Interactive Modular Arithmetic & Number Theory Research Suite

Unit Disk Model (𝔻)

The Poincaré disk model of hyperbolic geometry: {z ∈ ℂ : |z| < 1}. Points from the Farey sequence F_n—the set of all reduced fractions p/q with 0 ≤ p ≤ q ≤ n ordered by value—are mapped to angles 2πp/q on the unit circle ∂𝔻. The Farey triangle connecting these boundary points has the mediant property: for adjacent fractions p/q and r/s in F_n, we have |ps - qr| = 1 (the determinant condition). Prime numbers are positioned at angles 2πp/m where p is prime and m is the modulus, revealing Dirichlet's theorem: primes are equidistributed among residue classes coprime to m, each with asymptotic density 1/φ(m).

Upper Half-Plane via Cayley (ℍ)

The Cayley transform w = i(1+z)/(1-z) provides a conformal equivalence between 𝔻 and the upper half-plane ℍ = {w ∈ ℂ : Im(w) > 0}. This is one of the fundamental isometries of hyperbolic geometry, preserving the hyperbolic metric ds² = |dz|²/(1-|z|²) on 𝔻 and ds² = |dw|²/Im(w)² on ℍ. Geodesics in ℍ appear as semicircles orthogonal to the real axis (or vertical lines). The modular group PSL(2,ℤ) = SL(2,ℤ)/{±I} acts on ℍ via Möbius transformations z → (az+b)/(cz+d) where a,b,c,d ∈ ℤ and ad-bc = 1. This group is generated by S(z) = -1/z and T(z) = z+1, and its quotient ℍ/PSL(2,ℤ) is the modular curve, fundamental to the theory of modular forms and elliptic curves.

Full Complex Plane (ℂ)

The fourth panel extends the Cayley transform to visualize the entire Riemann sphere ℂ̂ = ℂ ∪ {∞}. Since the transform is defined everywhere except at z = -1, we see the complete partition:
• Interior |z| < 1 → Upper half-plane Im(w) > 0
• Unit circle |z| = 1 → Real axis Im(w) = 0
• Exterior |z| > 1 → Lower half-plane Im(w) < 0
The point z = 1 maps to ∞, z = -1 is the pole (undefined), and z = ±i map to the real axis at w = -1 and w = 1 respectively. This complete picture shows how Möbius transformations act as conformal automorphisms of ℂ̂, forming the group PSL(2,ℂ).

Nested Rings Structure (⊚)

Concentric rings represent the structure of (ℤ/mℤ)× for moduli m from min to max. Each ring m displays all residue classes k ∈ {0,1,...,m-1} at angles 2πk/m. Points are colored by gcd(k,m), revealing the multiplicative structure. Gold points (gcd = 1) form the group of units (ℤ/mℤ)×, whose order is given by Euler's totient φ(m). The Chinese Remainder Theorem states that if gcd(m₁,m₂) = 1, then ℤ/(m₁m₂)ℤ ≅ ℤ/m₁ℤ × ℤ/m₂ℤ, visible in the coprime point patterns. Connection modes visualize lifts and transitions: in a modular sequence Mₙ = M₀·bⁿ, a residue r at level n lifts to {r, r+Mₙ, r+2Mₙ, ..., r+(b-1)Mₙ} at level n+1. If gcd(r,M₀) = gcd(r,b) = 1, then coprimality is preserved: gcd(r,Mₙ₊₁) = 1.

Canvas 1: 𝔻 Unit Disk - Farey Triangle & Prime Distribution

Mathematical Foundation: The Farey sequence F_n is the ordered set of reduced fractions {p/q : 0 ≤ p ≤ q ≤ n, gcd(p,q) = 1}. These points are mapped to the unit circle at angles θ = 2πp/q, creating a dense set of rational points on ∂𝔻.

