eisenstein

Exact integer arithmetic for hexagonal lattice constraints via Eisenstein integers.

Zero unsafe code. Zero floating point. Zero dependencies. no_std compatible. rustc 1.75.0+.

What This Provides

• E12 — Eisenstein integer type a + bω where ω = (-1 + √-3)/2
  • Norm: a² - ab + b² (exact integer, no √-3 needed)
  • Hex distance, multiplication, D₆ symmetry
• HexDisk — Bounded hexagonal region of radius R
  • Vertex count: 3R² + 3R + 1
  • 6 neighbors via Eisenstein units
• EisensteinTriple — Parametric generator (m²-n², 2mn-n², m²-mn+n²)
  • Multiplication closure: N(z₁z₂) = N(z₁)N(z₂)
  • D₆ Weyl orbit invariance
  • ~6.8× denser than Pythagorean triples (59,841 vs 10,428 at c < 65,536)

Quick Start

use eisenstein::{E12, HexDisk, EisensteinTriple};

// Eisenstein integer
let z = E12::new(-5, 3);
assert_eq!(z.norm(), 49); // a²-ab+b² = 25+15+9 = 49

// Hex disk of radius 5
let disk = HexDisk::new(5);
assert_eq!(disk.vertex_count(), 91); // 3·25+3·5+1

// Parametric triple: m=7, n=4
let t = EisensteinTriple::new(7, 4);
assert_eq!(t.c(), 37); // m²-mn+n² = 49-28+16 = 37

Why Eisenstein Integers?

Eisenstein integers Z[ω] are the natural coordinate system for hexagonal lattices:

1. 6 units = 6 hex neighbors (no lookup table needed)
2. Norm multiplicativity gives exact integer constraint propagation
3. D₆ symmetry is built into the algebra (Weyl group of A₂)
4. ~6.8× triple density vs Pythagorean — far more solutions per search step
5. Laman rigidity — hex lattice has 1.5× edge redundancy (safety margin)

Applications

• Hex grid constraint propagation (game/simulation)
• Sensor fusion on hexagonal topologies
• Safety-critical integer-only constraint checking (DO-178C compatible)
• Lattice-based cryptography (structured lattices)
• Compressed sensing on hexagonal sampling grids

Properties Verified

Property	Status	Method
Norm multiplicativity	✅ 10,000 random multiplications, zero drift	cargo test
D₆ Weyl invariance	✅ All 6 rotations preserve norm	cargo test
Multiplication closure	✅ 100% (210/210 Python verification)	eisenstein_triples.py
Parametric form validity	✅ All m,n up to 9	eisenstein_triples.py
Laman redundancy (2D)	✅ → 1.5× as V → ∞	hex_zhc.py
Laman redundancy (3D FCC)	✅ → 2.0× as V → ∞	hex_zhc.py
O(V) holonomy check	✅ ~0.0009ms/vertex constant	hex_zhc.py

Storage

• Each coordinate: 4 bytes (i32)
• Max norm for |q|,|r| ≤ 4096: 3·4096² = 50,331,648 (fits in 26 bits, within i32 range)
• Hex disk R=36: 3,997 vertices, 11,082 edges

License

MIT OR Apache-2.0