[alisonkisk]

The article never explains why ∇2 means "average of neighbors". It's the divergence of the gradient, which is (one kind of) n-dimensional 2nd derivative.

In a one-dimensional function, the second derivative is 0 when there is no curvature, aka a straight line (of any slope), and any point on a line is equal to the average of its neighborhood. A plane also has this property, but in 2+ dimensions you can also make other shapes (like saddles), that are made up of lines (like a plane) but the lines are twisted relative to each other in interesting ways (like "string art"). You can also visualize (aka impose a coordinate system for) these surfaces as having positive curvature (concave up) in one direction, and exactly opposite negative curvature (convex up, or concave down) in the orthogonal direction, summing to 0.