Key Properties: The mediant property states that for adjacent fractions p/q, r/s in F_n, their mediant (p+r)/(q+s) first appears in F_{q+s}, and |ps - qr| = 1 (unimodular determinant). The Farey triangle connects these boundary points, with edges representing the SL(2,ℤ) action on ℚ/ℤ.

Prime Distribution: Primes p are placed at angles 2πp/M where M is the modulus. Dirichlet's theorem guarantees that primes are equidistributed among residue classes coprime to M, each with asymptotic density 1/φ(M). The visualization reveals this equidistribution through uniform angular spacing of prime points in coprime residue classes.

Applications: Farey sequences appear in continued fractions, Ford circles, the Stern-Brocot tree, rational approximation theory, and the geometric study of modular curves.

Canvas 2: ℍ Upper Half-Plane - Cayley Transform & Hyperbolic Geometry

Cayley Transform: The map w = i(1+z)/(1-z) is a conformal isomorphism from the Poincaré disk 𝔻 to the upper half-plane ℍ = {w ∈ ℂ : Im(w) > 0}. It preserves the hyperbolic metric: ds²_𝔻 = 4|dz|²/(1-|z|²)² corresponds to ds²_ℍ = |dw|²/Im(w)².

Hyperbolic Geodesics: In ℍ, geodesics are semicircles perpendicular to ℝ (or vertical lines). The hyperbolic distance between z₁, z₂ ∈ ℍ is d_ℍ(z₁,z₂) = arcosh(1 + |z₁-z₂|²/(2Im(z₁)Im(z₂))). Geodesics minimize this distance.

Modular Group Action: PSL(2,ℤ) = SL(2,ℤ)/{±I} acts on ℍ by Möbius transformations γ·z = (az+b)/(cz+d) where [[a,b],[c,d]] ∈ SL(2,ℤ). Generated by S(z) = -1/z and T(z) = z+1 with relation (ST)³ = 1. The fundamental domain 𝒟 = {z : |z| ≥ 1, |Re(z)| ≤ 1/2} tiles ℍ under PSL(2,ℤ).

Ford Circles: For rational p/q in lowest terms, the Ford circle has center (p/q, 1/(2q²)) and radius 1/(2q²). Ford circles corresponding to Farey neighbors are tangent (|ps-qr| = 1), providing geometric proof of Farey sequence properties.

Applications: Modular forms, elliptic curves, j-invariant, modular curves, automorphic forms, and connections to the Riemann Hypothesis through the Selberg trace formula.

Canvas 3: ⊚ Nested Modular Rings - Group Structure of (ℤ/mℤ)×

Group of Units: For each modulus m, (ℤ/mℤ)× = {k ∈ ℤ/mℤ : gcd(k,m) = 1} forms a multiplicative group of order φ(m). The visualization shows all residue classes 0 ≤ k < m on a circle, with coprime elements (units) highlighted.

Euler's Totient: φ(m) = m·∏_{p|m}(1 - 1/p) counts units. For prime powers: φ(p^k) = p^{k-1}(p-1). Multiplicativity: φ(mn) = φ(m)φ(n) when gcd(m,n) = 1. Euler's theorem: If gcd(a,m) = 1, then a^{φ(m)} ≡ 1 (mod m).

GCD Structure: Points are colored by gcd(k,m), revealing subgroup structure. Elements with gcd(k,m) = d correspond to m/d·(ℤ/dℤ)× embedded in ℤ/mℤ. The Chinese Remainder Theorem: if gcd(m₁,m₂) = 1, then ℤ/(m₁m₂)ℤ ≅ ℤ/m₁ℤ × ℤ/m₂ℤ, visible in the factorization patterns.

Residue Lifting: In modular towers Mₙ = M₀·b^n, a residue r at level n lifts to b residues {r, r+Mₙ, ..., r+(b-1)Mₙ} at level n+1. Coprimality is preserved: if gcd(r,M₀) = gcd(r,b) = 1, then all lifts remain coprime to Mₙ₊₁.

