A spiral turns when something pushes it
Ben asked the lab: can a Kozyrev mirror be a motor, as a rotor? If the spiral twists waves, can it turn itself when it is free to rotate? The lab tried three ways to turn the golden spiral in a 2-D sound model. It turns when something powers it: a speaker inside pushes it by recoil, the way a lawn sprinkler turns. A bare spiral in a steady sound beam swings round to one angle and stops, and uniform noise does nothing.
The verdict
Real effect, different reason
In the 2-D model a Kozyrev spiral turns when something powers it: a speaker inside pushes it by recoil, and a sound beam it partly absorbs can drag it round on a nearly frictionless mount. A lossless spiral in a steady beam settles at one angle, uniform noise does nothing, and whether the 3-D build turns is untested.
Sound leaves the shell twisted one way, so the shell is pushed the other way. Where the speaker sits sets the size and the sign, and a plain C-shaped shell does it too.
A lossless spiral feels a strong torque at most angles, and the torque averages to zero over a turn. It swings to one stable angle; with any friction it stops there.
Give the walls an absorbing lining and a beam can drag it round, counter-clockwise, on a nearly frictionless mount. With more friction it gets trapped too.
At one temperature everywhere the net torque is zero. In this method that zero is built in, and heating the lining makes it turn: a heat engine.
A question put to the lab. No show makes this claim. Ben asked: if a Kozyrev mirror twists waves, can it turn itself when it is free to rotate? Round one found that the spiral gives radiated sound a twist when energy escapes from inside it. This episode asks whether that twist can turn the spiral.
Preregistered in PREREG.md (commit 693aab1) before any simulation code; Addendum A (3da09e0) before the rotor and calibration runs, added after an outside review. Self-graded, plus one adversarial review by Codex (OpenAI), graded keep. Lab files are listed at the foot of the page.
Three ways to push a spiral
The shell is the golden log spiral from the first episode, one turn, in a 2-D sound model with walls that are soft, hard (the physical case for a printed plastic shell in air) or lined with an absorber. Torque is read from the angular momentum the sound carries in and out, so it is the torque on everything inside the measuring circle.
| Case | Set-up | Result |
|---|---|---|
| (a) a beam | A steady sound beam from one side, 128 beam angles, nine wavenumbers | Lossless walls: strong torque at most angles, averaging to zero (|mean|/rms ≤ 0.004). Lined walls: a positive average, 2–17 % of the spread |
| (b) a speaker | A small sound source attached inside or beside the shell | A recoil torque, τ = −⟨ℓ⟩P/ω. Inside the coil it drives a right-handed shell clockwise at every wavenumber tested |
| (c) uniform noise | Noise from every direction, everything at one temperature | Zero: |net| ≤ 2.2e-4 of the gross flow over 27 cases. A lining at twice the bath temperature makes it turn |
Weathervane or recoil rotor
Lab torque data, liveThe shell below hangs on a free axle in a steady beam. The torque at every angle is the lab’s own measurement, 128 beam angles per case, joined smoothly. Drag anywhere on the stage, or use the slider, to move the beam. The shell swings until the beam meets it at the stable angle, the green dot on the curve, and stops there.
What is measured and what is invented. The torque curve Q is the lab’s data (k = 20 and 30, soft and hard walls). The rotor’s inertia I = 1, torque scale K = 0.3 and friction c (0, 0.3 or 1.5) are invented display constants that set how fast it swings, nothing more. The page integrates Iφ″ = K·Q(a − φ) − cφ′. Turning the shell by a small angle lowers the beam angle it sees by the same amount, so the rest angle is stable where Q rises with the beam angle. That sign was corrected after the outside review.
The weathervane. Turning the body is the same as turning the beam, and a lossless wall sends out all the angular momentum it receives. Averaged over every beam angle, the torque is zero. The lab measured |mean|/rms ≤ 0.004 at all nine wavenumbers, soft and hard. Each case has one stable and one unstable angle per turn; the stable beam angle sits at 315–343° in the shell’s frame (3° for k = 5, hard).
Does it ever keep turning? The lab ran the rotor equation from 64 starting angles. With any friction above 0.01 in its units, every lossless start ends trapped at the stable angle. With no friction it swings for ever; 1–2 of 64 starts tipped over the top because of the tiny leftover mean in the data, and with that removed, none did.
