How small a container can hold n copies of a polyhedron?
Smallest containers found so far for n = 2–40 copies of a solid: each Platonic solid in itself and in its dual, every Archimedean and Catalan solid in itself, and regular prisms and antiprisms in themselves and in a cube. 2,145 packings in 55 problems, every one certified in exact arithmetic. Pick a family, a problem, then any n to inspect the packing in 3D.
s(n)
Smallest container found for each n (click a point to open that packing), the volume bound, and published values where they exist.
Increase from n − 1 to n
Bars: s(n)/s(n − 1) − 1 in percent (left axis). Line: s(n) − s(n − 1) (right axis). Both use the touching limits, without clearance. Shaded columns: no increase, the extra piece fits in the same container.
How the packings are found
Every piece starts as a soft ball and is sharpened into the solid under pressure, then the search hops between arrangements. The page walks through each step and links the code.
Why each number is proved
Every record carries an exact certificate: coordinates in number fields, rational rotations and a separating plane for every pair of pieces.
Beat a record
Paste a packing to check it in your browser, then submit it. A GitHub Action certifies it exactly and reports back.
Every construction, one page
All 2,145 record files with links to their coordinates, exact certificates and float checks on GitHub.