Applications: RSA cryptography (φ(pq) = (p-1)(q-1)), Diffie-Hellman key exchange, primitive roots, discrete logarithm problem, quadratic reciprocity, and structure theory of finite abelian groups.

Canvas 4: 🔄 Modular Reduction Projection - Channel Decomposition

Farey Channel Structure: Every rational p/q with 0 < p < q can be reduced by its GCD: if d = gcd(p,q), then p/q represents the same angle as (p/d)/(q/d). This reduction maps rationals to their "channels" based on the reduced denominator.

Dyadic Lifting: Starting from a base modulus M₀, create a geometric sequence M_n = M₀·2^n. A residue r mod M_n lifts to two residues {r, r+M_n} mod M_{n+1}. If gcd(r,M₀) = 1 and r is odd, both lifts remain coprime to all future M_k.

Zero-Residue Alignment: The 0-residue class (multiples of M₀) creates a "prime-avoiding channel" - no primes lie on this ray. As we lift through the tower, these channels bifurcate while preserving the prime-avoiding property, creating the sieve structure.

Applications: Sieve methods (Eratosthenes, Atkin), gap analysis between primes, twin prime conjecture approaches, and fractal sieve structures.

Canvas 5: ℂ Full Complex Plane - Complete Transform View

Riemann Sphere: The extended complex plane ℂ̂ = ℂ ∪ {∞} forms a compact Riemann surface (sphere). Möbius transformations act as conformal automorphisms of ℂ̂, forming the group Aut(ℂ̂) ≅ PSL(2,ℂ).

Complete Cayley Image: The full transform shows: |z| < 1 → Im(w) > 0 (upper half-plane), |z| = 1 → Im(w) = 0 (real axis), |z| > 1 → Im(w) < 0 (lower half-plane). The point z = 1 maps to w = ∞, and z = -1 is the pole (undefined).

Circle Preservation: Möbius transformations map generalized circles (circles and lines in ℂ̂) to generalized circles. This property characterizes Möbius transformations among all holomorphic maps.

Applications: Complex analysis, Riemann mapping theorem, uniformization theorem, conformal field theory, string theory, and Teichmüller theory.

Canvas 6: ⚙ Primitive Roots - Cyclic Structure of (ℤ/Mℤ)×

Primitive Root Definition: An element g ∈ (ℤ/Mℤ)× is a primitive root if its multiplicative order is φ(M), i.e., g^{φ(M)} ≡ 1 (mod M) and g^k ≢ 1 (mod M) for 0 < k < φ(M). When primitive roots exist, (ℤ/Mℤ)× is cyclic.

Existence Theorem: Primitive roots exist mod M if and only if M ∈ {1, 2, 4, p^k, 2p^k} where p is an odd prime. For these moduli, (ℤ/Mℤ)× ≅ ℤ/φ(M)ℤ is cyclic, and there are exactly φ(φ(M)) primitive roots.

Power Sequence Visualization: The sequence {g^n mod M : n = 0,1,2,...,φ(M)-1} cycles through all units, landing at each exactly once. Points are plotted at angles 2π·g^n/M, creating a spiraling pattern that demonstrates the cyclic group structure.

Order Structure: By Lagrange's theorem, the order of any element divides φ(M). Elements are organized by their orders, with primitive roots having maximal order φ(M). The visualization color-codes elements by order.

Applications: Discrete logarithm problem, Diffie-Hellman protocol, index calculus, cyclic codes, pseudorandom number generation, and construction of finite fields.

Canvas 7: 🔵 Coprime Circle - Visual Coprimality Patterns

Coprimality Visualization: For a given modulus M, all integers 1 ≤ k < M are placed on a circle at angles 2πk/M. Points are colored green if gcd(k,M) = 1 (coprime) and red otherwise, immediately revealing the density φ(M)/M.

