Ridge vs Lasso
Where the Hookean analogy breaks, on purpose — and why that break gives you exact zeros.
Shrink a single noisy estimate $y$ (mass $\phi$) toward zero and you get two textbook estimators depending on the shape of the penalty. Both are “a device pulling the bead toward the origin”; they differ in what kind of device.
Ridge is an ordinary Hookean spring of stiffness $\lambda$ anchored at $0$: force $-\lambda z$, energy $\tfrac12\lambda z^2$. Its pull weakens as the bead nears zero, so equilibrium $z^\ast_{\text{ridge}} = \phi y/(\phi+\lambda)$ approaches zero only in the limit $\lambda \to \infty$ — shrinkage, never exact sparsity.
Lasso replaces the spring with a constant-tension device — mechanically, a cable over a pulley with a hanging weight $\lambda$, always pulling toward $0$ with the same force regardless of distance. Its potential is the cone $\lambda|z|$, non-Hookean precisely at the kink the introduction's caveat warns about. Once the data-spring's pull weakens below $\lambda$, the constant-tension device wins outright and the bead sticks at exactly zero: $z^\ast_{\text{lasso}} = \mathrm{sign}(y)\max(|y| - \lambda/\phi,\, 0)$ — soft-thresholding.
Raise $\lambda$ past $\phi|y|$ and watch: the ridge bead keeps sliding, slowing down but always moving; the lasso bead stops dead at $0$ and stays there for every larger $\lambda$ — the shaded dead zone on its rail. This one demo's simulation is honestly not raw $F=ma$: the kink at $z=0$ makes naive force integration chatter, so the lasso bead is animated via proximal gradient flow (an exact implicit step through the kink each frame) rather than the site's usual damped spring integrator. The equilibrium it lands on is exactly the one above — only the transient's telling is different.
The dictionary calls Gaussian priors Hookean for a reason: a Laplace prior on $z$ is exactly what makes the penalty $\lambda|z|$, and its density has a corner at zero that a Gaussian's never does. Every other demo on this site works because the log-density is a parabola everywhere; this is the one demo about what happens the moment it isn't.