what is the Laplacian

Drag the probe over the field and a small ring of six neighbor samples appears at that spot. Three numbers show below: the value at the center, the average of the six neighbors, and ∇²f, proportional to their difference. On the peak at the upper right the neighbors' average is lower than the center, so ∇²f goes negative and the peak is blue. In the hollow at the lower left the neighbors' average is higher, so ∇²f goes positive and the hollow is gold. This is the place to check by hand that the Laplacian is one number for how far you sit from the neighbors' average. Slide the ring slowly between the peak and the hollow and the gap that was negative crosses to positive along a ridge, and on that ridge the center value matches the neighbors' average.

On the same field, see why the Laplacian is the divergence of the gradient. Drag the probe and six gradient arrows appear on the ring. Each arrow points up the steepest slope there. On a peak the arrows all gather toward the probe, because the uphill direction points at the peak inside. A flow that gathers inward has negative divergence, so ∇·∇f < 0. Over a hollow the arrows all reach outward and the divergence is positive, so ∇·∇f > 0. The screen shows ∇²f = ∇·(∇f) together with the live value. The arrows are the divergence applied once more to the gradient. Where the arrows gather to one place, that point is giving height away. Where they reach out in every direction, that point is taking height in from around it.

Now see the Laplacian of the whole field in color, not one point. The slider s changes the height of the upper-right peak from 0.5 to 3, and the background color is the map of ∇²f. Gold is a hollow (∇²f > 0, below the average) and blue is a peak (∇²f < 0, above the average), and the top of a peak and the bottom of a hollow are the strongest. Raise s and the upper-right peak rises, and the blue region there gets deeper and wider. Lower s and that peak fades, and the blue thins. The hollow at the lower left stays gold no matter what s does. One slider repaints the whole Laplacian map. A strong color means a large gap from the neighbors. A place close to gray already nearly matches its neighbors, so little remains to even out.

A field whose Laplacian is 0 everywhere is called harmonic. The buttons alternate two fields. One is h(x,y) = x·y, a saddle. At every point it is as concave across as it is convex up and down, so the two cancel and ∇²h = 0, and the ∇² sample readout at the center sits near 0. The other is b(x,y) = a Gaussian peak, a real bump, and ∇²b is a clear negative at the center. A harmonic field cannot hold a peak or a hollow inside. Every point equals the average of its neighbors, so the highest and lowest places are always on the boundary. That is the rule of steady heat, potential, and a membrane. So if you fix only the temperature on the rim, the inside follows. If you fix only the potential on the rim, the empty interior fills itself.

See why the Laplacian is the engine of diffusion. The heat equation is ∂f/∂t = ∇²f, which means that at every instant each point changes by its Laplacian. A hollow (∇²f > 0) rises and a peak (∇²f < 0) falls, so every place is pulled toward the neighbors' average. The grid starts bumpy, with two peaks on it, and each press of the step button moves every cell one step toward the neighbors' average. There is no automatic repeat. It moves only when you press. The more you press, the peaks sink, the valleys fill, and the color evens out. Reset returns to the first bumpy state. Counting the steps shows the Laplacian erasing differences, one stage at a time. Press for a long while and the first bumpiness disappears, and the whole grid remains as one calm color.