Devil's Staircase
What It Is
Numerical derivative of the Cantor function C(x), sampled on a uniform grid in a random sub-window of (0,1). C(x) is continuous, monotone, and singular --- derivative zero almost everywhere, with all of its increase concentrated on the ternary Cantor set (a fractal of measure zero). Emitting forward differences C(x_{i+1}) - C(x_i) exposes that Cantor measure: exactly zero on the deleted middle-thirds, sharp positive bursts on the Cantor set. Different pathology from Minkowski's ?'(x) --- ternary-Cantor geometry vs continued-fraction arithmetic. Raw C(x) emission was retired 2026-05-20: the monotone values collapsed to a near-linear ramp under [0,1] normalization (pixel-identical to Primes / Partition / Minkowski-? in rank-based geometries) and a stray '% 1.0' wrap left a spurious 1->0 cliff at the tail. Mirrors the Minkowski ?(x) -> ?'(x) fix.
Interpretation
Standard analysis sees: no strongly notable standard properties. The atlas finds no named structure, but the source is distinctively extreme on Ulam Spiral (Square):arm_density_variance (+9.2z) — beyond what the standard bank predicts for it.
Ulam Spiral (Square):arm_density_variance | +9.2z | bank-miss 2.0σ |
Visibility Graph:avg_clustering_coeff | -6.5z | bank-miss 4.1σ |
Zipf–Mandelbrot (8-bit):hapax_ratio | +5.8z | bank-miss 2.9σ |
Hodge–Laplacian:source_fraction | -5.1z | bank-miss 2.1σ |
Möbius-S³:phi_return_cv | +4.8z | bank-miss 2.1σ |
Visibility Graph:degree_r_squared | -4.7z | bank-miss 3.1σ |
D4 Triality:edge_ortho_polarity | -4.7z | bank-miss 1.3σ |
Chladni:temporal_burstiness | +4.6z | bank-miss 1.8σ |
Composition
Render Gallery
_(centered)/signed_log_z/Devils_Staircase.png)
_(centered)/xy_path/Devils_Staircase.png)
/barcode/Devils_Staircase.png)
/d_curve/Devils_Staircase.png)
/phi_spectrum/Devils_Staircase.png)
/default/Devils_Staircase.png)
/default/Devils_Staircase.png)
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Rainfall (ORD Hourly) | 4.47 | cross-domain |
| Sensor Event Stream | 4.97 | |
| OpenBSD ELF x86-64 | 5.24 | cross-domain |
Which Geometries Light Up
2-adic › 2-adic:distance_entropy | rank 297/298 | 0.5058 |
2-adic › 2-adic:mean_distance | rank 297/298 | 0.0603 |
Catch24 › Catch24:SB_TransitionMatrix_3ac_sumdiagcov | rank 3/298 | 0.2749 |
Catch24 › Catch24:SB_MotifThree_quantile_hh | rank 296/298 | 0.2422 |
Chladni › Chladni:nodal_clustering | rank 2/298 | 8.2619 |
Chladni › Chladni:temporal_burstiness | rank 2/298 | 0.7470 |
Chladni › Chladni:domain_ks_exponential | rank 5/298 | 0.7158 |
Hodge–Laplacian › Hodge–Laplacian:source_fraction | rank 296/298 | 0.0988 |
Hyperbolic (Poincaré) › Hyperbolic (Poincaré):spatio_temporal_corr | rank 296/298 | -0.3266 |
Laplacian › Laplacian:gradient_curvature_anticorrelation | rank 297/298 | -0.1257 |
Level Statistics › Level Statistics:spacing_gue_distance | rank 2/298 | 0.9613 |
Level Statistics › Level Statistics:spacing_poisson_distance | rank 2/298 | 0.9613 |
Mostow Rigidity › Mostow Rigidity:mean_turn_angle | rank 4/298 | 0.9432 |
Möbius-S³ › Möbius-S³:phase_profile_deviation | rank 2/298 | 0.7406 |
Möbius-S³ › Möbius-S³:phi_return_cv | rank 2/298 | 7.3937 |
Nonstationarity › Nonstationarity:vol_of_vol | rank 3/298 | 2.9184 |
Ordinal Partition › Ordinal Partition:markov_mixing | rank 1/298 | 1.0000 |
SL(2,ℝ) (Thurston) › SL(2,ℝ) (Thurston):lyapunov_exponent | rank 5/298 | 1.8957 |
Ulam Spiral (Square) › Ulam Spiral (Square):arm_density_variance | rank 2/298 | 6.4114 |
Ulam Spiral (Square) › Ulam Spiral (Square):polynomial_concentration | rank 4/298 | 0.1409 |
Ulam Spiral (Square) › Ulam Spiral (Square):diagonal_alignment | rank 298/298 | -0.6451 |
Visibility Graph › Visibility Graph:avg_clustering_coeff | rank 298/298 | 0.0000 |
Wasserstein › Wasserstein:concentration | rank 5/298 | 30.7623 |
Wasserstein › Wasserstein:entropy | rank 294/298 | 0.3728 |
Zariski › Zariski:nonsep_fraction | rank 5/298 | 0.9065 |
Zipf–Mandelbrot (16-bit) › Zipf–Mandelbrot (16-bit):gini_coefficient | rank 1/298 | 0.9780 |
Zipf–Mandelbrot (8-bit) › Zipf–Mandelbrot (8-bit):gini_coefficient | rank 1/298 | 0.9774 |
Zipf–Mandelbrot (8-bit) › Zipf–Mandelbrot (8-bit):hapax_ratio | rank 1/298 | 0.5035 |