Look-and-Say
symbolic-dynamics · 37 views
symbolic-dynamicssymbolic
What It Is
Conway's look-and-say sequence 1, 11, 21, 1211, 111221, 312211, ... emitted as a concatenated digit stream over {1,2,3}. Each term reads off the run-lengths of the previous ("one 1" -> 11, "two 1s" -> 21). For the standard seed the digits stay in {1,2,3}, and successive term lengths grow as lambda^n with lambda ~= 1.30357 (Conway's constant). Deterministic, self-describing cousin of Kolakoski.
Interpretation
Standard analysis sees: few distinct values; blue spectrum (high-frequency power); homoskedastic; stationary. The atlas finds no named structure, but the source is distinctively extreme on Cantor Set:mean_gap (-4.9z) — beyond what the standard bank predicts for it.
What standard analysis sees
tail heaviness0.33
asymmetry0.76
occupancy0.11
short-range corr0.16
long-range memory0.17
spectral colour0.90
periodicity0.49
complexity0.51
time-irreversibility0.83
volatility clustering0.12
multifractality0.34
dimensionality0.81
nonstationarity0.15
What the atlas adds
Atlas-extreme metrics the standard bank can’t predict for this source
Cantor Set:mean_gap | -4.8z | bank-miss 1.2σ |
Nonstationarity:change_quantiles_high | +4.5z | bank-miss 2.8σ |
Zariski:algebraic_residual | +3.2z | bank-miss 1.4σ |
Klein Bottle:wht_spectral_kurtosis | +2.9z | bank-miss 2.0σ |
Zariski:residual_slope | -2.7z | bank-miss 2.1σ |
Predictability:entropy_decay_rate | -2.5z | bank-miss 1.3σ |
Composition
dtypeuint8
range[1, 3]
unique values3 / 16384
mean ± std1.69 ± 0.765
Fixed alphabet — only 3 distinct symbols across 16384 samples.
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
_(centered)/signed_log_z/Look-and-Say.png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/Look-and-Say.png)
Heisenberg (Nil) (centered) > xy_path
Hodge–Laplacian > default
/barcode/Look-and-Say.png)
Inflation (Substitution) > barcode
/d_curve/Look-and-Say.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/Look-and-Say.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/Look-and-Say.png)
Ulam Spiral (Sacks) > default
/default/Look-and-Say.png)
Ulam Spiral (Square) > default
Visibility Graph > default
Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Phi-Squared β-Shift | 4.13 | |
| DNA Thermus | 4.18 | cross-origin |
| DNA Centromere | 4.28 | cross-origin |
Which Geometries Light Up
Cantor Set › Cantor Set:bit_plane_autocorrelation | rank 3/306 | 0.9988 |
Cantor Set › Cantor Set:max_gap | rank 305/306 | 0.0006 |
Cantor Set › Cantor Set:mean_gap | rank 305/306 | 0.0000 |
Cayley › Cayley:delta_hyp_norm | rank 3/306 | 0.2628 |
Nonstationarity › Nonstationarity:change_quantiles_high | rank 2/306 | 0.2448 |
Zariski › Zariski:algebraic_residual | rank 1/306 | 0.0735 |
Zariski › Zariski:residual_slope | rank 306/306 | -5.7294 |
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