Pattern Recognition: For prime M, all points except M itself are green (φ(M) = M-1). For composite M, the pattern reveals prime factors: non-coprime points appear at intervals corresponding to divisors of M. For M = p^k, non-coprime points are spaced every p positions.

Connection Structure: Optional edges connect coprime pairs, revealing the graph structure of (ℤ/Mℤ)×. The complete graph on φ(M) vertices shows all unit interactions, with visual clustering indicating subgroup structure.

Applications: Visual verification of coprimality, clock arithmetic, modular multiplication tables, and understanding the structure of cyclic groups.

Canvas 8: 📊 Phi Ratios Chart - φ(n)/n Convergence Behavior

Totient Density Function: The ratio φ(n)/n represents the probability that a randomly chosen integer k with 1 ≤ k ≤ n is coprime to n. Using the formula φ(n) = n·∏_{p|n}(1-1/p), we see this equals ∏_{p|n}(1-1/p).

Prime Values: For prime p, φ(p)/p = (p-1)/p → 1 as p → ∞. These are marked as red dots on the chart, showing that primes have maximum coprime density.

Prime Power Values: For p^k, φ(p^k)/p^k = 1 - 1/p, constant in k. This creates horizontal "shelves" in the chart at heights 1/2 (powers of 2), 2/3 (powers of 3), 4/5 (powers of 5), etc.

Multiplicative Pattern: For n = ∏ p_i^{a_i}, we have φ(n)/n = ∏(1-1/p_i). Highly composite numbers with many small prime factors have low ratios, while primorials have systematically decreasing ratios.

Asymptotic Results: The average value of φ(n)/n is asymptotically 6/π² ≈ 0.6079 (related to probability of coprimality). The minimum values occur at primorials, approaching 0.

Canvas 9: 📈 Convergence to 6/π² - Mertens' Third Theorem

Probability of Coprimality: For randomly chosen integers a,b ≤ N, the probability that gcd(a,b) = 1 is asymptotically 6/π² ≈ 0.6079 as N → ∞. This is equivalent to asking: what fraction of all pairs (a,b) with a,b ≤ N are coprime?

Derivation via Euler Product: The probability equals ∏_p (1-1/p²) where the product is over all primes. This Euler product telescopes to 1/ζ(2) = 6/π², where ζ(s) = ∑_{n≥1} 1/n^s is the Riemann zeta function and ζ(2) = π²/6 (Basel problem).

Visualization Method: The cumulative count of coprime pairs (a,b) with 1 ≤ a < b ≤ n is ∑_{k=1}^n φ(k). The ratio of this count to the total number of pairs n(n-1)/2 converges to 6/π².

Mertens' Third Theorem: ∑_{n≤x} φ(n) = (3/π²)x² + O(x log x). This stronger result shows not just convergence but the precise asymptotic density with error term.

Connections: Related to the probability that a random polynomial over ℤ is irreducible, the density of squarefree integers (also 6/π²), and probabilistic number theory.

Canvas 10: 🎯 Coprime Pairs Grid - GCD Matrix Visualization

GCD Matrix: The grid displays gcd(i,j) for 1 ≤ i,j < M as a heatmap. Coprime pairs (gcd = 1) are bright, while pairs with common factors are dimmer. The visual intensity is proportional to 1/gcd(i,j).

Diagonal Structure: The main diagonal (i = j) shows gcd(i,i) = i, highlighted in gold. These are the only "self-GCD" values. The matrix is symmetric: gcd(i,j) = gcd(j,i).

Prime Patterns: Row/column i corresponding to prime p shows: gcd(i,j) ∈ {1,p}. The row is bright except at multiples of p, creating a characteristic sparse pattern. Prime powers show similar but denser patterns.

Totient Interpretation: Each row i has exactly φ(i) bright cells (coprime pairs). The total number of bright cells in the matrix equals ∑_{i=1}^{M-1} φ(i), which asymptotically equals (3/π²)M².

