2. Plastic Constant (φ₂)

Dimension: 2D (hue + saturation)
Uniformity: Optimal for 2D
Source: φ₂ ≈ 1.324717... (root of x³ = x + 1)
Formula:
- h = (seed + n/φ₂) mod 1
- s = (seed + n/φ₂²) mod 1

3. Halton Sequence

Dimension: nD (direct RGB or HSL)
Uniformity: Good for any dimension
Source: Prime bases (2, 3, 5, 7...)
Formula: halton(n, base) = ∑ dᵢ/baseⁱ⁺¹

4. R-sequence (Recursive)

Dimension: nD
Uniformity: Near-optimal
Source: φ_d (d-dimensional golden ratio)
Formula: α_d = roots of x^(d+1) = x + 1

5. Kronecker Sequence

Dimension: 1D
Uniformity: Optimal (equidistributed)
Source: Any irrational α
Formula: {nα} mod 1

6. Sobol Sequence

Dimension: nD (up to 1000+)
Uniformity: Excellent for high dimensions
Source: Direction numbers
Formula: Gray code XOR with direction vectors

7. Pisot Sequence

Dimension: nD
Uniformity: Quasiperiodic
Source: Pisot-Vijayaraghavan numbers (algebraic integers)
Formula: θⁿ rounded to nearest integer

8. Continued Fractions

Dimension: 1D
Uniformity: Geodesic in hyperbolic geometry
Source: Continued fraction expansion
Formula: [a₀; a₁, a₂, ...] convergents

Bijection Property

All sequences are bijective on index: Given (color, seed), you can recover n.

This enables:

Reafference: "I generated color C at index n"
Inverse queries: "What index produced this color?"
Temporal reconstruction: "When did I see this color?"

Integration with Gay.jl

These sequences extend the existing gay-mcp MCP server tools:

gay_golden_thread: Current φ-based generation
gay_plastic_thread: New φ₂-based 2D generation
gay_halton_color: Direct RGB via Halton
gay_r_sequence: n-dimensional R-sequence
gay_sobol_color: High-dimensional Sobol
gay_invert_color: Recover index from color

Related Skills

gay-mcp: Deterministic color generation foundation
reafference: Self-recognition via prediction matching
golden-thread: Original φ spiral implementation
phenomenal-bisect: Temperature τ bisection using colors
crystal-family: Crystallographic color assignments
bidirectional-awareness: Skill graph visualization colors

GF(3) Trit Assignment

Trit: 0 (ERGODIC)

Low-discrepancy sequences are infrastructure for uniform space coverage - neither generative (+1) nor analytical (-1), but foundational coordination (0).

References

Niederreiter, H. (1992). Random Number Generation and Quasi-Monte Carlo Methods
Kuipers, L. & Niederreiter, H. (1974). Uniform Distribution of Sequences
Pisot, C. & Salem, R. (1963). Algebraic Numbers and Fourier Analysis
Arnoux, P. & Ito, S. (2001). Pisot substitutions and Rauzy fractals
Series, C. (1985). The geometry of Markoff numbers (continued fractions)

Usage Example

using LowDiscrepancySequences

# Golden angle (current method)
color1 = golden_angle_color(69, seed=42)

# Plastic constant (2D: hue + saturation)
color2 = plastic_color(69, seed=42)

# Halton (direct RGB)
color3 = halton_color(69)

# R-sequence (3D)
color4 = r_sequence_color(69, dim=3, seed=42)

# Invert: recover index
n = invert_color(color2, method=:plastic, seed=42)
@assert n == 69

Connections to Geodesics

Continued fractions provide geodesic paths in hyperbolic geometry (PSL(2,ℝ) action on ℍ²). This connects to:

Geodesic skill representations (shortest execution paths)
Hyperbolic geometry of skill space
Non-backtracking paths (prime geodesics)

The Farey sequence F_n = {p/q : gcd(p,q)=1, 0≤p≤q≤n} gives rational approximations to irrationals via continued fractions, mirroring the discrete approximations to geodesic flows.

SDF Interleaving

This skill connects to Software Design for Flexibility (Hanson & Sussman, 2021):

Primary Chapter: 7. Propagators

Concepts: propagator, cell, constraint, bidirectional, TMS

GF(3) Balanced Triad

low-discrepancy-sequences (○) + SDF.Ch7 (○) + [balancer] (○) = 0

Skill Trit: 0 (ERGODIC - coordination)

Secondary Chapters

Ch3: Variations on an Arithmetic Theme
Ch4: Pattern Matching
Ch6: Layering

Connection Pattern

Propagators flow constraints bidirectionally. This skill propagates information.

low-discrepancy-sequences

Low-Discrepancy Sequences

Purpose

Sequences Implemented

1. Golden Angle (φ)