Oscilloscope Math — our demo

One example among many. Don't reuse its story, arc, shots, props or timings.

Demo: The Gibbs Phenomenon 《吉布斯現象》 (51.1 s) · oscilloscope-math.mp4 · source in demo/

How do you roll a square out of circles? A beam writes a perfect square wave; the question's letters turn into a circle; one spinning circle draws a sine, smaller and faster circles hang on its tip, and the wave turns square. Then STOP, and the camera dives into one corner: there is a spike, and adding a hundred, then four hundred circles only makes it thinner, never shorter. The overshoot is measured at 8.95 % of the jump; the upper limit of the sum becomes ∞, the camera pulls back over the whole wave with a light on every corner, and only then is the phenomenon named. English voice (Kokoro am_michael), Chinese and English subtitles. The numbers on screen are computed by the film; their mathematical sources are in demo/FACTS.md.

Story & structure

Logline: the circles can build a square wave, but never its corners.

Arc: hook (the ideal square and the question) → build (one circle, a sine; a circle three times as fast and a third the size; then odd ones only, faster and faster) → twist (STOP, the dive, the spike that will not shrink, measured) → ending (∞, the pull-back, the name, power-off). The promise is a square; the ending is the square you can never quite reach.

Native moves spent: the beam writing (the cold-open square, every title), text into geometry (question → circle, chapter title → axis), epicycle chain (shots 2–3), match cut (tip arc → wave crest), knob zoom (the dive), cursor measurement (the overshoot), STOP and power-off.

Shots

# Time Shot Camera Native move / note
1 0.0–4.6 A dot ignites; the beam writes an ideal square wave (1 Hz, ±1 V) follows the writing point at 1.7×, then pulls back to 0.9× (3.0–4.6) the beam writing
1b 4.5–8.3 Title "How do you roll a square out of circles?" traced and filled by the beam above the wave locked a 3.8 s hold
1c 8.3–9.6 Every letter morphs into the same circle (staggered rings); the square fades push to the circle (0.9× → 1.5×) text into geometry
2 9.6–13.4 The radius grows from the centre; RUN at 10.15; the circle spins, a dashed line carries the tip's height to the trace pushes onto the rotation, then rides the tip at 2.6× epicycle, afterglow, tip trail
2b 13.4 At the top of its turn the tip's arc cuts to the crest of the sine, same place, same curve hard match cut
2c 13.4–16.4 Circle, connector and sine together; readouts light up pull back to 0.92× readouts from the same values
3 14.6–19.2 Chapter title "Any wave is a sum of circles" / 任何波，都是圓的疊加 written top left; at 18.4 its letters fly down and straighten into the time axis; the graticule brightens locked title becomes the axis
3b 19.4–25.4 The formula lights glyph by glyph (Σ upper limit = number of circles); a circle ×3 · r = 1/3 grows on the tip, its label hanging beside it; the new trace loses its shoulders push and follow the small circle (1.9×), back to 0.92×
3c 25.35–30.45 ×5, ×7, then 21 circles faster and faster (N = 25); the wave turns square while the older trace on the right keeps the history pull out to 0.7× acceleration
4 30.45–32.8 STOP: the picture freezes, the music stops dive into the newest rising corner, 0.7× → 14×; the graticule refines, 250 ms/div → 50 ms/div knob zoom
4b 32.8–43.0 The spike. Amber cursors at 1.000 V and the peak, Δ = 0.179 V; circles added to 100 and 400 (35.05, 36.55): the spike thins and moves towards the edge, the reading holds; at 37.3 "8.95 % of the 2 V jump" locked so the thinning is seen cursor measurement; 5.7 s hold
5 43.0–45.6 Σ → ∞; the cursors go pull back 14× → 0.55× over the whole wave and the chain
5b 45.0–50.3 Amber lights on every corner's spike, a wave of light running along them; title "The Gibbs phenomenon" / 吉布斯現象 traced by the beam, "overshoot → 8.95 % for every N, even ∞" under it locked the name comes last
5c 46.6–50.3 End card (below) locked
5d 50.3–51.1 Power-off: the picture collapses to a bright line, then a dot CRT switch-off

Voice and subtitles

Ten lines (demo/voice/lines.json, Kokoro am_michael, speed 0.82, about 2.5 words a second), each checked with asr_check.py; one subtitle cue per line, bilingual, one line per language, exported to oscilloscope-math.srt from the same table. Numbers are spelled out in the voice text and written as digits in the subtitles. The STOP and the dive (30.45–31.0) and the power-off are wordless.

