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. 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 Persistent Homology:max_h1_lifetime, Nonstationarity:adf_pvalue. It sits beside fBm (Persistent) in the atlas (standard-bank rank 21) — a neighbor conventional features miss.
Multi-Scale Wasserstein:w_max_ratio | +9.4z | bank-miss 1.9σ |
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) | 4.06 | cross-origin |
| Riemann-Hardy-Littlewood | 4.19 | cross-origin |
| Geometric Brownian Motion | 4.28 | cross-origin |
Which Geometries Light Up
Catch24 › Catch24:CO_Embed2_Dist_tau_d_expfit_meandiff | rank 1/306 | 98.7170 |
Catch24 › Catch24:SP_Summaries_welch_rect_area_5_1 | rank 2/306 | 0.9999 |
Catch24 › Catch24:SC_FluctAnal_2_dfa_50_1_2_logi_prop_r1 | rank 3/306 | 0.8200 |
Catch24 › Catch24:SB_BinaryStats_mean_longstretch1 | rank 4/306 | 5461.9000 |
Catch24 › Catch24:DN_Mean | rank 5/306 | 0.7499 |
Catch24 › Catch24:FC_LocalSimple_mean3_stderr | rank 306/306 | 0.0047 |
Chladni › Chladni:plate_low_mode_fraction | rank 1/306 | 0.9957 |
Chladni › Chladni:captured_power_fraction | rank 305/306 | 0.0006 |
H² × ℝ (Thurston) › H² × ℝ (Thurston):boundary_dynamics | rank 2/306 | 0.1367 |
H² × ℝ (Thurston) › H² × ℝ (Thurston):depth_height_corr | rank 5/306 | 0.6158 |
Moiré › Moiré:moire_phi_response | rank 305/306 | -0.1184 |
Multi-Scale Wasserstein › Multi-Scale Wasserstein:w_max_ratio | rank 1/306 | 3293.5651 |
SL(2,ℝ) (Thurston) › SL(2,ℝ) (Thurston):trace_autocorrelation | rank 1/306 | 0.9998 |
SL(2,ℝ) (Thurston) › SL(2,ℝ) (Thurston):cocycle_growth_burstiness | rank 2/306 | 0.7111 |
Septagonal (Danzer) › Septagonal (Danzer):z_conjugate | rank 306/306 | -0.6836 |
Spectral Analysis › Spectral Analysis:spectral_bandwidth | rank 305/306 | 0.0004 |
S² × ℝ (Thurston) › S² × ℝ (Thurston):great_circle_dispersion | rank 304/306 | 0.0068 |
Visibility Graph › Visibility Graph:assortativity | rank 2/306 | 0.5839 |
Wavelet Cascade › Wavelet Cascade:cascade_coupling | rank 3/306 | 0.8737 |