Least squares as springs
Regression is a rigid rod that stopped arguing with the data.
Each data point is tied to a rigid rod by a spring of natural length zero and unit stiffness, running in a frictionless vertical guide so it can only stretch up and down. The stored energy is then exactly $\tfrac12\sum_i r_i^2$ — half the residual sum of squares. Release the rod and it relaxes until the spring forces and torques balance, which is exactly the normal equations: the resting position is the ordinary least squares line (dashed, computed in closed form). Drag the points and the rod chases the new fit; pluck it to see it oscillate back. The idea follows Joshua Loftus's “Least Squares as Springs”; for drawing springs properly, see Hadley Wickham's spring geom case study in the ggplot2 book.
In the dictionary: stiffness is precision, so weighted least squares is nothing but unequal springs, and adding a ridge penalty is one more spring tying the rod to the prior. Generalise from a rod to a network of beads and you are back at the fusion demo and the Kalman filter.