TITLE Pendulum Wave PROMPT A physically exact simulation of the classic pendulum-wave demonstration: fifteen pendulums of carefully graded lengths hang in a row from a wooden frame; released together they drift out of phase into travelling waves, apparent chaos, then split into two and three interleaved groups, and after exactly sixty seconds swing back into perfect unison. Each bob chimes softly as it passes the centre. Concept: order, disorder and return, produced by nothing but arithmetic. Mood: calm, precise, a physics cabinet in a sunlit hall. Palette: plaster cream #F2ECDF background, dark walnut/iron charcoal #2A2622 for frame and type, brass #B8893A for the bobs (with highlight #E3C27A and shadow #7A5A22), and cool slate blue #6B7F8E for shadows, traces and secondary type. Light page. Physics: pendulum i (i = 0…14) completes (51 + i) oscillations in 60 seconds, so its period is 60 / (51 + i) s and its length is g·(T/2π)²: draw the lengths true to scale relative to one another. Angle θ_i(t) = θ0 · cos(2π t / T_i), exact and analytic, so the 60-second return is perfect and the page can run forever without drift; time derives from a clock that can be paused and scrubbed. Rendering (Canvas 2D, crisp at high DPI): two coordinated views of the same system. (1) Front view, large, drawn in a shallow oblique projection so the swing towards and away from the viewer reads as a sideways sweep with correct depth ordering: the frame's top beam with fifteen suspension points, a centimetre scale engraved on the right post, fine strings, brass spherical bobs with a specular highlight and soft contact-less shadows cast on the back wall (offset, blurred, slate), the bobs' lengths forming a graceful curve; motion blur trails behind fast bobs. (2) Top/side "wave" view beneath: looking down the row so the bobs' displacements read as a single undulating line of dots, with a thin spline through them and a fading history, which is where the wave patterns become obvious. A circular 60-second dial at the side shows the cycle with engraved marks at the famous moments: 0 unison, 15 s, 20 s (three groups), 30 s (two groups in antiphase), 40 s, 45 s, 60 s unison; a hand sweeps it, and the current pattern is named in small type ("unison", "travelling wave", "five groups", "four groups", "three groups", "alternating", "apparent chaos", "returning"), together with the live fraction of a swing by which each bob leads its neighbour (t/60); the engraved marks are also buttons that jump the clock to that moment. Sound (off until the viewer turns it on with a designed brass toggle): each pendulum is assigned a note of a pentatonic scale across two octaves (longest = lowest); each time a bob crosses the centre line a soft struck-bar tone plays (sine + a little inharmonic partial, 1.2 s exponential decay, slight stereo pan by position). The result: unison chords at the start, cascading arpeggios during the waves, rhythmic alternation at 30 s, a great chord on the return. Signature moment: the return at 60 seconds: all fifteen bobs sweep back into a single line, the chord lands, the dial flashes softly, and a thin brass ring pulses outward from the frame; then it begins again. The first three seconds: the bobs are held to one side by a drawn release bar (a hinged iron rod with brass knobs across the strings), which swings up and over the beam on load, and they begin in perfect unison. Interaction: play/pause; a scrubber along the bottom styled as a brass rule from 0 to 60 s that the viewer can drag to any moment (the system jumps there exactly); speed ×½, ×1, ×2, ×4; drag the release bar back down onto the strings (or press R, or the "release again" key) to gather the bobs and reset; while one bob is soloed only its chime sounds; a control for number of pendulums (9, 15, 21) which rebuilds the frame; click any bob to solo-highlight it and see its period and length in a small engraved tooltip ("No. 8 · 58 swings per minute · T 1.034 s · 26.6 cm"; pendulums are numbered from 1, longest first, and each suspension point on the beam is also a small focusable button). Keyboard: space, arrows scrub. Typography: restrained scientific-instrument style: title in small tracked serif capitals (Baskerville/Georgia) on a brass plaque shape; captions in italic serif; numerals tabular. A short paragraph in a narrow column explains the trick in three sentences. Responsive: on phones the front view fills the width (strings shorter, bobs smaller), wave view beneath, dial reduced to a slim arc. Reduced motion: runs at ×½ with no blur trails and no pulse; the scrubber remains. DESCRIPTION A museum-cabinet instrument on sunlit plaster: a dark walnut frame carries a row of brass bobs on fine strings, their lengths cut so that each makes one more swing per minute than its neighbour. A hinged release bar holds them to one side, lifts, and the row leaves in unison, then slides into a travelling wave, tangles, sorts itself into four, three and two interleaved groups, and at exactly sixty seconds sweeps back into one line while a brass ring spreads from the frame and the dial glows. Beneath the frame the same row is seen from above as a single undulating line with a fading wake; at the side a brass-bezelled sixty-second dial names the pattern of the moment and states the arithmetic behind it; along the bottom a brass rule lets the viewer slide to any instant. Everything is computed from the closed-form angle, so any moment can be visited exactly and the minute repeats forever without drift. VISUAL TECHNIQUES - Canvas 2D front view in oblique projection with per-frame depth sorting, so bobs pass in front of and behind one another and the frame posts - Pendulum lengths computed from g(T/2π)² and drawn true to scale, with an engraved centimetre scale on the post - Pre-rendered brass sphere sprite: offset radial gradient, window-pane specular, cool bounce light from the wall, dark rim - Soft slate wall shadows for bobs, strings, frame and bar, with parallax and blur that grow as a bob nears the viewer - Analytic motion blur: ghost exposures sampled at earlier instants of the exact solution rather than from a frame buffer - Top view with a Catmull-Rom spline through the bobs and a fading history of fifteen earlier curves, also sampled analytically - SVG dial generated in code: brass bezel gradient, sixty engraved ticks, diamond marks and tiny dot diagrams for the famous moments, sweeping hand; a slim arc version for phones - Brass rule scrubber with engraved ticks, numerals and a glass slide-rule rider; brass plaque title with chamfered corners and rivets - Plaster wall built from an SVG turbulence grain, a slow-drifting skewed window-light pattern and a slate vignette - Expanding thin brass rings and a dial flash choreographed to the sixty-second return - Web Audio struck-bar voices (sine plus inharmonic partial, exponential decay, stereo pan by position) on a two-octave pentatonic scale INTERACTION MODEL On load the release bar holds the bobs aside, then swings up over the beam and the minute begins. The round brass button or the space bar pauses and resumes. Dragging anywhere on the brass rule, or pressing the left and right arrows (Shift for five seconds, Home and End for the ends), moves the clock to an exact moment; the marks on the dial and on the phone arc jump to 0, 15, 20, 30, 40 and 45 seconds. The rate keys choose ×½, ×1, ×2 or ×4, and the pendulum keys rebuild the frame with 9, 15 or 21 pendulums and release them afresh. Dragging the release bar by its brass knob down onto the strings gathers the bobs back to the start and lets them go again; pressing R or the "release again" key does the same. Touching a bob, or the peg above it on the beam, singles it out: the others fade, a brass label gives its number, swings per minute, period and length, and only its chime sounds; touching it again, empty wall, or Escape clears the choice. The brass chimes switch turns the sound on: each bob strikes its note whenever it crosses the centre, and the whole chord rolls at the return. With reduced motion requested the apparatus runs at half speed without blur trails or rings.