Takagi Function
special-functions · 37 views
special-functionsfractal/singular
What It Is
Takagi (blancmange) function T(x) = sum_{n>=0} s(2^n x)/2^n with s(x) the distance to the nearest integer. Continuous everywhere, differentiable nowhere; range [0, 2/3]. Sister to Weierstrass/Riemann-H-L but built from triangle waves on dyadic doublings instead of trigonometric series --- a fundamentally different self-similar mechanism. The Takagi graph has Hausdorff dimension 1 (the canonical nowhere-differentiable dimension-1 example; box-count grows like (1/δ)·log(1/δ), a log correction on dimension 1, not dimension 2).
Interpretation
Standard analysis sees: left-skewed; smooth / autocorrelated; long-range memory (persistent); strongly periodic; volatility-clustering (bursty); low-dimensional; nonstationary / drifting. The atlas additionally detects Multi-Scale Wasserstein:w_max_ratio, Nonstationarity:adf_pvalue, Persistent Homology:max_h1_lifetime, Multifractal Spectrum:hurst_estimate.
What standard analysis sees
tail heaviness0.68
asymmetry0.01
occupancy0.47
short-range corr1.00
long-range memory0.99
spectral colour0.22
periodicity0.90
complexity0.29
time-irreversibility0.63
volatility clustering0.99
multifractality0.52
dimensionality0.06
nonstationarity0.94
What the atlas adds
Multi-Scale Wasserstein:w_max_ratio+9.3z
scale-coherent dispersion mismatch (max/min Wasserstein across scales)
Nonstationarity:adf_pvalue+3.3z
unit-root nonstationarity (ADF cannot reject random-walk null)
Persistent Homology:max_h1_lifetime+3.2z
reconstructed phase space contains a persistent topological loop (S¹) — a smooth simple limit cycle
Multifractal Spectrum:hurst_estimate+2.6z
self-affinity H = τ(2)/2 (alternate to Hölder hurst)
Atlas-extreme metrics the standard bank can’t predict for this source
H² × ℝ (Thurston):boundary_dynamics | +5.2z | bank-miss 1.1σ |
Hodge–Laplacian:source_fraction | -2.4z | bank-miss 1.0σ |
Composition
dtypefloat64
range[0.0003315, 0.6667]
unique values14854 / 16384
mean ± std0.5 ± 0.167
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/Takagi_Function.png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/Takagi_Function.png)
Heisenberg (Nil) (centered) > xy_path
Hodge–Laplacian > default
/barcode/Takagi_Function.png)
Inflation (Substitution) > barcode
/d_curve/Takagi_Function.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/Takagi_Function.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/Takagi_Function.png)
Ulam Spiral (Sacks) > default
/default/Takagi_Function.png)
Ulam Spiral (Square) > default
Visibility Graph > default
Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| fBm (Persistent) | 4.05 | cross-origin |
| Riemann-Hardy-Littlewood | 4.19 | cross-origin |
| Geometric Brownian Motion | 4.27 | cross-origin |
Which Geometries Light Up
Catch24 › Catch24:CO_Embed2_Dist_tau_d_expfit_meandiff | rank 1/307 | 98.7170 |
Catch24 › Catch24:SP_Summaries_welch_rect_area_5_1 | rank 2/307 | 0.9999 |
Catch24 › Catch24:SC_FluctAnal_2_dfa_50_1_2_logi_prop_r1 | rank 3/307 | 0.8200 |
Catch24 › Catch24:SB_BinaryStats_mean_longstretch1 | rank 4/307 | 5461.9000 |
Catch24 › Catch24:DN_Mean | rank 5/307 | 0.7499 |
Catch24 › Catch24:FC_LocalSimple_mean3_stderr | rank 307/307 | 0.0047 |
Chladni › Chladni:plate_low_mode_fraction | rank 1/307 | 0.9957 |
Chladni › Chladni:captured_power_fraction | rank 306/307 | 0.0006 |
H² × ℝ (Thurston) › H² × ℝ (Thurston):boundary_dynamics | rank 2/307 | 0.1367 |
H² × ℝ (Thurston) › H² × ℝ (Thurston):depth_height_corr | rank 5/307 | 0.6158 |
Moiré › Moiré:moire_phi_response | rank 306/307 | -0.1184 |
Multi-Scale Wasserstein › Multi-Scale Wasserstein:w_max_ratio | rank 1/307 | 3293.5651 |
SL(2,ℝ) (Thurston) › SL(2,ℝ) (Thurston):trace_autocorrelation | rank 1/307 | 0.9998 |
SL(2,ℝ) (Thurston) › SL(2,ℝ) (Thurston):cocycle_growth_burstiness | rank 2/307 | 0.7111 |
Septagonal (Danzer) › Septagonal (Danzer):z_conjugate | rank 307/307 | -0.6836 |
Spectral Analysis › Spectral Analysis:spectral_bandwidth | rank 306/307 | 0.0004 |
S² × ℝ (Thurston) › S² × ℝ (Thurston):great_circle_dispersion | rank 305/307 | 0.0068 |
Visibility Graph › Visibility Graph:assortativity | rank 2/307 | 0.5839 |
Wavelet Cascade › Wavelet Cascade:cascade_coupling | rank 3/307 | 0.8737 |
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