The Pendulum That Rotates by Multiplying

Watching the bob for an hour teaches you something the textbook glosses over: the swing plane doesn’t lurch or accelerate. It advances by the same small angle, tick after tick, hour after hour. In Calgary the rate works out to 15°·sin(latitude) per hour — about 11.7°/hr at 51°N — and it never varies. That regularity is the whole trick, and it happens to be a computing trick too.

If you wanted to plot the swing direction each hour, the naive way is to recompute the azimuth with sin and cos every step. But if every step is the same rotation, you only need the trig once. Store the direction as a pair (x, y) — a unit complex number, a phasor — and multiply by a fixed rotor each tick. Complex multiplication is rotation. The demoscene crowd leaned on this hard in the ’80s, spinning starfields on machines where a sine call cost more than the whole frame budget.

const lat = 51.05 * Math.PI / 180;         // Calgary
const d = -15 * Math.sin(lat) * Math.PI / 180;  // radians per hour
const c = Math.cos(d), s = Math.sin(d);    // trig once, outside the loop

let [x, y] = [1, 0];                        // swing azimuth as a phasor
for (let h = 0; h <= 6; h++) {
  const deg = (Math.atan2(y, x) * 180 / Math.PI + 360) % 360;
  console.log(`hour ${h}: ${deg.toFixed(1)}°`);
  [x, y] = [x * c - y * s, x * s + y * c]; // one multiply = one rotation
}

hour 0: 0.0°
hour 1: 348.3°
hour 2: 336.7°
hour 3: 325.0°

Erlang says the same thing with tail recursion, the evolving phasor carried in the arguments the way the pendulum carries its phase:

run() ->
    Lat = 51.05 * math:pi() / 180,
    D = -15 * math:sin(Lat) * math:pi() / 180,
    step({1.0, 0.0}, {math:cos(D), math:sin(D)}, 0, 6).

step(_, _, H, Max) when H > Max -> ok;
step({X, Y}, {C, S}, H, Max) ->
    Deg = math:fmod(math:atan2(Y, X) * 180 / math:pi() + 360, 360),
    io:format("hour ~p: ~.1f~n", [H, Deg]),
    step({X*C - Y*S, X*S + Y*C}, {C, S}, H + 1, Max).

Both print identical numbers, because both are doing the identical thing the lead bob is doing four metres above the basement floor: applying one fixed rotor, over and over, and letting the angle accumulate. The pendulum never computes its heading from scratch. Neither should you.