Pell Word

exotic · 36 views
exotic

What It Is

Silver-mean Sturmian: substitution a→aab, b→a. Substitution matrix [[2,1],[1,0]] has eigenvalue 1+√2 ≈ 2.414 (silver mean). Algebraic-symbolic sibling of Fibonacci Word but with silver-mean rather than golden-mean rotation; serves as a silver-mean contrast class against the golden-quasi anchors.

Interpretation

Standard analysis sees: few distinct values; anti-correlated (alternating); anti-persistent; blue spectrum (high-frequency power); strongly periodic; homoskedastic; monofractal; stationary. The atlas finds no named structure, but the source is distinctively extreme on Projective ℙ²:collinearity (-4.2z) — beyond what the standard bank predicts for it.

What standard analysis sees
tail heaviness0.32
asymmetry0.82
occupancy0.01
short-range corr0.08
long-range memory0.04
spectral colour0.98
periodicity0.85
complexity0.23
time-irreversibility0.49
volatility clustering0.03
multifractality0.04
dimensionality0.46
nonstationarity0.06
What the atlas adds
Atlas-extreme metrics the standard bank can’t predict for this source
Projective ℙ²:collinearity-4.2zbank-miss 1.2σ
Sol (Thurston):path_length+4.0zbank-miss 1.6σ
Marginal:tail_occupancy_r1_5+3.8zbank-miss 2.7σ
E8 Lattice:coset_transition+3.6zbank-miss 3.1σ
Sol (Thurston):sol_step_persistence-3.0zbank-miss 1.4σ
Projective ℙ²:mean_distance+2.7zbank-miss 1.0σ
Isochronicity:frequency_shear+2.6zbank-miss 1.4σ
Lorentzian:crossing_density+2.4zbank-miss 1.9σ

Composition

dtypeuint8
range[0, 255]
unique values2 / 16384
mean ± std74.7 ± 116

Binary sequence — two distinct symbols.

Render Gallery

Native > phase_plot
Native > phase_plotraw
Native > timeseries
Native > timeseriesraw
Native > triple_lattice
Native > triple_latticeraw
Boltzmann > lag_map
Boltzmann > lag_map
Boltzmann > spin_grid
Boltzmann > spin_grid
Cayley > default
Cayley > default
Chladni > plate_bottom
Chladni > plate_bottom
Chladni > plate_top
Chladni > plate_top
Heisenberg (Nil) (centered) > signed_log_z
Heisenberg (Nil) (centered) > signed_log_z
Heisenberg (Nil) (centered) > xy_path
Heisenberg (Nil) (centered) > xy_path
Hodge–Laplacian > default
Hodge–Laplacian > default
Inflation (Substitution) > barcode
Inflation (Substitution) > barcode
Inflation (Substitution) > d_curve
Inflation (Substitution) > d_curve
Information Theory > default
Information Theory > default
Julia Set > escape_histogram
Julia Set > escape_histogram
Klein Bottle > default
Klein Bottle > default
Laplacian > default
Laplacian > default
Logarithmic Spiral > default
Logarithmic Spiral > default
Möbius-S³ > mobius_cylinder
Möbius-S³ > mobius_cylinder
Möbius-S³ > s2_base_map
Möbius-S³ > s2_base_map
Ordinal Partition > default
Ordinal Partition > default
Penrose (Quasicrystal) > phi_spectrum
Penrose (Quasicrystal) > phi_spectrum
Persistent Homology > barcode
Persistent Homology > barcode
Persistent Homology > h1_diagram
Persistent Homology > h1_diagram
Predictability > default
Predictability > default
Recurrence Quantification > default
Recurrence Quantification > default
Spectral Analysis > default
Spectral Analysis > default
Spectral Graph > default
Spectral Graph > default
Spirograph > mode_spectrum
Spirograph > mode_spectrum
Spirograph > phasor_trajectory
Spirograph > phasor_trajectory
Symplectic > flux_grid
Symplectic > flux_grid
Symplectic > phase_space
Symplectic > phase_space
Ulam Spiral (Sacks) > default
Ulam Spiral (Sacks) > default
Ulam Spiral (Square) > default
Ulam Spiral (Square) > default
Visibility Graph > default
Visibility Graph > default
Wavelet Cascade > default
Wavelet Cascade > default

Atlas Position

Nearest neighborDistance
Penrose Substitution3.89
Tribonacci Word4.24
Fibonacci Word4.49

Open in Atlas →

Which Geometries Light Up

AutoRegressive › AutoRegressive:ar_coef_1rank 295/298-1.0002
Cayley › Cayley:delta_hyp_normrank 295/2980.0000
E8 Lattice › E8 Lattice:coset_transitionrank 4/2980.2023
Fisher Information › Fisher Information:log_det_fisherrank 5/298137.4467
Fisher Information › Fisher Information:trace_fisherrank 5/298229604.8322
Hyperbolic (Poincaré) › Hyperbolic (Poincaré):curvature_structurerank 296/2981.0683
H² × ℝ (Thurston) › H² × ℝ (Thurston):hyperbolic_triangle_arearank 298/2980.0000
Isochronicity › Isochronicity:frequency_shearrank 5/2980.8529
Lorentzian › Lorentzian:crossing_densityrank 4/2980.5858
Marginal › Marginal:tail_occupancy_r1_5rank 1/2980.2929
Mostow Rigidity › Mostow Rigidity:margulis_ratiorank 3/2980.9247
Möbius-S³ › Möbius-S³:phi_vertex_excessrank 1/2980.2000
Navier-Stokes › Navier-Stokes:intermittency_autocorrrank 295/298-0.4142
Nonstationarity › Nonstationarity:metric_volatilityrank 1/2983.2873
Projective ℙ² › Projective ℙ²:mean_distancerank 2/2980.3755
Projective ℙ² › Projective ℙ²:distance_stdrank 3/2980.2466
Projective ℙ² › Projective ℙ²:collinearityrank 297/2980.9297
Sol (Thurston) › Sol (Thurston):path_lengthrank 4/29830989515.1580
Sol (Thurston) › Sol (Thurston):sol_step_persistencerank 298/298-0.2310
Symplectic › Symplectic:windowed_area_cvrank 295/2980.0333
Time Reversibility › Time Reversibility:ordinal_reversal_distancerank 297/2980.0002
Time Reversibility › Time Reversibility:increment_skewness_maxrank 298/2980.0003
Visibility Graph › Visibility Graph:assortativityrank 294/298-0.3586
Wavelet Cascade › Wavelet Cascade:cascade_couplingrank 296/298-0.5556
Zariski › Zariski:residual_convexityrank 1/29814.5042
Zariski › Zariski:heyting_gaprank 2/2980.9048
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Structure Atlas — Open Atlas · GitHub