Kolakoski Sequence

What It Is

Kolakoski K = 1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,... --- the unique infinite sequence over {1,2} starting with 1 whose run-length encoding equals itself. Density of 1s conjectured to be exactly 1/2 (Keane 1991, still open in 2026). Aperiodic, self-similar in run-length structure, complexity class unknown.

Interpretation

Standard analysis sees: bounded / light-tailed; few distinct values; anti-correlated (alternating); anti-persistent; blue spectrum (high-frequency power); homoskedastic; monofractal; stationary. The atlas finds no named structure, but the source is distinctively extreme on Nonstationarity:change_quantiles_mid (+5.5z) — beyond what the standard bank predicts for it.

Composition

Binary sequence — two distinct symbols.

Render Gallery

Native > phase_plotraw

Native > timeseriesraw

Native > triple_latticeraw

Boltzmann > lag_map

Boltzmann > spin_grid

Cayley > default

Chladni > free_plate_nodal

Chladni > plate_bottom

Chladni > plate_top

Heisenberg (Nil) (centered) > signed_log_z

Heisenberg (Nil) (centered) > xy_path

Hodge–Laplacian > default

Inflation (Substitution) > barcode

Inflation (Substitution) > d_curve

Information Theory > default

Julia Set > escape_histogram

Klein Bottle > default

Laplacian > default

Logarithmic Spiral > default

Möbius-S³ > mobius_cylinder

Möbius-S³ > s2_base_map

Ordinal Partition > default

Penrose (Quasicrystal) > phi_spectrum

Persistent Homology > barcode

Persistent Homology > h1_diagram

Predictability > default

Recurrence Quantification > default

Spectral Analysis > default

Spectral Graph > default

Spirograph > mode_spectrum

Spirograph > phasor_trajectory

Symplectic > flux_grid

Symplectic > phase_space

Ulam Spiral (Sacks) > default

Ulam Spiral (Square) > default

Visibility Graph > default

Wavelet Cascade > default

Atlas Position

Open in Atlas →

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