Thue-Morse
symbolic-dynamics · 37 views
symbolic-dynamicssymbolic
What It Is
Binary substitution sequence (0→01, 1→10) --- perfectly balanced yet aperiodic, with a singular-continuous, highly peaked Fourier spectrum (not flat like noise) despite being entirely deterministic
Interpretation
Standard analysis sees: bounded / light-tailed; few distinct values; anti-correlated (alternating); anti-persistent; blue spectrum (high-frequency power); monofractal; stationary. The atlas finds no named structure, but the source is distinctively extreme on Nonstationarity:change_quantiles_mid (+5.0z) — beyond what the standard bank predicts for it.
What standard analysis sees
tail heaviness0.00
asymmetry0.41
occupancy0.08
short-range corr0.10
long-range memory0.04
spectral colour0.97
periodicity0.48
complexity0.40
time-irreversibility0.51
volatility clustering0.22
multifractality0.00
dimensionality0.64
nonstationarity0.00
What the atlas adds
Atlas-extreme metrics the standard bank can’t predict for this source
Nonstationarity:change_quantiles_mid | +5.0z | bank-miss 2.0σ |
Gottwald-Melbourne:k_variance | +4.4z | bank-miss 1.4σ |
Wavelet Cascade:cascade_coupling | -2.9z | bank-miss 1.1σ |
Lorentzian:crossing_density | +2.8z | bank-miss 1.2σ |
Nonstationarity:vol_of_vol | +2.7z | bank-miss 2.8σ |
Nonstationarity:trajectory_dim | -2.4z | bank-miss 1.4σ |
D4 Triality:triplet_temporal | -2.3z | bank-miss 1.6σ |
Bispectrum:bicoherence_gini | -2.3z | bank-miss 2.3σ |
Composition
dtypefloat64
range[0, 1]
unique values2 / 16384
mean ± std0.5 ± 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 > free_plate_nodal
Chladni > plate_bottom
Chladni > plate_top
_(centered)/signed_log_z/Thue-Morse.png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/Thue-Morse.png)
Heisenberg (Nil) (centered) > xy_path
Hodge–Laplacian > default
/barcode/Thue-Morse.png)
Inflation (Substitution) > barcode
/d_curve/Thue-Morse.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/Thue-Morse.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/Thue-Morse.png)
Ulam Spiral (Sacks) > default
/default/Thue-Morse.png)
Ulam Spiral (Square) > default
Visibility Graph > default
Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Kolakoski Sequence | 2.99 | |
| Collatz Cycle Word | 3.99 | cross-origin |
| Periodic Word | 4.33 |
Which Geometries Light Up
Catch24 › Catch24:DN_Spread_Std | rank 2/306 | 0.5000 |
Catch24 › Catch24:SC_FluctAnal_2_dfa_50_1_2_logi_prop_r1 | rank 302/306 | 0.1600 |
Catch24 › Catch24:SC_FluctAnal_2_rsrangefit_50_1_logi_prop_r1 | rank 306/306 | 0.1200 |
Gottwald-Melbourne › Gottwald-Melbourne:k_variance | rank 5/306 | 0.7330 |
H3 Icosahedral › H3 Icosahedral:nn_enrichment | rank 1/306 | 5.8534 |
H3 Icosahedral › H3 Icosahedral:temporal_coherence | rank 1/306 | 0.6408 |
Heisenberg (Nil) (centered) › Heisenberg (Nil) (centered):xy_spread | rank 303/306 | 0.9718 |
Higher-Order Statistics › Higher-Order Statistics:bicoherence_max | rank 302/306 | 0.0163 |
Higher-Order Statistics › Higher-Order Statistics:c3_energy | rank 303/306 | 0.0001 |
Klein Bottle › Klein Bottle:linear_complexity | rank 3/306 | 1.0398 |
Lorentzian › Lorentzian:crossing_density | rank 3/306 | 0.6667 |
Nonstationarity › Nonstationarity:change_quantiles_mid | rank 3/306 | 0.6667 |
Nonstationarity › Nonstationarity:trajectory_dim | rank 303/306 | 0.2054 |
Projective ℙ² › Projective ℙ²:collinearity | rank 303/306 | 0.9338 |
Spherical S² › Spherical S²:angular_spread | rank 1/306 | 1.5708 |
Spherical S² › Spherical S²:hemisphere_balance | rank 2/306 | 0.9999 |
Spherical S² › Spherical S²:concentration | rank 306/306 | 0.0001 |
Wavelet Cascade › Wavelet Cascade:cascade_coupling | rank 305/306 | -0.6000 |
Zipf–Mandelbrot (8-bit) › Zipf–Mandelbrot (8-bit):gini_coefficient | rank 302/306 | 0.0000 |
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