Fibonacci Word
symbolic-dynamics · 37 views
symbolic-dynamicssymbolic
What It Is
Sturmian sequence --- the simplest aperiodic word, with golden-ratio quasiperiodicity and complexity function p(n) = n + 1 (minimal for non-periodic sequences)
Interpretation
Standard analysis sees: bounded / light-tailed; left-skewed; few distinct values; anti-correlated (alternating); anti-persistent; blue spectrum (high-frequency power); strongly periodic; homoskedastic; monofractal; stationary. The atlas additionally detects Penrose (Quasicrystal):long_range_order.
What standard analysis sees
tail heaviness0.07
asymmetry0.07
occupancy0.02
short-range corr0.05
long-range memory0.03
spectral colour0.99
periodicity0.94
complexity0.25
time-irreversibility0.52
volatility clustering0.02
multifractality0.02
dimensionality0.66
nonstationarity0.04
What the atlas adds
Penrose (Quasicrystal):long_range_order+7.7z
quasiperiodic long-range order (sharp φ-ratio ACF peak)
Atlas-extreme metrics the standard bank can’t predict for this source
Moiré:moire_phi_response | +5.1z | bank-miss 2.3σ |
Zariski:heyting_gap | +4.3z | bank-miss 1.2σ |
Wavelet Cascade:scale_gini_dispersion | +4.2z | bank-miss 3.2σ |
Hyperbolic (Poincaré):temporal_variance | +4.1z | bank-miss 1.9σ |
Septagonal (Danzer):cubic_coherence | -3.9z | bank-miss 1.4σ |
H3 Icosahedral:nn_enrichment | +3.9z | bank-miss 3.0σ |
Boltzmann:dominant_lag | +3.9z | bank-miss 1.8σ |
Sol (Thurston):path_length | +3.7z | bank-miss 1.5σ |
Composition
dtypefloat64
range[0, 1]
unique values2 / 16384
mean ± std0.618 ± 0.486
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/Fibonacci_Word.png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/Fibonacci_Word.png)
Heisenberg (Nil) (centered) > xy_path
Hodge–Laplacian > default
/barcode/Fibonacci_Word.png)
Inflation (Substitution) > barcode
/d_curve/Fibonacci_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/Fibonacci_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/Fibonacci_Word.png)
Ulam Spiral (Sacks) > default
/default/Fibonacci_Word.png)
Ulam Spiral (Square) > default
Visibility Graph > default
Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Penrose Substitution | 2.23 | |
| Collatz Cycle Word | 2.85 | cross-origin |
| Periodic Word | 4.01 |
Which Geometries Light Up
Boltzmann › Boltzmann:dominant_lag | rank 5/307 | 987.0000 |
Catch24 › Catch24:CO_HistogramAMI_even_2_5 | rank 306/307 | 0.0000 |
D4 Triality › D4 Triality:triplet_temporal | rank 305/307 | -0.8320 |
H3 Icosahedral › H3 Icosahedral:nn_enrichment | rank 5/307 | 3.9834 |
Laplacian › Laplacian:poisson_recovery_error | rank 3/307 | 15.4566 |
Moiré › Moiré:moire_phi_response | rank 5/307 | 0.2548 |
Navier-Stokes › Navier-Stokes:ess_rescaling_gain | rank 2/307 | 0.9996 |
Ordinal Partition › Ordinal Partition:forbidden_transitions | rank 5/307 | 0.8667 |
Penrose (Quasicrystal) › Penrose (Quasicrystal):phi_tower | rank 5/307 | 0.7638 |
Septagonal (Danzer) › Septagonal (Danzer):cubic_coherence | rank 305/307 | -0.2314 |
Septagonal (Danzer) › Septagonal (Danzer):z_primary | rank 305/307 | -0.4963 |
Sol (Thurston) › Sol (Thurston):integrated_walk_smoothness | rank 304/307 | -0.1382 |
Time Reversibility › Time Reversibility:ordinal_reversal_distance | rank 303/307 | 0.0002 |
Visibility Graph › Visibility Graph:assortativity | rank 306/307 | -0.4416 |
Wavelet Cascade › Wavelet Cascade:scale_gini_dispersion | rank 2/307 | 0.3023 |
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