Tribonacci Word
symbolic-dynamics · 37 views
symbolic-dynamicssymbolic
What It Is
Cubic-Pisot Sturmian-like: substitution a→ab, b→ac, c→a. Substitution matrix [[1,1,0],[1,0,1],[1,0,0]] has eigenvalue Tribonacci constant ≈1.839 (real root of x³=x²+x+1). Ternary alphabet, cubic algebraic ratio --- a degree-3 Pisot contrast class against degree-2 Fibonacci/Pell.
Interpretation
Standard analysis sees: few distinct values; anti-correlated (alternating); anti-persistent; blue spectrum (high-frequency power); strongly periodic; homoskedastic; monofractal; stationary. The atlas additionally detects Multi-Scale Wasserstein:w_max_ratio.
What standard analysis sees
tail heaviness0.43
asymmetry0.80
occupancy0.09
short-range corr0.03
long-range memory0.05
spectral colour0.98
periodicity0.85
complexity0.28
time-irreversibility0.48
volatility clustering0.11
multifractality0.05
dimensionality0.51
nonstationarity0.10
What the atlas adds
Multi-Scale Wasserstein:w_max_ratio+4.1z
scale-coherent dispersion mismatch (max/min Wasserstein across scales)
Atlas-extreme metrics the standard bank can’t predict for this source
Julia Set:potential_variance | +4.6z | bank-miss 1.9σ |
Julia Set:potential_smoothness | +4.5z | bank-miss 1.9σ |
Modular Residue:cycle_fraction | +3.2z | bank-miss 1.5σ |
Composition
dtypeuint8
range[0, 255]
unique values3 / 16384
mean ± std78.8 ± 95.3
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/Tribonacci_Word.png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/Tribonacci_Word.png)
Heisenberg (Nil) (centered) > xy_path
Hodge–Laplacian > default
/barcode/Tribonacci_Word.png)
Inflation (Substitution) > barcode
/d_curve/Tribonacci_Word.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/Tribonacci_Word.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/Tribonacci_Word.png)
Ulam Spiral (Sacks) > default
/default/Tribonacci_Word.png)
Ulam Spiral (Square) > default
Visibility Graph > default
Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Periodic Word | 3.91 | |
| Collatz Cycle Word | 4.00 | cross-origin |
| Pell Word | 4.22 |
Which Geometries Light Up
Catch24 › Catch24:SC_FluctAnal_2_dfa_50_1_2_logi_prop_r1 | rank 307/307 | 0.1200 |
Multi-Scale Wasserstein › Multi-Scale Wasserstein:w_max_ratio | rank 4/307 | 1526.7181 |
Sol (Thurston) › Sol (Thurston):integrated_walk_smoothness | rank 303/307 | -0.1236 |
← Thue-Morsein signal-synthesis
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