low-discrepancy-sequences skill
Deterministic color generation via low-discrepancy sequences with bijective index recovery.
Extends beyond the golden angle (φ) with multiple low-discrepancy sequences for uniform color space coverage. All sequences maintain bijectivity: given a color and seed, you can recover the index n.
All sequences are bijective on index: Given (color, seed), you can recover n.
This enables:
These sequences extend the existing gay-mcp MCP server tools:
gay_golden_thread: Current φ-based generationgay_plastic_thread: New φ₂-based 2D generationgay_halton_color: Direct RGB via Haltongay_r_sequence: n-dimensional R-sequencegay_sobol_color: High-dimensional Sobolgay_invert_color: Recover index from colorTrit: 0 (ERGODIC)
Low-discrepancy sequences are infrastructure for uniform space coverage - neither generative (+1) nor analytical (-1), but foundational coordination (0).
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
Continued fractions provide geodesic paths in hyperbolic geometry (PSL(2,ℝ) action on ℍ²). This connects to:
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.