A rectangle, filled entirely by squares, no two the same size.

It sounds like something you could solve on a napkin. It is not. The squares must cover the rectangle with no gaps and no overlaps, none may repeat a size, and — the condition that defeats every naive attempt — no group of them may quietly form a smaller rectangle inside. Below nine squares it cannot be done at all.

The way out is not to draw. Every squared rectangle is secretly an electrical circuit: each horizontal join is a node holding a voltage, each square a wire carrying a current equal to its own side. Kirchhoff's law and Ohm's law are the tiling constraints. So Squaresmith never places a square. It takes a surface — a 3D model, or a photograph lifted into relief by depth estimation — treats that surface's own structure as the circuit, powers it on, and reads the squares off the currents.

The seed decides everything. Change the mesh and every tile moves.

Each piece carries its Bouwkamp code — the notation mathematicians use to write a squared rectangle down. The image is an artwork and a proof at once.

The correspondence is Brooks, Smith, Stone and Tutte's, published in 1940 under the invented name Blanche Descartes. aiyopasta's film “What's Hiding In The 8,000 Squared Mosaic?” (https://www.youtube.com/watch?v=0fH80JF2mDM) runs a Stanford bunny through exactly this route and comes out with eight thousand squares.

Works from this series are in production, and will appear here as they are finished.