Applications: Visual GCD computation, verification of coprimality, identification of common divisors, and pattern recognition in number theory. Related to the Möbius function and inclusion-exclusion.

Canvas 11: 🔢 Nested Density Rings - Sequential φ(n)/n Visualization

Concentric Ring Structure: Each ring n (from 2 to M) displays all integers 1 ≤ k < n on a circle of radius proportional to n. Coprime points (gcd(k,n) = 1) are shown in lime green, revealing the density φ(n)/n geometrically.

Visual Density Comparison: Ring opacity/brightness is proportional to φ(n)/n. Prime rings are almost fully bright (only one missing point). Highly composite rings are dimmer, reflecting lower coprime density. This allows instant visual comparison across moduli.

Radial Alignment: A vertical ray from center intersects each ring at the point k = ⌊n/2⌋. Following this ray outward shows how a particular residue class evolves across increasing moduli, demonstrating the lifting structure.

Pattern Recognition: Composite n with small prime factors show clear "gaps" - empty sectors where all multiples of those primes lie. Prime powers p^k show regular p-fold rotational symmetry in their gap structure.

Applications: Visual verification of coprime density trends, understanding how (ℤ/nℤ)× grows with n, identifying highly composite vs prime moduli, and seeing the self-similar structure of coprimality across scales.

Canvas 12: 🌟 Dirichlet Characters - Multiplicative Characters & L-Functions

Dirichlet Character Definition: A Dirichlet character modulo M is a group homomorphism χ: (ℤ/Mℤ)× → ℂ× extended to all of ℤ/Mℤ by setting χ(n) = 0 if gcd(n,M) > 1. The character is completely multiplicative: χ(mn) = χ(m)χ(n) for all m,n.

Principal Character: The principal character χ₀ is defined by χ₀(n) = 1 if gcd(n,M) = 1 and χ₀(n) = 0 otherwise. This is the trivial homomorphism on (ℤ/Mℤ)×. The visualization shows this character: green points (χ = 1) for coprime n, faded points (χ = 0) for non-coprime n.

Character Group: The set of all Dirichlet characters mod M forms a group under pointwise multiplication, isomorphic to the dual group (ℤ/Mℤ)^×. There are exactly φ(M) distinct characters. For cyclic (ℤ/Mℤ)× (when primitive roots exist), characters correspond to powers of a primitive root of unity.

Order Coloring: Points are colored by the multiplicative order of n in (ℤ/Mℤ)×. This order equals the smallest positive k such that n^k ≡ 1 (mod M). For any character χ, we have χ(n)^{ord(n)} = 1, so χ(n) is an ord(n)-th root of unity. The color coding reveals this structure.

Dirichlet L-Function: For a character χ, the Dirichlet L-function is L(s,χ) = ∑_{n≥1} χ(n)/n^s for Re(s) > 1. This generalizes the Riemann zeta function (L(s,χ₀) = ζ(s)∏_{p|M}(1-p^{-s})). The Euler product is L(s,χ) = ∏_p (1-χ(p)p^{-s})^{-1}.

Orthogonality Relations: Characters satisfy beautiful orthogonality: ∑_{n mod M} χ(n)ψ̄(n) = φ(M)δ_{χ,ψ} (character orthogonality) and ∑_χ χ(n)χ̄(m) = φ(M)δ_{n,m} (where δ is 1 if arguments equal, 0 otherwise). These are analogs of Fourier orthogonality.

Applications to Prime Distribution: Dirichlet's theorem on primes in arithmetic progressions uses characters: π(x;M,a) ~ x/(φ(M)log x) is proved via L(1,χ) ≠ 0 for non-principal χ. The Generalized Riemann Hypothesis (GRH) concerns zeros of L(s,χ): all non-trivial zeros should satisfy Re(s) = 1/2.

Advanced Connections: Class field theory, Artin L-functions, automorphic forms, Hecke characters, algebraic number theory, Langlands program, and the study of Galois representations.