The Laplacian Measures the Gap from the Average
Stand at a point and average the values of its immediate neighbors. If that average sits higher than the value where you stand, you are in a dip, hollowed out below your neighbors. If the average is lower, you are on a raised peak. This gap is exactly the Laplacian. It is the divergence of the gradient (∇·∇f), namely ∂²f/∂x² + ∂²f/∂y², and in the end it is one number measuring how far you stray from the average of your neighbors. Negative on a peak, positive in a dip. This single quantity drives how heat spreads, how a stretched membrane settles, even the rule by which an electric potential arranges itself. It is the operator that closes the vector-calculus bundle (gradient, divergence, curl) with one second derivative at a point.
Drag the probe over the field. Wherever you set it, a small ring of six neighbor sample dots wraps around it. Three numbers appear below: the center value, the average of the six neighbors, and the ∇²f that is proportional to the gap between them. Put it over the peak (upper right) and the neighbor average sits below the center, so ∇²f turns negative, a blue peak. Put it over the dip (lower left) and the neighbor average is higher, so ∇²f turns positive, a gold dip. Feel for yourself by hand that the Laplacian is, in the end, one number: how far you stray from the average of your neighbors.
Now, 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 around it. Each arrow points up the steepest local slope. Over a peak the arrows all converge toward the probe, because uphill points inward toward the peak. An inward-gathering flow has negative divergence, so ∇·∇f < 0. Over a dip the arrows all reach outward, so divergence is positive, ∇·∇f > 0. The screen shows "∇²f = ∇·(∇f)" alongside its live value. See by arrows that the Laplacian is divergence (lesson 3) acting once more on the gradient (lesson 2).
Now see the Laplacian not at one point but across the whole field, as color. The slider s changes the height of the upper-right peak from 0.5 to 3. The background color is the map of ∇²f itself. Gold is a dip (∇²f > 0, below the average), blue is a peak (∇²f < 0, above the average), and the summit of the peak and the floor of the dip glow strongest. Turn s up and the upper-right peak rises, so its blue region deepens and widens. Turn s down and that peak subsides, so the blue fades. The lower-left dip stays gold regardless of s. Watch one slider repaint the entire Laplacian map.
A field whose Laplacian is zero everywhere is called harmonic. Switch between two fields with the buttons. One is h(x,y) = x·y, a saddle. At every point it is concave one way exactly as much as it is convex the other, so the two cancel and ∇²h = 0. The ∇² readout for the sample at the center reads near zero. The other is b(x,y), a Gaussian bump, a genuine raised hill, so at the center ∇²b is clearly negative. Here is the key: a harmonic field can hold no interior peak or dip. Since every point equals the average of its neighbors, the highest or lowest place is always on the boundary alone. This is the deep rule behind steady-state heat, electric potential, and a stretched membrane.
See why the Laplacian is the engine of diffusion. The heat equation is ∂f/∂t = ∇²f. It says each point's value changes each instant by its Laplacian. A dip (∇²f > 0) rises, a peak (∇²f < 0) falls, so everywhere is pulled toward the neighbor average. A grid starts bumpy, with two peaks laid on it. Each press of the step button nudges every cell one stride toward its neighbor average. There is no automatic looping. It moves one step only when you press. With each press the peaks lower and the troughs fill and the color settles even. Reset returns to the original bumpy state. Counting the steps, watch the Laplacian erase the differences stage by stage.