Drop a hundred discs into a bounded region and let them shove each other apart until nothing overlaps. What settles out is not decoration — it is a solved geometry. Today’s PAZ ACTION sketch (“2D Bricks Distribution”) lets you run exactly that: seed a cloud of circles, watch them relax into a jammed packing, remix the radii, download the result. The move looks playful. The mathematics underneath is four centuries deep.

Start with the number every packing chases. Fill an infinite plane with equal circles and the best you can do is the hexagonal lattice, density π/√12 ≈ 0.9069 — Gauss proved that lattice bound; Thue and Fejes Tóth closed the general case. Push to three dimensions and the analogue is the Kepler conjecture of 1611: no sphere packing beats face-centred cubic at π/√18 ≈ 0.7405. It stayed unproven until Thomas Hales delivered a computer-assisted proof, formally verified in 2014 by the Flyspeck project. The kissing number — how many spheres can touch one central sphere without overlap — is exactly 12 in 3D. These are not aesthetics. They are the ceiling on how much load-bearing material you can pour into a given void.

Now the lineage that makes packing generative rather than static. When unequal circles pack tightly, the tangent gaps obey the Descartes Circle Theorem: for four mutually tangent circles with curvatures (signed inverse radii) k₁…k₄, (k₁+k₂+k₃+k₄)² = 2(k₁²+k₂²+k₃²+k₄²). Iterate that into every gap and you get the Apollonian gasket — a self-similar aggregation that is pure recursion, no manual placement. And the dual of any circle packing is a Voronoi diagram: as PAZ’s own Voronoi — En Ingenieria concept panel puts it, the diagram “minimises the surface (or material) needed to enclose a given set of seeds,” which is why trabecular bone and soap foam settle into it. Pack circles, take the dual, and Lloyd’s algorithm relaxes the whole field toward a centroidal tessellation — the standard mesh-quality move in FEA.

←TODAY: A browser sketch relaxes a disc cloud into a π/√12-dense packing you can download tonight. →3012: Facades and infill are grown from a stated packing density, not drawn panel by panel. Fulcrum: The packing IS the derivation — the density is the spec, so a stranger can rebuild your wall from the number, not the file.

On a working desk this is not abstract. Variable-radius packing is how you resolve a facade with mixed opening sizes, how you seed acoustic-diffuser cells, how you zone additive-manufacturing infill: dense discs where the stress field is high, sparse where it is low. The trap is the beautiful guess — a generated packing with no recorded objective is a form you cannot defend in a structural review once the plugin goes dark. Make the tool tell you which quantity it minimised: overlap energy, gap area, or wall mass.

Atelier: Offices adopting generative packing report the same failure mode — the sketch ships, the density is never written down, and six months later nobody can regenerate it. Your Monday move: pick one facade or infill study and log its packing density (target and achieved) as a project parameter beside the geometry, so the wall is reproducible from the number. If the team wants the theory under the slider, McNeel Europe’s Grasshopper Level 3 online course (May 27–29, 2026, Rhino 8, EUR 395, instructor Ping-Hsiang Chen, a former Zaha Hadid senior architect with 10+ years in computational design) covers exactly the physics-based relaxation and form-finding this depends on.

Hack: Relax an overlapping disc cloud into a jammed packing by pushing each tangent pair apart by half its overlap — the same repulsion the sketch runs, in five lines. The intention: feel that packing is just overlap-energy descent, not magic.

for a, b in combinations(circles, 2):
    d = b.c - a.c
    overlap = a.r + b.r - d.length()
    push = d.normalized() * max(overlap, 0) * 0.5
    a.c -= push; b.c += push            # split the gap

Run that loop 30–50 times and the field converges to non-overlapping — weight push by a stress scalar and dense regions self-select. That is weighted relaxation in one line.

Open the ACTION sketch, raise the radius spread, and watch the Descartes gaps fill themselves. Then write down the density before you close the tab — the file will not survive you, but the number will.