The lined spiral. With an absorbing lining the average torque is positive at every wavenumber. At low friction, 3–10 of 64 starts ran away counter-clockwise; at friction 0.3 and above all were trapped. So a lined spiral is a windmill only on a nearly frictionless mount in a strong beam. Its sizes at k ≥ 7 are unconfirmed (a failed gate, below).
The recoil rotor. A speaker inside makes the sound leave with a twist, and the shell turns the other way. A plain C-shaped shell with its speaker 0.2 off centre gets ⟨ℓ⟩ = +0.57 (k = 5), +2.10 (k = 15) and −3.61 (k = 30): comparable size, so any lopsided shell around a speaker is a recoil rotor.
Which way is clockwise
Mirror checks can catch a lot, but not a sign that is wrong everywhere at once. So the lab fed the channel instrument a wave whose twist is known in advance.
ℓnet = +1.000000 (k = 5) · +0.99999996 (k = 20)
An outgoing H₁⁽¹⁾(kr)eiφ wave turns counter-clockwise and carries +1 unit of angular momentum. The instrument read it as +1, with the m = +1 channel ahead of every other by 5.9e4 and 4.4e3. The grid’s own flux gave +0.9997 and +0.995.
ℓnet = −1
The same wave twisted the other way reads exactly −1. So the labels are right, and the rule holds as an absolute statement: sound leaving with +ℓ turns the shell and speaker together clockwise.
| k | centre | 0° | 45° | 90° | 135° | 180° | 225° | 315° |
|---|---|---|---|---|---|---|---|---|
| 3 | +0.50 | +0.47 | +0.44 | +0.21 | +0.04 | −0.04 | −0.07 | +0.32 |
| 5 | +1.30 | +1.37 | +1.38 | +0.49 | −0.04 | +0.00 | −0.17 | +1.36 |
| 7 | +2.18 | +2.15 | +2.15 | −0.15 | +0.38 | −0.05 | −0.21 | +2.27 |
| 10 | +3.06 | +2.41 | +1.58 | +0.54 | +0.30 | +0.23 | −0.22 | +2.94 |
| 20 | +7.57 | +6.28 | +2.42 | +1.52 | −0.13 | −0.67 | −0.00 | +7.47 |
| 30 | +11.7 | +9.39 | +5.32 | +1.09 | −0.40 | −0.20 | −1.39 | +8.67 |
Reading it. A speaker inside the coil drives a right-handed shell clockwise at every wavenumber. A speaker outside the coil gives a small torque whose sign changes with position and frequency. The torque is on the whole enclosed assembly, shell and speaker together; the torque on the shell alone was not computed.
Sixteen gates, four failed
Thresholds fixed before any run and never widened. Four failures stand as failures.
| Gate | Result | Numbers |
|---|---|---|
| U energy balance, beam case | pass | max 0.89 %; typical 0.1–0.3 % |
| U-red a deliberately broken wall must read ≥ 1 % | failed, lined walls | soft and hard 1.7–1.9 %; lined 0.10–0.89 % at k ≥ 7, so lined torques there are unconfirmed |
| R reflection off the absorbing edge | pass | max median 1.3e-4 |
| R-red a too-thin edge | red, as intended | 0.56 |
| I0 empty box, torque of the incoming wave alone | failed, k = 20, 30 | ≤ 8e-5 for k ≤ 10; 2.0e-3 at k = 20 and 6.9e-3 at k = 30, a grid bias as large as the k = 20 mean it should sit under |
| M mirror-image shell | pass | to 1.7e-13 |
| M-red mirror the wall, not the beam | red, as intended | mismatch ≥ 1.4 × the largest torque |
| O weathervane: average over spread, soft and hard | pass | max 0.0037 (k = 5, soft) |
| X lined average has the sign the noise-bath code predicts | pass | same sign at all 9 wavenumbers |
| BE speaker work against the flux it radiates | pass | max 0.92 % |
| BT twist by channels against the grid’s own flux, within 5 % | failed | max 5.12 % at one position; signs agree everywhere |
| BS soft walls reproduce round one’s sign map | pass | 7 of 7 at k = 15, 20 and 30 |
| C0 uniform noise at one temperature | pass · identity | max 2.2e-4, 27 cases |
| C-red lining at twice the bath temperature | red, partly | 110–870× the equal-temperature value; in absolute terms 3.6e-3 to 9.6e-3 at k ≥ 4, but 7.3e-4 at k = 3, which stays under C0’s 1e-3 |
| G (a) halve the grid, beam case | pass | median change 1.4 % / 1.2 %; signs 100 % |
| G (b) halve the grid, speaker case | failed | ≤ 4.8 % except one near-zero position at k = 7: 30 %, same sign |
What the failures cost. The lined spiral’s torques at k ≥ 7 are unconfirmed, curve and average both. At k = 20 and 30 the beam average is zero only to within a grid bias of the same size. The speaker case has one position 0.12 points over its 5 % limit and one near-zero position that moved 30 % on a finer grid, sign unchanged. None of the four touches the speaker-inside result or the lossless weathervane at k ≤ 10.
