Four beads, four heights, one arrival time

At its defaults it draws four beads, four heights, one arrival time. One arch of a cycloid of radius 1, drawn with four beads on it at heights 0.061, 0.235, 0.592, 2.000 — a range of 33.0 to one. Each bead's time to slide, from rest and without friction, to the bottom of the arch is computed as a quadrature of ds/v along the curve as drawn, and the four answers are 1.003205 s, 1.003205 s, 1.003205 s, 1.003205 s: identical to 6.9e-13 of themselves. The closed form for this curve is π√(a/g) = 1.003205 s, which the quadrature reproduces without being told it. A bead let go at the cusp travels 5.7 times as far as the lowest one and arrives with it, because the extra distance is exactly paid for by the extra speed the extra height buys.

cycloid-path is one function in lib/figures/mechanics.js — motion, force, energy and rotation. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

One arch of a cycloid of radius 1, drawn with four beads on it at heights 0.061, 0.235, 0.592, 2.000 — a range of 33.0 to one. Each bead's time to slide, from rest and without friction, to the bottom of the arch is computed as a quadrature of ds/v along the curve as drawn, and the four answers are 1.003205 s, 1.003205 s, 1.003205 s, 1.003205 s: identical to 6.9e-13 of themselves. The closed form for this curve is π√(a/g) = 1.003205 s, which the quadrature reproduces without being told it. A bead let go at the cusp travels 5.7 times as far as the lowest one and arrives with it, because the extra distance is exactly paid for by the extra speed the extra height buys.

One arch of a cycloid of radius 1, drawn with four beads on it at heights 0.061, 0.235, 0.592, 2.000 — a range of 33.0 to one. Each bead's time to slide, from rest and without friction, to the bottom of the arch is computed as a quadrature of ds/v along the curve as drawn, and the four answers are 1.003205 s, 1.003205 s, 1.003205 s, 1.003205 s: identical to 6.9e-13 of themselves. The closed form for this curve is π√(a/g) = 1.003205 s, which the quadrature reproduces without being told it. A bead let go at the cusp travels 5.7 times as far as the lowest one and arrives with it, because the extra distance is exactly paid for by the extra speed the extra height buys.

Five beads on a circular arc of radius 4, released at 9°, 27°, 54°, 90°, 126° from the bottom, with the time each takes to reach the bottom divided by the small-amplitude quarter period. The ratios are 1.00154, 1.01406, 1.05852, 1.18034, 1.42829: the bead released at 126° takes 42.6 per cent longer than the one released at 9°. This is the same computation as the cycloid's, on the same axes, with the same quadrature — the only thing changed is the curve. A circular arc is what a string and a bob give for free, and it is not the curve for which the times agree.

Time to slide to the bottom, against how far up the curve the bead starts, for a cycloid and for a circular arc of radius 4, each divided by its own small-amplitude value so the two can share an axis. The cycloid is a horizontal line: over the whole range, out to release at the cusp itself, its time moves by 8.5e-13 of itself, which is the noise of the quadrature and not a physical variation. The circle rises by 18.0 per cent by the time the bead starts at the quarter point, and the rise is slow at first — it is second order in the amplitude, which is why a pendulum keeps passable time in spite of not being isochronous at all. Both curves are measured by the same quadrature over the drawn geometry; the circle's answer is checked against the elliptic integral and agrees to 2.1e-10.

A pendulum of length 4.00 hung from a cusp between two cycloidal cheeks. As it swings, the upper part of the string lies along a cheek and only the remainder is straight, so the bob's distance from the point of support shortens as the swing widens. The path the bob traces is drawn here point by point from that construction, and it is a cycloid — the same size as the cheeks — to 5.5e-15. That is the fact the whole device rests on: the evolute of a cycloid is another cycloid. Huygens found it while trying to make a clock that would keep time at sea, where a ship's motion changes a pendulum's amplitude constantly and a circular pendulum's period with it. The bob is drawn at one position, with the string's wrapped part along the cheek and its free part running straight from the point of contact, which is 3.410 long where the full string is 4.00.

Time to slide to the bottom, against how far up the curve the bead starts, for a cycloid and for a circular arc of radius 2.4, each divided by its own small-amplitude value so the two can share an axis. The cycloid is a horizontal line: over the whole range, out to release at the cusp itself, its time moves by 8.5e-13 of itself, which is the noise of the quadrature and not a physical variation. The circle rises by 18.0 per cent by the time the bead starts at the quarter point, and the rise is slow at first — it is second order in the amplitude, which is why a pendulum keeps passable time in spite of not being isochronous at all. Both curves are measured by the same quadrature over the drawn geometry; the circle's answer is checked against the elliptic integral and agrees to 2.1e-10.

physicscheck asserts something about cycloid-path that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.