Weierstrass
special-functions · 37 views
special-functionsfractal/singular
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 finds no named structure, but the source is distinctively extreme on Bispectrum:off_diagonal_ratio (+4.5z) — beyond what the standard bank predicts for it. It sits beside fBm (Antipersistent) in the atlas (standard-bank rank 55) — a neighbor conventional features miss.
What standard analysis sees
tail heaviness0.40
asymmetry0.26
occupancy0.80
short-range corr0.77
long-range memory0.85
spectral colour0.12
periodicity0.89
complexity0.49
time-irreversibility0.54
volatility clustering0.76
multifractality0.38
dimensionality0.43
nonstationarity0.71
What the atlas adds
Atlas-extreme metrics the standard bank can’t predict for this source
Bispectrum:off_diagonal_ratio | +4.5z | bank-miss 2.1σ |
AutoRegressive:ar_coef_2 | -4.1z | bank-miss 1.8σ |
Spirograph:petal_symmetry | +3.1z | bank-miss 1.4σ |
Bispectrum:coupling_frequency_centroid | -2.8z | bank-miss 1.9σ |
AutoRegressive:ar_coef_1 | +2.5z | bank-miss 3.7σ |
Symplectic:phase_volume_explored | +2.4z | bank-miss 1.4σ |
Laplacian:laplacian_evolutionary_index | -2.2z | bank-miss 1.6σ |
Septagonal (Danzer):z_conjugate | +2.2z | bank-miss 2.0σ |
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 > free_plate_nodal
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 | |
|---|---|---|
| Langevin Double-Well | 3.71 | cross-origin |
| Stochastic Resonance | 3.71 | cross-origin |
| fBm (Antipersistent) | 3.87 | cross-origin |
Which Geometries Light Up
AutoRegressive › AutoRegressive:ar_coef_2 | rank 304/307 | -2.6622 |
Bispectrum › Bispectrum:bicoherence_gini | rank 1/307 | 0.9881 |
Bispectrum › Bispectrum:off_diagonal_ratio | rank 2/307 | 0.9771 |
Bispectrum › Bispectrum:coupling_frequency_centroid | rank 304/307 | 0.2858 |
Catch24 › Catch24:SC_FluctAnal_2_rsrangefit_50_1_logi_prop_r1 | rank 1/307 | 0.8420 |
Laplacian › Laplacian:curvature_autocorrelation | rank 305/307 | -0.3218 |
Symplectic › Symplectic:phase_volume_explored | rank 1/307 | 0.8749 |
← Takagi Functionin geophysical
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