis or at ∞)
are the endpoints at infinity of each geodesic.
The geodesics connecting consecutive Farey neighbors form a tessellation of triangles.
These are ideal triangles: all three vertices are at infinity
(on the x-axis), yet the triangle has finite hyperbolic area π.
Two rationals p/q and r/s are Farey neighbors if |ps - qr| = 1.
Geometrically, this means their Ford circles are tangent. The geodesic connecting
them passes through the point of tangency.
The Unified Picture
Everything connects: the Stern-Brocot tree encodes the structure
of Farey neighbors. The Calkin-Wilf tree gives another enumeration
via continued fractions. Minkowski's question-mark function maps
between them continuously.
Ford circles visualize the adjacency structure as tangent circles.
And here in the Poincaré half-plane, we see the geometric truth: these aren't
arbitrary constructions, they're fundamental features of hyperbolic space.
The modular group PSL(2,ℤ) acts on all these objects: it permutes rationals, maps
Ford circles to Ford circles, geodesics to geodesics, and tiles the hyperbolic plane.
Number theory and hyperbolic geometry are two views of the same structure.