Euler Totient Ratio
number-theory · 37 views
number-theoryarithmetic
What It Is
Euler totient ratio φ(n)/n --- the fraction of integers below n that are coprime to n, dense in (0,1) with a fractal structure governed by prime factorization
Interpretation
Standard analysis sees: anti-correlated (alternating); anti-persistent; blue spectrum (high-frequency power); strongly periodic; monofractal; stationary. The atlas finds no named structure, but the source is distinctively extreme on Heisenberg (Nil) (centered):area_length_ratio (+4.8z) — beyond what the standard bank predicts for it. It sits beside Logistic r=3.65 (Banded Chaos) in the atlas (standard-bank rank 21) — a neighbor conventional features miss.
What standard analysis sees
tail heaviness0.21
asymmetry0.71
occupancy0.41
short-range corr0.02
long-range memory0.05
spectral colour0.92
periodicity0.93
complexity0.38
time-irreversibility0.51
volatility clustering0.50
multifractality0.06
dimensionality0.37
nonstationarity0.10
What the atlas adds
Atlas-extreme metrics the standard bank can’t predict for this source
Heisenberg (Nil) (centered):area_length_ratio | +4.8z | bank-miss 1.9σ |
Heisenberg (Nil) (centered):xy_spread | +4.7z | bank-miss 2.2σ |
Gottwald-Melbourne:k_variance | +4.1z | bank-miss 1.7σ |
Bispectrum:off_diagonal_ratio | -3.3z | bank-miss 1.3σ |
Projective ℙ²:cross_ratio_std | +2.9z | bank-miss 1.2σ |
S² × ℝ (Thurston):great_circle_dispersion | +2.4z | bank-miss 1.4σ |
Möbius-S³:spinorial_asymmetry | +2.2z | bank-miss 3.3σ |
Composition
dtypefloat64
range[0.1918, 1]
unique values16071 / 16384
mean ± std0.608 ± 0.242
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/Euler_Totient_Ratio.png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/Euler_Totient_Ratio.png)
Heisenberg (Nil) (centered) > xy_path
Hodge–Laplacian > default
/barcode/Euler_Totient_Ratio.png)
Inflation (Substitution) > barcode
/d_curve/Euler_Totient_Ratio.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/Euler_Totient_Ratio.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/Euler_Totient_Ratio.png)
Ulam Spiral (Sacks) > default
/default/Euler_Totient_Ratio.png)
Ulam Spiral (Square) > default
Visibility Graph > default
Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Noisy Period-2 | 4.64 | cross-origin |
| Logistic r=3.65 (Banded Chaos) | 5.14 | cross-origin |
| Fibonacci Quasicrystal | 5.33 | cross-origin |
Which Geometries Light Up
Bispectrum › Bispectrum:off_diagonal_ratio | rank 304/306 | 0.1706 |
Chladni › Chladni:plate_low_mode_fraction | rank 303/306 | 0.0217 |
H4 600-Cell › H4 600-Cell:lattice_closure | rank 1/306 | 1.0000 |
SL(2,ℝ) (Thurston) › SL(2,ℝ) (Thurston):mean_trace | rank 305/306 | -1.9706 |
Symplectic › Symplectic:phase_reflection_symmetry | rank 304/306 | -0.0545 |
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