Continued Fractions
number-theory · 37 views
number-theoryarithmetic
What It Is
CF digits of random quadratic irrationals --- bounded partial quotients with hidden periodicity
Interpretation
Standard analysis sees: right-skewed; homoskedastic. The atlas finds no named structure, but the source is distinctively extreme on Nonstationarity:change_quantiles_high (+4.7z) — beyond what the standard bank predicts for it. It sits beside Markov Chain (10-state) in the atlas (standard-bank rank 32) — a neighbor conventional features miss.
What standard analysis sees
tail heaviness0.80
asymmetry0.89
occupancy0.19
short-range corr0.15
long-range memory0.38
spectral colour0.65
periodicity0.27
complexity0.61
time-irreversibility0.55
volatility clustering0.10
multifractality0.47
dimensionality0.47
nonstationarity0.47
What the atlas adds
Atlas-extreme metrics the standard bank can’t predict for this source
Nonstationarity:change_quantiles_high | +4.7z | bank-miss 1.6σ |
Attractor Reconstruction:lyap_sum | -3.6z | bank-miss 2.3σ |
Composition
dtypefloat64
range[0.6931, 5.293]
unique values149 / 16384
mean ± std1.51 ± 1.09
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/Continued_Fractions.png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/Continued_Fractions.png)
Heisenberg (Nil) (centered) > xy_path
Hodge–Laplacian > default
/barcode/Continued_Fractions.png)
Inflation (Substitution) > barcode
/d_curve/Continued_Fractions.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/Continued_Fractions.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/Continued_Fractions.png)
Ulam Spiral (Sacks) > default
/default/Continued_Fractions.png)
Ulam Spiral (Square) > default
Visibility Graph > default
Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Sandpile | 3.52 | cross-origin |
| Earthquake Depths | 3.91 | cross-origin |
| Markov Chain (10-state) | 3.98 | cross-origin |
Which Geometries Light Up
Mostow Rigidity › Mostow Rigidity:mean_turn_angle | rank 303/307 | 0.0639 |
Nonstationarity › Nonstationarity:change_quantiles_high | rank 3/307 | 0.2440 |
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