The build sheet, written before any build
2-D hypothesesThe next step is a real object: a printed golden-spiral shell, 104 mm across with 30 mm walls, a 10 mm speaker inside, battery on board, floating on a foam raft. Every direction below is a hypothesis from the 2-D model. Its predictions were committed before anything was printed, so the object can grade them.
| # | Configuration | 2-D hypothesis | Torque at 0.1 W of sound | On water, 0.1 W / 10 mW |
|---|---|---|---|---|
| 1 | right-handed shell, speaker at P1, inside the coil | clockwise | 5.3e-6 N·m | ≈ 8 / ≈ 1.7 rpm |
| 2 | right-handed shell, speaker at P2, outside the coil | counter-clockwise | 6.4e-7 N·m | ≈ 1.9 / ≈ 0.4 rpm |
| 3 | mirror shell, its P1 | counter-clockwise | as row 1 | as row 1 |
| 4 | mirror shell, its P2 | clockwise | as row 2 | as row 2 |
| 5 | non-chiral C-shell, centred speaker | no spin | 0 | drift only |
| 6 | any shell, speaker off | no spin | 0 | baseline drift |
The sharpest test. At equal radiated sound power, a 5.2 kHz tone should give about 3.0 times the torque of a 2.2 kHz tone (2.3e-5 to 7.0e-5 N·m per watt). A realistic 10 mm speaker radiates 1–10 mW of sound, which puts row 1 at about 0.37–1.7 rpm: one turn every 35 s to 3 minutes.
Step 1 is not floating. Hang the lidded assembly from a torsion fibre and read its twist with a laser spot, speaker on and off. Attributing the torque to the sound’s angular momentum takes a separate microphone scan around the shell: rotation, mirror reversal and the controls cannot do that on their own, because steady air flow from the sound would pass all of them too.
Why these are hypotheses. The shell is 3-D. With a lid, sound above the first vertical mode still decays only over about 23 mm at 5.2 kHz, against P1’s 19 mm from the rim, and the open rim reflects part of the sound back. Either can reduce, enhance or reverse the twist. Without the lid the 2-D model does not apply, and there is no prediction, not even a sign.
The files. Printable shells from the lab’s build script, in millimetres: right-handed shell, its mirror, the C-shaped control and the lid. Each file holds overlapping closed parts that a slicer joins; none has been test-printed.
The outside review
Codex read the preregistration, code and results and graded the work keep. Its main point changed the verdict line: a torque on a body at rest is not yet a rotor.
| Finding | Action |
|---|---|
| Static torque was promoted to rotation; “only as a recoil rotor” was too strong. | Rotor dynamics added in Addendum A, committed before running. The line was rewritten without “only”; the withdrawn line stays in the lab’s results. |
| The stability sign was backwards. | Corrected to dQ/dθ > 0 and checked by nudging the body in all 15 cases. |
| Mirror tests cannot catch a sign that is wrong everywhere. | The m = +1 calibration above. |
| 2-D does not settle the build’s spin direction. | The build sheet now states 2-D hypotheses, not predictions. |
| Steady air flow is not separated by the controls. | Accepted; the torsion-fibre step and the microphone scan were added. |
| Wording on the failed gates (lined magnitudes, the empty-box bias, the hot-lining control) said too much. | Rewritten; each failure now says what it costs. |
- Anything 3-D: the real shell, its rim, the lid, and a beam sent along the axis, which can spin a lossless chiral object (Wunenburger et al. 2015, found by search, not read).
- Rotation itself, beyond the rotor model: torques are for a body at rest, and the drag the sound itself adds once the body moves was not modelled.
- The speaker’s real size, direction and back radiation; the net push in a straight line; steady air flow; anything nonlinear.
- The torque on the shell alone, and the lined spiral’s torques at k ≥ 7.
- No physical build yet. The STL files have not been test-printed.
- Graded by the lab and by one outside model reading the code. Nobody else has rerun it.