# Start English (spoken) 中文
1 0.9 Here is a square wave. Flat tops, sharp corners. 這是一個方波：平頂，直角。
2 5.6 So how do you roll it out of circles? 那麼，怎樣用圓把它滾出來？
3 9.95 Start with one spinning circle. 先讓一個圓轉起來。
4 12.65 Follow its tip. Its height draws a sine wave. 盯住它的尖端：它的高度畫出一條正弦波。
5 18.95 Hang a smaller one on its tip: three times as fast, a third the size. 在尖端再掛一個小圓：轉速是三倍，大小是三分之一。
6 25.1 Then a fifth, a seventh, odd ones only, until the wave is square. 再加五倍、七倍，只要奇數倍，直到波形變方。
7 31.0 But look closely at a corner. 可是，湊近看一個角。
8 33.5 There's a spike. Add more circles, and it only gets thinner. 這裡冒出一個尖。圓加得再多，它也只是變窄。
9 38.25 It never gets shorter. About 9%, every time. 它從來不會變矮：每次都高出約 9%。
10 42.8 Even infinitely many circles leave it there. It's called the Gibbs phenomenon. 就算有無窮多個圓，它也還在。這叫吉布斯現象。

Score structure

96 BPM, A minor turning to A major at the end; the lead's timbre follows the circles on screen.

Section Time Palette Notes
Cold open 0–9.5 square-timbre plucks in a question figure each bar, high bowed vibraphone (E5 A5 B5), relay and warm-up chirp, a buzz while the beam writes, a tick per corner and per glyph the square on screen is a square in the ear
One circle 9.5–16.6 the pluck turns to a pure sine; 8th-note pulse from RUN; A–F–C–G the match cut lands on a tick
Circles 16.6–30.45 each new circle: a detent click and a bell at its own harmonic (330, 550, 770 Hz); the pluck gains odd harmonics; 16ths in the rush; synth bass with 2nd/3rd harmonics; a 110 Hz signal tone that is the same partial sum the title-to-axis morph falls with a noise sweep and a thump
STOP 30.45 relay clack, the music cut dead (reverb tail only) the film's first silence
Corner 31–42.6 a low drone (55/110/165 Hz), a soft clock on the beat, one high glockenspiel on "spike", rising blip bursts as circles are added
Ending 43.05–50.3 the infinity bell chord, A-major arpeggio in square timbre, glockenspiel answering the lights on the spikes, held chord under the end card
Power-off 50.3 falling zap, thump, silence

Mix (mix.py): the voice is compressed and set by RMS about 10 dB over the music while it speaks; the music's 300 Hz–4 kHz band is cut under the voice and the foley steps back; −14 LUFS from mux.sh. All cue times come from timeline.json through film.py events, so the sound follows any retiming.

Palette & props

End card

On the last frame, bottom right on the glass, typed by the beam after a hairline: Oscilloscope Math 示波器數學 (STIX italic and Noto Serif SC, 28 px), then LemoLab × Claude Opus 5.5 (JetBrains Mono, 30 px). The film's title is the name written top left in the same frame, "The Gibbs phenomenon" / 吉布斯現象. Typed 46.6–47.4 s, held to 50.3 s, then the power-off. The "LemoLab × Claude Opus 5.5" sign-off belongs to this library's demo only; a user's film carries no LemoLab credit and no copy of this card.

Build notes

demo/
  build.sh         one command: Manim venv and fonts (if missing) → voice (if missing) → speech check → picture (WORKERS slices)
                   → phosphor pass → events, cues → .srt → mix → master → readcheck → poster, styleframe
  film.py          frame(t): world, camera, chain, trace, titles (WorldGlyphs), HUD, amber layer, subtitles, end card;
                   commands render / events / subs / texts
  post.py          phosphor pass: channel colours, bloom, shoulder, vignette, dither, CRT power-off
  mix.py           voice + score + signal tone + foley from out/events.json -> out/mix.wav
  timeline.json    every cue in film seconds (voice starts, births, STOP, dive, titles, end card, power-off)
  voice/           lines.json (Kokoro am_michael; one asr override: Whisper writes "9%"), subs.json (bilingual cues)
  tex/formulas.json  the formula as MathJax SVG templates (any upper limit, and ∞); tools/tex2svg.mjs regenerates it
  make_fonts.py    static instances of JetBrains Mono, STIX Two Text and the Noto Serif SC subset
  tools/gibbs.py   the overshoot for any N, checked against (2/π)·Si(π)
  tools/readcheck.html  bridge page so core/render/readcheck.mjs can read the texts film.py exports
  FACTS.md         the mathematics on screen and its sources