Heartbeat
y = 3 · exp(−40(mod(x − t, 4) − 2)²) − 0.6 · exp(−6(mod(x − t, 4) − 2.6)²)
Open in the app The dials and keys named below are the app's.
What it draws
Everything hangs on mod(x − t, 4), a ramp that climbs from 0 to 4 and starts again. Because it is fed x − t, the pattern repeats every 4 units along the line and travels to the right at one unit per second. One pulse passes any fixed point every 4 seconds at Speed 1.
The two terms are Gaussian bumps placed on that ramp. 3·exp(−40·(u − 2)²) is the spike, centred at u = 2, with a width at half height of 0.263. −0.6·exp(−6·(u − 2.6)²) is the trough that follows it, 0.6 further along. The trough is two and a half times as wide, at 0.680, and a fifth as deep.
How the two bumps meet
The trough is broad enough to reach back under the spike. At the spike's centre it subtracts 0.069, so the tallest point of the curve is 2.931 rather than 3, and it sits a little to the left of u = 2. The lowest point is −0.600 at u = 2.6. Between beats the line is flat.
The ramp's jump from 4 back to 0 would show as a break. At both ends of it, though, the spike is smaller than 10^(−60) and the trough smaller than 0.000005: a discontinuous formula draws as a continuous one.
Why it reads as a pulse
A narrow spike with a shallow depression trailing it is the signature the eye files as a heartbeat. The resemblance to a cardiac trace is deliberate, and the formula is a drawing rather than a model of anything a heart does. It is a useful shape all the same: a periodic train of well separated Gaussians is what a sampling pulse or a radar train looks like.
Try
- Front in the deck: the trace side on, running right, which is the way a monitor draws it.
- Double the rate with Speed at 2: a beat every 2 seconds, with the pulses still 4 apart along the line.
- Blunt the spike by changing −40 to −10, which widens it at half height from 0.263 to 0.527.
- Turn the trough into a second bump by changing −0.6 to 0.6: a tall thin peak followed by a low round one.
- Raise Ribbon depth to 40: each spike trails away through the ribbon as a slanted ridge, and the slant is its speed.