Sturmian Fractals

The Collatz dropping sign rule (Parts 4–7 of the Dropping Zeta Spectrum working notes) is a Sturmian cutting sequence of slope $\log_2 3$ — the same complexity class as the Fibonacci word and the symbolic side of irrational rotation. When you feed a Sturmian word into a turtle-graphics program, the curve you get out is a direct geometric fingerprint of the sequence's structure.

This page lets you play with the recipe interactively.

Sequence log₂3 Sturmian (Collatz dropping sign)Fibonacci word (slope 1/φ)Stopping-Class parity P_o mod 2Custom rational slope p/q

Turn angle: 120.0°

Length: 2,896

Recipe Wikipedia (turn on 0, alt L/R)Dragon (left on 0, right on 1) Mode 2D3D (drag to rotate) Color Time directionHeading axisDistance from originSolid

Auto-spin Compare A vs B

log₂3 Sturmian

Length: 2,896 symbols

+1 density (A): 0.5849

Bounds:

What you're seeing

The turtle starts at the purple dot (0, 0) facing right. For each symbol of the chosen sequence it draws a unit segment forward, then maybe turns by the chosen angle. The orange dot is where the curve ends. Color along the path encodes time direction (purple → cyan → yellow).

Two turtle recipes are available:

• Wikipedia recipe (default) — the canonical Fibonacci-word-fractal rule: turn 90° (or whatever angle you've chosen) only on symbol 0, alternating left/right by the symbol's index parity. This is the rule that produces the famous self-similar Fibonacci curve.
• Dragon recipe — turn at every step, left on 0 and right on 1. For Sturmian sequences this typically produces bounded diamond-tiling shapes; for high-entropy sequences it produces 2D-Brownian-motion-like blobs.

What the four sequences are

Sequence	What each symbol means	Where it comes from
log₂3 Sturmian	Gap symbol of the $o$-th Collatz dropping class. By the closed form on the L-probe page (argument outlined there; the class counts checked for every $o \le 400$) it is the sign of the χ_6 sum over that class: the number of its residue classes whose destinations are 2 modulo 3, minus the number whose destinations are 1 modulo 3. At $o = 3$ that difference is 0	Closed form in Parts 4–7 of the working notes (see the L-probe page)
Fibonacci word	Letter $i$ of the standard Fibonacci word $0100101001001\ldots$	Reference Sturmian — slope $1/\varphi$
Stopping-Class parity	$P_o \bmod 2$ where $P_o$ = parity-class count of class $R_{k_o}$ (OEIS A100982)	An observation from Part 8: over the first 20,000 terms it looks close to coin flips (share of 1s 0.476); nothing is proved
Custom rational $p/q$	Cutting sequence of slope $p/q$	Periodic with period $q$ (so not Sturmian); for a deep convergent of $\log_2 3$ it matches the $\log_2 3$ word for a long first stretch (52 symbols for $19/12$, 358 for $84/53$)

Note that none of these are individual Collatz orbits — they are meta-sequences indexed by class number, one symbol per equivalence class. The "Collatz orbit of $n=27$" panel from the static gallery image lives elsewhere.

Things worth poking at

• Angle 120° on the Sturmian sequences gives beautiful triangular tilings.
• Angle 90° on log₂3 collapses to the regular rectangular tiling — that's the original "boring" panel.
• Stopping-Class parity at any angle, dragon recipe: looks like Brownian motion / DLA. That visual is the Part 8 observation: over the range computed, the sequence looks close to coin flips.
• Custom rational at small q (e.g. $3/2$, $8/5$): periodic-ish shapes. At deep convergents like $19/12$ or $84/53$, the rational sequence is visually identical to log₂3 for short lengths.
• 3D mode: drag to rotate, scroll to zoom. The 3D Wikipedia recipe alternates yaw / pitch on 0-symbols by index mod 4 — turning the planar fractal into a Hilbert-3D-curve-like structure.

Reference: the static gallery

Companion static images with the same recipe, generated by scripts/collatz_fibonacci_fractal.py, scripts/collatz_fractal_variants.py:

• data/collatz_fibonacci_fractal.png — original 2×2 (Fibonacci ref, log₂3 sign, Stopping-Class parity, orbit of $n=27$)
• data/collatz_fractal_angles.png — 2×4 angle scan at 60° / 72° / 90° / 120°
• data/collatz_fractal_dragon.png — dragon recipe on all four sequences