Penrose Substitution
exotic · 36 views
exotic
What It Is
Robinson-triangle 1D inflation: substitution a→aba, b→ab. Substitution matrix [[2,1],[1,1]] has eigenvalue (3+√5)/2 = φ² ≈ 2.618 (golden squared). Canonical 1D cross-section through the rhombic Penrose tiling along the golden direction --- distinct from Fibonacci Word's φ-scale substitution; same golden family but at a different golden-power resonance.
Interpretation
Standard analysis sees: bounded / light-tailed; few distinct values; anti-correlated (alternating); anti-persistent; blue spectrum (high-frequency power); strongly periodic; homoskedastic; monofractal; stationary. The atlas additionally detects quasicrystalline order.
What standard analysis sees
tail heaviness0.06
asymmetry0.75
occupancy0.03
short-range corr0.05
long-range memory0.03
spectral colour0.96
periodicity0.93
complexity0.25
time-irreversibility0.49
volatility clustering0.01
multifractality0.01
dimensionality0.66
nonstationarity0.04
What the atlas adds
quasicrystalline order+5.0z
aperiodic long-range order at incommensurate (golden-ratio / Fibonacci) spacing — structure repeats at irrational ratios, not a fixed period
Atlas-extreme metrics the standard bank can’t predict for this source
Moiré:moire_phi_response | +7.0z | bank-miss 2.6σ |
Wavelet Cascade:scale_gini_dispersion | +4.2z | bank-miss 2.2σ |
Projective ℙ²:collinearity | -4.2z | bank-miss 1.6σ |
Hyperbolic (Poincaré):temporal_variance | +4.1z | bank-miss 1.4σ |
H3 Icosahedral:nn_enrichment | +4.0z | bank-miss 3.1σ |
Boltzmann:dominant_lag | +3.9z | bank-miss 1.7σ |
D4 Triality:triplet_temporal | -3.2z | bank-miss 2.3σ |
H3 Icosahedral:temporal_coherence | +3.2z | bank-miss 3.5σ |
Composition
dtypeuint8
range[0, 255]
unique values2 / 16384
mean ± std97.4 ± 124
Binary sequence — two distinct symbols.
Render Gallery
Native > phase_plotraw
Native > timeseriesraw
Native > triple_latticeraw
Boltzmann > lag_map
Boltzmann > spin_grid
Cayley > default
Chladni > plate_bottom
Chladni > plate_top
_(centered)/signed_log_z/Penrose_Substitution.png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/Penrose_Substitution.png)
Heisenberg (Nil) (centered) > xy_path
Hodge–Laplacian > default
/barcode/Penrose_Substitution.png)
Inflation (Substitution) > barcode
/d_curve/Penrose_Substitution.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/Penrose_Substitution.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/Penrose_Substitution.png)
Ulam Spiral (Sacks) > default
/default/Penrose_Substitution.png)
Ulam Spiral (Square) > default
Visibility Graph > default
Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Fibonacci Word | 2.40 | |
| Pell Word | 3.89 | |
| Collatz Parity | 4.47 | cross-domain |
Which Geometries Light Up
AutoRegressive › AutoRegressive:ar_coef_1 | rank 294/298 | -1.0000 |
Boltzmann › Boltzmann:frustration | rank 1/298 | 1.0000 |
Catch24 › Catch24:CO_HistogramAMI_even_2_5 | rank 298/298 | 0.0000 |
D4 Triality › D4 Triality:triplet_temporal | rank 297/298 | -0.8326 |
H3 Icosahedral › H3 Icosahedral:nn_enrichment | rank 5/298 | 3.9833 |
H3 Icosahedral › H3 Icosahedral:temporal_coherence | rank 5/298 | 0.3077 |
Laplacian › Laplacian:poisson_recovery_error | rank 4/298 | 15.1189 |
Moiré › Moiré:moire_phi_response | rank 2/298 | 0.3403 |
Multi-Scale Wasserstein › Multi-Scale Wasserstein:w_slope | rank 5/298 | 2.4607 |
Navier-Stokes › Navier-Stokes:ess_quality | rank 1/298 | 0.9996 |
Penrose (Quasicrystal) › Penrose (Quasicrystal):phi_tower | rank 1/298 | 0.9785 |
Projective ℙ² › Projective ℙ²:mean_distance | rank 1/298 | 0.4113 |
Projective ℙ² › Projective ℙ²:distance_std | rank 2/298 | 0.2489 |
Projective ℙ² › Projective ℙ²:collinearity | rank 298/298 | 0.9289 |
Septagonal (Danzer) › Septagonal (Danzer):cubic_coherence | rank 295/298 | -0.2246 |
Septagonal (Danzer) › Septagonal (Danzer):z_primary | rank 295/298 | -0.4744 |
Spectral Analysis › Spectral Analysis:spectral_slope | rank 4/298 | 1.6957 |
Time Reversibility › Time Reversibility:ordinal_reversal_distance | rank 294/298 | 0.0002 |
Visibility Graph › Visibility Graph:assortativity | rank 295/298 | -0.3653 |
Wavelet Cascade › Wavelet Cascade:scale_gini_dispersion | rank 3/298 | 0.3021 |
Zariski › Zariski:residual_convexity | rank 5/298 | 14.4188 |
p-Variation › p-Variation:var_p2 | rank 3/298 | 0.7639 |
← / → within domain · ⇧← / ⇧→ alphabetical · ⇧← / ⇧→ inside an open render = same view across sources