Phyllotaxis
number-theory · 37 views
number-theoryarithmetic
What It Is
Golden-angle sunflower spiral --- successive seeds placed at 137.508° (= 360°/phi²), the most irrational angle. Quantized angular positions carry genuine quasicrystalline spectral structure with peaks at golden-ratio frequencies
Interpretation
Standard analysis sees: rich, high-entropy values; anti-correlated (alternating); anti-persistent; blue spectrum (high-frequency power); strongly periodic; time-irreversible (slow rise, sharp collapse); homoskedastic; monofractal; low-dimensional; stationary. The atlas additionally detects Penrose (Quasicrystal):long_range_order, Time Reversibility:ordinal_reversal_distance.
What standard analysis sees
tail heaviness0.25
asymmetry0.43
occupancy0.99
short-range corr0.07
long-range memory0.07
spectral colour0.87
periodicity0.88
complexity0.20
time-irreversibility0.08
volatility clustering0.01
multifractality0.06
dimensionality0.08
nonstationarity0.05
What the atlas adds
Penrose (Quasicrystal):long_range_order+7.8z
quasiperiodic long-range order (sharp φ-ratio ACF peak)
Time Reversibility:ordinal_reversal_distance+3.1z
time-reversal symmetry breaking via Bandt-Pompe ordinal patterns
Atlas-extreme metrics the standard bank can’t predict for this source
Moiré:moire_phi_response | +6.0z | bank-miss 2.3σ |
Septagonal (Danzer):cubic_coherence | -4.5z | bank-miss 1.4σ |
Laplacian:laplacian_evolutionary_index | -3.3z | bank-miss 1.8σ |
D4 Triality:triplet_temporal | -2.8z | bank-miss 1.4σ |
p-Variation:volatility_clustering | -2.5z | bank-miss 1.0σ |
Hodge–Laplacian:solenoidal_fraction | -2.0z | bank-miss 2.0σ |
Composition
dtypefloat64
range[8.762e-05, 6.283]
unique values16384 / 16384
mean ± std3.14 ± 1.81
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/Phyllotaxis.png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/Phyllotaxis.png)
Heisenberg (Nil) (centered) > xy_path
Hodge–Laplacian > default
/barcode/Phyllotaxis.png)
Inflation (Substitution) > barcode
/d_curve/Phyllotaxis.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/Phyllotaxis.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/Phyllotaxis.png)
Ulam Spiral (Sacks) > default
/default/Phyllotaxis.png)
Ulam Spiral (Square) > default
Visibility Graph > default
Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Circle Map Quasiperiodic | 1.96 | cross-origin |
| Standard Map K=0.5 (Mixed) | 3.98 | cross-origin |
| Critical Circle Map (Silver Mean) | 4.14 | cross-origin |
Which Geometries Light Up
Cayley › Cayley:local_linearity | rank 1/306 | 1.0000 |
Fisher Information › Fisher Information:effective_dimension | rank 4/306 | 16.0000 |
Fisher Information › Fisher Information:log_det_fisher | rank 303/306 | 44.3614 |
Fisher Information › Fisher Information:trace_fisher | rank 303/306 | 256.0002 |
Hodge–Laplacian › Hodge–Laplacian:solenoidal_fraction | rank 305/306 | 0.2384 |
Hodge–Laplacian › Hodge–Laplacian:curl_div_ratio | rank 306/306 | -1.2723 |
Inflation (Substitution) › Inflation (Substitution):acf_geometric | rank 3/306 | 0.8888 |
Laplacian › Laplacian:gradient_curvature_anticorrelation | rank 3/306 | 1.0000 |
Laplacian › Laplacian:curvature_autocorrelation | rank 302/306 | -0.3090 |
Laplacian › Laplacian:laplacian_evolutionary_index | rank 303/306 | -0.3090 |
Laplacian › Laplacian:cross_scale_curvature_coherence | rank 305/306 | -0.5878 |
Modular Residue › Modular Residue:cycle_fraction | rank 2/306 | 1.0000 |
Modular Residue › Modular Residue:occupancy_entropy | rank 2/306 | 1.0000 |
Moiré › Moiré:moire_phi_response | rank 4/306 | 0.2960 |
Penrose (Quasicrystal) › Penrose (Quasicrystal):phi_tower | rank 3/306 | 0.8291 |
Septagonal (Danzer) › Septagonal (Danzer):cubic_coherence | rank 305/306 | -0.2676 |
Septagonal (Danzer) › Septagonal (Danzer):z_primary | rank 305/306 | -0.5634 |
Sol (Thurston) › Sol (Thurston):path_length | rank 302/306 | 9236.8820 |
Spectral Graph › Spectral Graph:weyl_exponent | rank 3/306 | 2.1615 |
Wasserstein › Wasserstein:entropy | rank 4/306 | 5.0000 |
Wasserstein › Wasserstein:concentration | rank 304/306 | 1.0031 |
Wasserstein › Wasserstein:dist_from_uniform | rank 304/306 | 0.0000 |
p-Variation › p-Variation:volatility_clustering | rank 306/306 | -0.6180 |
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