JetPacker is a continuously-updated, independently-verified
catalogue of geometric packing, covering, and tiling
problems in 2D and 3D. For each combination of shapes and
count N it publishes the best known arrangement, verified by
re-running the optimizer's kernel and reproducible from a single
integer seed. The flagship class asks for the tightest fit of
N copies of an inner shape inside a container shape; other
classes minimize area or volume, pack fixed-width strips, maximize
how many copies fit, and solve the discrete puzzle families (bin
packing, cutting stock, nesting, tilings, and exact-cover
assemblies). Inner and container shapes include the regular polygons
(triangle, square, pentagon, hexagon, … up to the 100-gon), the
right-isosceles triangle (“tan”), the 1×2 domino,
the circle, and the L-tromino, and in 3D the sphere and the five
Platonic solids (cube, tetrahedron, octahedron, dodecahedron,
icosahedron). Each result lists the optimal scale, the exact
closed-form value when the configuration tiles or has one, and a
downloadable JSON layout.
Created and maintained by Frederic Laing.
Problem classes
JetPacker spans fourteen classes of geometric packing, covering, and
tiling problems:
Smallest container
How small can a fixed-shape container be while still holding
N copies of another shape, without overlap? (2D and 3D)
Minimum area
How small an area can enclose N copies of a shape when the
rectangle's proportions are free? Reported as s = √area. (2D)
Minimum volume
How small a volume can enclose N copies of a solid when the
box's proportions are free? Reported as s = ∛volume. (3D)
Strip packing
How short a strip of fixed width can hold N copies of a
shape, when the length is free? (2D)
Maximize count
How many copies of a shape fit inside one fixed container of a given
size? Here the count N is the quantity being maximized. (2D)
Bin packing
Distribute a set of items among as few identical fixed-size bins as
possible. (2D and 3D)
Cutting stock
Cut a required multiset of item lengths from identical stock units
while minimizing the number of units or the trim waste. (2D)
Irregular nesting
Place arbitrary, often non-convex outlines as tightly as possible,
usually onto a fixed-width strip. (2D)
Dense packing
Pack identical disks or spheres as densely as possible across the
plane or space, maximizing the covered fraction. (2D)
Polyomino & polycube packing
Pack copies of one polyomino into a rectangle, or one polycube into
a box, filling it exactly with no gaps and no overlaps. (2D and 3D)
Plane tiling
Tile the whole plane with copies of a single shape so that the
copies cover everything with no gaps and no overlaps. (2D)
Assembly puzzles
Fill one fixed target region exactly with a prescribed set of
distinct pieces, each used once (exact cover); for example the
twelve pentominoes in a rectangle or the seven Soma pieces in a
3×3×3 cube. (2D and 3D)
Pallet loading
How many identical a×b boxes fit on an A×B rectangular
pallet, axis-aligned with 90-degree rotation allowed? (2D)
Sphere-cluster packing
Pack N rigid clusters of fused spheres (polyspheres) into a
3D container, minimizing the container scale. (3D)
Example result pages: 9 squares in a square
(/p/r4/r4/9), 6 triangles in a hexagon
(/p/r3/r6/6). A machine-readable sitemap of
every result is at
/api/sitemap.xml; AI agents, see
/llms.txt.