Weierstrass
exotic · 36 views
exotic
What It Is
Weierstrass function --- continuous everywhere but differentiable nowhere, the first known mathematical 'monster'. Self-similar roughness at all scales
Interpretation
Standard analysis sees: red spectrum (low-frequency / 1-over-f power); strongly periodic. The atlas additionally detects multi-frequency phase coupling. It sits beside fBm (Antipersistent) in the atlas (standard-bank rank 64) — a neighbor conventional features miss.
What standard analysis sees
tail heaviness0.39
asymmetry0.26
occupancy0.80
short-range corr0.77
long-range memory0.85
spectral colour0.12
periodicity0.89
complexity0.49
time-irreversibility0.55
volatility clustering0.76
multifractality0.38
dimensionality0.43
nonstationarity0.71
What the atlas adds
multi-frequency phase coupling+4.4z
quadratic phase coupling between distinct frequencies — multiple incommensurate tones interacting (quasiperiodic / multi-tone)
Atlas-extreme metrics the standard bank can’t predict for this source
AutoRegressive:ar_coef_9 | +8.3z | bank-miss 2.2σ |
AutoRegressive:ar_coef_8 | -6.7z | bank-miss 1.7σ |
AutoRegressive:ar_coef_2 | -3.9z | bank-miss 2.3σ |
AutoRegressive:ar_coef_6 | +3.2z | bank-miss 1.3σ |
Spirograph:petal_symmetry | +3.2z | bank-miss 2.2σ |
AutoRegressive:ar_coef_1 | +2.6z | bank-miss 4.2σ |
AutoRegressive:ar_coef_10 | -2.5z | bank-miss 1.9σ |
AutoRegressive:ar_coef_5 | -2.4z | bank-miss 1.7σ |
Composition
dtypefloat64
range[-1.935, 1.926]
unique values16384 / 16384
mean ± std0.0243 ± 0.814
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/Weierstrass.png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/Weierstrass.png)
Heisenberg (Nil) (centered) > xy_path
Hodge–Laplacian > default
/barcode/Weierstrass.png)
Inflation (Substitution) > barcode
/d_curve/Weierstrass.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/Weierstrass.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/Weierstrass.png)
Ulam Spiral (Sacks) > default
/default/Weierstrass.png)
Ulam Spiral (Square) > default
Visibility Graph > default
Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Stochastic Resonance | 3.76 | |
| Langevin Double-Well | 3.82 | cross-domain |
| fBm (Antipersistent) | 3.96 | cross-domain |
Which Geometries Light Up
AutoRegressive › AutoRegressive:ar_coef_1 | rank 3/298 | 2.7065 |
Bispectrum › Bispectrum:bicoherence_concentration | rank 1/298 | 0.9880 |
Bispectrum › Bispectrum:off_diagonal_ratio | rank 2/298 | 0.9721 |
Bispectrum › Bispectrum:coupling_frequency_centroid | rank 295/298 | 0.2865 |
Catch24 › Catch24:SC_FluctAnal_2_rsrangefit_50_1_logi_prop_r1 | rank 3/298 | 0.8440 |
Laplacian › Laplacian:curvature_autocorrelation | rank 296/298 | -0.3218 |
Symplectic › Symplectic:phase_volume_explored | rank 1/298 | 0.8727 |
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