Takagi Function
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 with graph Hausdorff dimension 2 (filling the plane up to log corrections).
Interpretation
Standard analysis sees: left-skewed; smooth / autocorrelated; long-range memory (persistent); strongly periodic; volatility-clustering (bursty); low-dimensional; nonstationary / drifting. The atlas finds no named structure, but the source is distinctively extreme on Multi-Scale Wasserstein:w_max_ratio (+8.3z) — beyond what the standard bank predicts for it. It sits beside fBm (Persistent) in the atlas (standard-bank rank 23) — a neighbor conventional features miss.
Multi-Scale Wasserstein:w_max_ratio | +8.3z | bank-miss 2.1σ |
Composition
Render Gallery
_(centered)/signed_log_z/Takagi_Function.png)
_(centered)/xy_path/Takagi_Function.png)
/barcode/Takagi_Function.png)
/d_curve/Takagi_Function.png)
/phi_spectrum/Takagi_Function.png)
/default/Takagi_Function.png)
/default/Takagi_Function.png)
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| fBm (Persistent) | 4.12 | cross-domain |
| Riemann-Hardy-Littlewood | 4.23 | |
| Magnetic Pendulum (3-Magnet) | 4.50 | cross-domain |
Which Geometries Light Up
Cantor Set › Cantor Set:bit_plane_autocorrelation | rank 5/298 | 0.9958 |
Catch24 › Catch24:CO_Embed2_Dist_tau_d_expfit_meandiff | rank 1/298 | 98.8975 |
Catch24 › Catch24:SP_Summaries_welch_rect_area_5_1 | rank 2/298 | 0.9999 |
Catch24 › Catch24:DN_Mean | rank 4/298 | 0.7499 |
Catch24 › Catch24:SC_FluctAnal_2_dfa_50_1_2_logi_prop_r1 | rank 4/298 | 0.8064 |
Catch24 › Catch24:SB_BinaryStats_mean_longstretch1 | rank 5/298 | 5361.2000 |
Catch24 › Catch24:FC_LocalSimple_mean3_stderr | rank 298/298 | 0.0047 |
Chladni › Chladni:plate_low_mode_fraction | rank 3/298 | 0.9950 |
Chladni › Chladni:captured_power_fraction | rank 297/298 | 0.0006 |
Fractal (Mandelbrot) › Fractal (Mandelbrot):potential_roughness | rank 298/298 | 0.0030 |
H² × ℝ (Thurston) › H² × ℝ (Thurston):boundary_dynamics | rank 2/298 | 0.1364 |
H² × ℝ (Thurston) › H² × ℝ (Thurston):depth_height_corr | rank 5/298 | 0.6158 |
H² × ℝ (Thurston) › H² × ℝ (Thurston):radial_temporal_memory | rank 5/298 | 4.9372 |
Inflation (Substitution) › Inflation (Substitution):return_concentration | rank 294/298 | 0.0851 |
Moiré › Moiré:moire_phi_response | rank 295/298 | -0.1108 |
Multi-Scale Wasserstein › Multi-Scale Wasserstein:w_max_ratio | rank 2/298 | 2908.9163 |
SL(2,ℝ) (Thurston) › SL(2,ℝ) (Thurston):trace_autocorrelation | rank 1/298 | 0.9998 |
SL(2,ℝ) (Thurston) › SL(2,ℝ) (Thurston):lyapunov_std | rank 2/298 | 0.7111 |
Septagonal (Danzer) › Septagonal (Danzer):z_conjugate | rank 297/298 | -0.6394 |
Spectral Analysis › Spectral Analysis:spectral_bandwidth | rank 297/298 | 0.0004 |
S² × ℝ (Thurston) › S² × ℝ (Thurston):great_circle_dispersion | rank 296/298 | 0.0068 |
Visibility Graph › Visibility Graph:assortativity | rank 4/298 | 0.5839 |
Wavelet Cascade › Wavelet Cascade:cascade_coupling | rank 3/298 | 0.8713 |
p-Variation › p-Variation:var_p2 | rank 298/298 | 0.0000 |