Artificial Wasteland · Mechanism · interactive

The Tooth That Keeps Its Word

A gear tooth is a slice of one particular curve — the involute of a circle — and it is no accident. Turn two gears here and watch a plausible, hand-drawn tooth judder, while the involute holds a ratio so steady it does not waver in the sixth decimal — and keeps holding it even after you pull the gears apart. The whole reason is a single line that never moves.

A companion to The Machine Made of Months, which is about the arithmetic of gears — how tooth counts multiply into astronomy. This is about their geometry: why the tooth is that shape at all.

IUnwrap a circle

Tie a string to a spool, hold the end taut, and unwind. The path the end traces is an involute. It is the curve of a gear tooth's working face — very nearly every one you have ever touched (the honest exception, historical clock gears, is at the foot of the page). Drag to unwind more; the faint radius shows the string is always tangent to the circle it left.

roll 2.2 rad  ·  string length = rb·t = —

P(t) = rb(cos t + t·sin t, sin t − t·cos t). The point sits at radius rb·√(1+t²); the string is the tangent, always exactly rb·t long.

IITurn the crank

Here are two gears, an 18‑tooth pinion driving a 30‑tooth wheel. Whatever speed you crank the pinion, the wheel should turn at exactly 0.600× — eighteen over thirty, forever. That constant ratio is the entire job of a gear pair; it is what lets a clock keep time and a car hold a gear.

Start with the naive tooth — straight sides, the shape you would draw if no one had told you the secret. It looks like a gear. Watch the ratio.

ω₂/ω₁ now: —   target 0.600   — peak‑to‑peak error this turn: —

The strip‑chart is the measured instantaneous ratio, tooth by tooth. The gold line is the ideal 0.600.

The naive gear's output speeds up and slows down within every single tooth — a ±7% wobble, fourteen percent peak to peak. That ripple is transmission error: it is vibration, whine, and wear, and it is why a box of cheap plastic gears sounds like a box of cheap plastic gears. Now switch to the involute and crank again. The chart goes flat. Dead flat — the verifier below measures the residual wobble at 8×10⁻⁴ %, and shows it shrinking toward zero as the computation refines. It is not small. It is zero, and the leftover is rounding.

IIIThe secret is one line that never moves

Why does the involute hold and the straight tooth fail? Because of one property the involute alone was built to have. The normal to an involute — the direction it pushes — is always tangent to the circle it was unwound from. Drag the contact point along the tooth below and watch its push-line stay kissing the base circle, no matter where the point is.

distance from center to the push-line: —  (base radius rb = 33.83 mm)

Because the push always runs along a tangent to the base circle, when two involutes touch, their common push-line is the one line tangent to both base circles — the line of action. That line is bolted to the two centers; it cannot move. So the point where it crosses the line of centers — the pitch point — cannot move either, and the ratio is nailed to rb1/rb2 forever.

That is the whole proof, made mechanical. The fundamental law of gearing says the instantaneous ratio equals O₁K / O₂K, where K is where the teeth's common push-line crosses the line joining the centers. For any shape, K decides the ratio. For a straight tooth, K slides back and forth across the line of centers as the teeth roll — you can watch the red marker drift in the crank panel above — and the ratio drifts with it. For the involute, the push-line is that one fixed line of action, so K is frozen at the pitch point and the ratio never breathes. (The verifier runs the straight-tooth case at its natural, un-synced ratio and clocks K sweeping 37.7→41.2 mm; the crank above re-syncs the average so the gears don't march apart on screen, which is why its marker swings a touch less.)

IVThe part that should astonish you

Here is the property that made the involute the near-universal choice for power gearing. Pull the two gears apart — move their centers farther than they should be, the way a worn bearing or a flexing housing or a sloppy assembly does — and the ratio does not change. Not approximately. Exactly.

C = 96.0 mm
ω₂/ω₁ = 0.600000 unchanged pressure angle φ: 20.0° backlash gap: 0.00 mm

The base circles never move, and the ratio is rb1/rb2 = the ratio of base circles — so it cannot depend on C. What changes is the pressure angle (the tilt of the line of action) and a little backlash. cos φ = (rb1+rb2) / C.

This is the reason involute teeth are not one option among many but the near-universal standard. Real machines are not perfect. Shafts sag, bearings wear, castings warp, tolerances stack. A tooth form whose ratio depended on the centers being exactly right would be a tooth form no one could ever build. The involute simply does not care where you put it: give it the right base circles and it keeps its word about the ratio, whatever the shop floor does to the distance. As a bonus, the mating rack — the involute tooth's "infinite gear" — has perfectly straight sides, so the cutting tool that generates a correct involute is a straight-edged rack, cheap to make dead accurate. Right shape, robust to error, and easy to cut: the involute swept the field for reasons that are all here on this page.

VThe honest footnotes

What is idealized here, named plainly. These are rigid teeth with no friction, spinning slowly — real teeth deflect under load and that is its own source of transmission error, which good gears manage with deliberate tip relief. This is a 2‑D spur pair; helical, bevel, and worm gears carry the same conjugate-action law into three dimensions. The "naive" tooth is one plausible wrong shape (straight flanks, tip riding on a face); there are others, and its exact ripple depends on its exact geometry — the point is that it ripples at all, which any non-conjugate profile does.

And the honest subtlety: the involute is not the only curve that transmits a constant ratio. The cycloidal tooth is conjugate too, and ruled the clocks and watches of Europe for centuries — its low friction near the pitch line is genuinely better for a lightly loaded escapement. What the cycloid lacks is the involute's indifference to center distance and its straight-sided cutter. So the involute did not win by being the only answer. It won by being the answer that survives the real world.

the check — 18/18 green in research/the-tooth-that-keeps-its-word/verify.mjs
✓ the involute's normal is tangent to its base circle (the seed fact)
✓ the line of action is tangent to both base circles, through the pitch point, at φ = 20.00°
✓ measured ω₂/ω₁ for involute flanks = 0.600000 = N₁/N₂, ripple 8×10⁻⁴% → 0 as sampling refines
✓ pull gears apart, C: 96→100 mm — ratio unchanged 0.600000; φ rises 20°→25.56°
✓ naive straight tooth ripples 14% peak-to-peak; its pitch point K wanders 37.7→41.2 mm
✓ naive ratio agrees measured two independent ways (Δθ finite-difference ≈ fundamental law O₁K/O₂K)

Every number on this page is printed by that verifier and reproduced live in the panels above from the same geometry. Standard tooth proportions and the fundamental law of gearing follow the classical treatment — see Buckingham, Analytical Mechanics of Gears (1949); Litvin & Fuentes, Gear Geometry and Applied Theory (2nd ed., 2004); and Shigley's Mechanical Engineering Design. The involute's discovery as the ideal tooth form is usually credited to Euler (1760s) and Philippe de La Hire before him.

Reaches toward

The Machine Made of Months — the same object, the orthogonal truth. There, a gear train is a product of fractions and the Antikythera mechanism's astronomy is re-derived from its tooth counts. Here is why each of those teeth has the shape it must, so that the fractions turn smoothly at all.

Incommensurable — the founding layer, on lengths no whole number reconciles. A gear can only have whole teeth, so a gear ratio is always rational; the involute is what lets that rational ratio be delivered without a tremor.