Kolakoski Sequence
exotic · 36 views
exotic
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.
What standard analysis sees
tail heaviness0.01
asymmetry0.44
occupancy0.06
short-range corr0.11
long-range memory0.10
spectral colour0.94
periodicity0.48
complexity0.39
time-irreversibility0.47
volatility clustering0.04
multifractality0.14
dimensionality0.53
nonstationarity0.03
What the atlas adds
Atlas-extreme metrics the standard bank can’t predict for this source
Nonstationarity:change_quantiles_mid | +5.5z | bank-miss 3.1σ |
H3 Icosahedral:nn_enrichment | +4.6z | bank-miss 1.6σ |
H3 Icosahedral:temporal_coherence | +4.2z | bank-miss 1.9σ |
Projective ℙ²:collinearity | -4.0z | bank-miss 1.6σ |
Spirograph:petal_symmetry | +3.4z | bank-miss 1.3σ |
Lorentzian:crossing_density | +2.9z | bank-miss 2.0σ |
Modular Residue:occupancy_entropy | -2.4z | bank-miss 1.0σ |
Composition
dtypeint8
range[1, 2]
unique values2 / 16384
mean ± std1.5 ± 0.5
Binary sequence — two distinct symbols.
Render Gallery
Native > phase_plotraw
Native > timeseriesraw
Native > triple_latticeraw
Boltzmann > lag_map
Boltzmann > spin_grid
Cayley > default
Chladni > plate_bottom
Chladni > plate_top
_(centered)/signed_log_z/Kolakoski_Sequence.png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/Kolakoski_Sequence.png)
Heisenberg (Nil) (centered) > xy_path
Hodge–Laplacian > default
/barcode/Kolakoski_Sequence.png)
Inflation (Substitution) > barcode
/d_curve/Kolakoski_Sequence.png)
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
/phi_spectrum/Kolakoski_Sequence.png)
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
/default/Kolakoski_Sequence.png)
Ulam Spiral (Sacks) > default
/default/Kolakoski_Sequence.png)
Ulam Spiral (Square) > default
Visibility Graph > default
Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Thue-Morse | 3.17 | |
| L-System (Dragon Curve) | 4.44 | |
| Symbolic Henon | 4.58 |
Which Geometries Light Up
Catch24 › Catch24:DN_Spread_Std | rank 4/298 | 0.5000 |
Catch24 › Catch24:SC_FluctAnal_2_rsrangefit_50_1_logi_prop_r1 | rank 295/298 | 0.1220 |
H3 Icosahedral › H3 Icosahedral:nn_enrichment | rank 3/298 | 4.4933 |
H3 Icosahedral › H3 Icosahedral:temporal_coherence | rank 3/298 | 0.3912 |
Laplacian › Laplacian:cross_scale_curvature_coherence | rank 294/298 | -0.5001 |
Lorentzian › Lorentzian:crossing_density | rank 2/298 | 0.6668 |
Mostow Rigidity › Mostow Rigidity:distance_rigidity | rank 2/298 | 0.9844 |
Nonstationarity › Nonstationarity:change_quantiles_mid | rank 3/298 | 0.6666 |
Projective ℙ² › Projective ℙ²:collinearity | rank 296/298 | 0.9324 |
Spherical S² › Spherical S²:angular_spread | rank 3/298 | 1.5708 |
Spherical S² › Spherical S²:concentration | rank 296/298 | 0.0015 |
Spirograph › Spirograph:petal_symmetry | rank 5/298 | 0.6658 |
Time Reversibility › Time Reversibility:increment_skewness_max | rank 294/298 | 0.0007 |
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