Given three probability measures μ1,μ2,μ3 on the plane (here a square, a disk, and an L-tetromino, each as a uniform density on a 128×128 grid), their *Wasserstein barycenter* with weights w∈Δ2 is the measure ν∗(w)=argνmini=1∑3wiW22(ν,μi),
where W2 is the quadratic optimal-transport distance. Unlike pixelwise convex combination — which would simply superimpose the shapes — the barycenter respects mass transport: each particle of ν∗ is a weighted average of three coupled particles, one from each shape. The result is a genuine *morph*. We approximate it cheaply via displacement interpolation: sample ∼900 Lloyd-relaxed points per shape, couple them by a common radial canonical order, and average positions xk(w)=∑iwixk(i) before splatting back to a density field rendered in viridis. The weight (w1,w2,w3) traces the boundary of the 2-simplex over a 12-second loop; a small triangle widget shows its current location. Inspired by Gabriel Peyré's optimal-transport visualizations.