Collatz Cycle Word

What It Is

Lower-Christoffel / Sturmian word of slope log2(3): the balanced binary sequence s_i = floor((i+1)d) - floor(i*d) with d = 1/(1+log2(3)) ~ 0.3868 (the 1-density). Like the Fibonacci Word it has minimal complexity p(n) = n+1 (the fewest distinct length-n blocks an aperiodic sequence can have), but its irrational slope is log2(3) --- the growth constant of the Collatz / 3n+1 map --- rather than the golden ratio, making it the symbolic cycle/parity word of that dynamics. Stationary and non-monotone. Included as a contrast to the golden-slope Fibonacci Word: is the log2(3) slope geometrically special, or just another minimal-complexity Sturmian?

Composition

Binary sequence — two distinct symbols.

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

Heisenberg (Nil) (centered) > signed_log_z

Heisenberg (Nil) (centered) > xy_path

Hodge–Laplacian > default

Inflation (Substitution) > barcode

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

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

Ulam Spiral (Sacks) > default

Ulam Spiral (Square) > default

Visibility Graph > default

Wavelet Cascade > default

Atlas Position

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