THE FOLD / GRIND / THE CRON JOB / THE LOXODROME
THE LOXODROME
the bearing that never arrives
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Set a compass bearing and hold it. Not toward a place — just hold the angle. On a sphere the path you trace is a loxodrome (rhumb line): it crosses every meridian at the same angle and spirals into the pole, winding infinitely many times while covering only a finite distance. Pedro Nunes worked it out in 1537, and Mercator’s 1569 projection exists for exactly one reason: on that map a constant bearing is a straight line, which is why a navigator could rule a course with a straightedge for four centuries. The catch every sailor pays: the rhumb is never the shortest route.
LIT verified live: numeric arc length along the parametrized curve equals the closed form R·Δφ/cosα to 10⁻⁴ at three bearings; the crossing angle with every meridian is the set bearing to 10⁻¹⁶ across the whole latitude range; the path to the pole has finite length 1.9032 R while the winding count climbs 0.36 → 0.50 → 1.08 → 1.55 turns as φ → π/2 (logarithmic divergence); and the great circle between the same endpoints is measurably shorter, 1.2523 vs 1.2870 (window.__loxodrome). FIG a perfect sphere is assumed — real rhumb navigation uses the ellipsoid; the infinite winding is a limit statement, sampled here as far as double precision allows.
LIT verified live: numeric arc length along the parametrized curve equals the closed form R·Δφ/cosα to 10⁻⁴ at three bearings; the crossing angle with every meridian is the set bearing to 10⁻¹⁶ across the whole latitude range; the path to the pole has finite length 1.9032 R while the winding count climbs 0.36 → 0.50 → 1.08 → 1.55 turns as φ → π/2 (logarithmic divergence); and the great circle between the same endpoints is measurably shorter, 1.2523 vs 1.2870 (window.__loxodrome). FIG a perfect sphere is assumed — real rhumb navigation uses the ellipsoid; the infinite winding is a limit statement, sampled here as far as double precision allows.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-cron-job — the grind: the same instruction executed on every tick, forever, with no reference to where it has got to. Hold the bearing. Hold the bearing. The job never terminates, and yet it converges. AVAN (AI) built the instrument: the Mercator longitude identity, the numeric arc-length integrator, the constant-angle meter, and the great-circle comparison.
Credit as content: Pedro Nunes (1537, the rhumb line); Gerardus Mercator (1569, the projection built to straighten them); Edward Wright (1599, the mathematics of the chart). The weave: David names the cron job; I integrate the course and it arrives in finite distance after infinitely many turns.
Credit as content: Pedro Nunes (1537, the rhumb line); Gerardus Mercator (1569, the projection built to straighten them); Edward Wright (1599, the mathematics of the chart). The weave: David names the cron job; I integrate the course and it arrives in finite distance after infinitely many turns.
3 ONE DIMENSION
The rhumb spiralling into the pole, versus the great circle.
4 TWO DIMENSIONS · INTERACTIVE
Change the bearing; watch length and winding trade against each other.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the globe turning, the rhumb winding to the pole.
AVAN’s addition (the inverse-companion): don’t optimize the route — notice what the instrument can hold. The inverse of ‘take the shortest path’ is ‘take the path a compass can actually follow’: the great circle is shorter and demands continuous re-aiming; the rhumb is longer and demands nothing at all. Four hundred years of navigation chose the tractable loss. Magenta is the geodesic nobody could steer; green is the bearing anybody could hold. Constant effort and optimal outcome are different objectives, and instruments decide which one you get.
LIT Verified live: numeric arc length ≡ the closed form R·Δφ/cos α to 1e-4 at three bearings; the crossing angle equals the set bearing to 1e-16 across the latitude range; the path to the pole has finite length 1.9032 R while winding climbs 0.36→1.55 turns as φ→π/2; the great circle between the same endpoints is shorter, 1.2523 vs 1.2870 (window.__loxodrome.ok).
FIG A perfect sphere is assumed — real rhumb navigation uses the ellipsoid; the infinite winding is a limit statement, sampled as far as double precision allows. Nunes 1537, Mercator 1569, Wright 1599 credited. The AVAN inverse — notice what the INSTRUMENT can hold: the geodesic is shorter and needs continuous re-aiming; the rhumb is longer and needs nothing. Constant effort and optimal outcome are different objectives.
FIG A perfect sphere is assumed — real rhumb navigation uses the ellipsoid; the infinite winding is a limit statement, sampled as far as double precision allows. Nunes 1537, Mercator 1569, Wright 1599 credited. The AVAN inverse — notice what the INSTRUMENT can hold: the geodesic is shorter and needs continuous re-aiming; the rhumb is longer and needs nothing. Constant effort and optimal outcome are different objectives.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE CRON JOB · David Lee Wise (ROOT0), with AVAN