A unit circle is the set of points satisfying x² + y² = 1 — but the same parameters (radius = 1, circumference = 2π, rotation θ ∈ [0, 2π)) manifest in many different geometric shapes and coordinate systems. The same mathematical object, viewed through different coordinate systems. The Farey modular residue rings visualized here are precisely one such lens: the classical circle becomes a discrete, modular shell system — and the structures below illuminate why this connection is so deep.
| Geometry / System | Shape of "Unit Circle" | Key Equation |
|---|---|---|
| Euclidean | Circle | x² + y² = 1 |
| Manhattan (L¹) | Diamond | |x| + |y| = 1 |
| Chebyshev (L∞) | Square | max(|x|,|y|) = 1 |
| Polygon limit | Regular n-gon → circle | n → ∞ |
| Complex / Fourier | Exponential rotation | z = eiθ |
| Topology (mod 1) | Wrapped number line | ℝ/ℤ |
| Lattice shells | Integer rings | x² + y² = r² |
| Modular residues | Discrete Farey rings | r/m ∈ Farey(N) |
Each rational r/m is placed on ring m at angle θ = 2πr/m, corresponding to the m-th roots of unity
These points form a regular m-gon on the unit circle. The full structure across all rings 1 ≤ m ≤ N is a modular lattice on concentric circles.
The multiplicative units modulo m are the residues coprime to m:
Their count is Euler’s totient φ(m). These are the Farey fractions — the blue points in the visualiser. They define the primitive angular directions not eliminated by the divisibility structure of m.
Each modulus generates a regular polygon. The primorial sequence eliminates composite directions one prime at a time:
What remains after each step are exactly the angular rays where primes can occur. Use the hierarchy navigator below to walk through each level interactively.
All primes greater than p must lie on the unit-group rays of the primorial modulus M = 2·3·…·p:
For M=30 this gives 8 rays; for M=210, 48 rays. The gap overlay marks which pairs carry twin primes (gap=2), cousin primes (gap=4), or sexy primes (gap=6).
Twin prime candidates (p, p+2) correspond to residue pairs (r, r+2) both coprime to M. Their angular separation on ring m is:
As M grows through primorial extensions, admissible twin-prime ray pairs lift to the refined lattice. The candidate directions are preserved through sieve refinement — visible as parallel chord pairs in the gap overlay.
By the Franel–Landau theorem, the Riemann Hypothesis is equivalent to the summed deviations of the Farey fractions from equal spacing satisfying:
The near-uniform angular distribution visible at finite N is consistent with RH. It is not a proof — only a geometric shadow of the analytic statement. The N-body panel and gap overlay make the discrepancy geometry directly visible.
The proportion of primitive residues mod m is given by the Euler product; along the primorial sequence M = 2·3·5·…·p it decays to zero (though not for general m — for prime m it approaches 1):
The lattice refines as m grows while allowed directions simultaneously thin — a precise quantification of prime sparsity. Averaged over all m ≤ N this density converges to 6/π² = 1/ζ(2). The hierarchy navigator tracks φ(N)/N live.
The ring structure is the discrete skeleton of a Laurent series. For a rational function with poles at coprime prime residues r₁,r₂,… mod N, the complex plane splits into annular convergence regions — one per gap between consecutive pole radii.
The analogy drawn here: gcd(r,m)=1 points play the role of irreducible partial-fraction terms, while gcd(r,m)>1 points reduce to lower-denominator rings — just as non-lowest-terms fractions reduce. This is a structural analogy, not a theorem: no result identifies the 6/π² coprime density with a fraction of Laurent basis elements. What is rigorous is the classical annulus picture above (a rational function converges in the annuli between consecutive pole radii) and the coprime density itself.
The N-body panel uses these same rₖ as orbit positions — bodies in the annular band between poles. The explorer below shows the decomposition live as N changes.
Riemann’s explicit formula gives the prime-power counting function Π₀(x) = Σpk≤x 1/k exactly, with one oscillatory correction per nontrivial zero ρ of ζ(s). (π(x) is then recovered by Möbius inversion: π(x) = Σn μ(n)/n · Π(x1/n).) The known zeros all have Re(ρ) = ½; that this holds for every zero is precisely RH. Writing ρ = ½ + iγn:
Each conjugate pair ρ, ρ¯ produces a real oscillation of amplitude ≈ 2√x/(|ρ| log x) and phase γn log x. The Farey discrepancy DN grows no faster than N½+ε exactly when every zero sits on the critical line — the Franel–Landau equivalence to RH (§6). The N-body panel places bodies at residue positions rk = pk mod N — a loose visual analogy to the xρ phase evaluations, not a formal correspondence. Similarly, the pairing of conjugate zeros (γn, −γn) visually parallels the (r, N−r) mirror involution on the coprime ring.