Close Packing

Packaging-wise, we live in a rectangular world: big box stores with products stacked on square pallets... retail space, measured in square and cubic units ... rectangular boxes on shelves, stacked efficiently in rows with no wasted space. 

But rectangular boxes are not the only shapes that can stack efficiently. There are lots of other interesting shapes that can do this.

Toblerone’s triangular prisms (above) will fit together snugly in a packing carton. As will the hexagonal prisms of Droste (below).
Add a rectangular toothpaste-tube-box to these two examples and you have the three regular prisms that fit together like three dimensional tiles. But this is just the tip of the polyhedral iceberg. The real story of close packing starts with spheres.

Maximum interior volume + minimum surface area = the sphere. If that were our only calculation then all packaging should be spherical. Put your “sphere packs” together, however, and right away there are inefficient gaps! In the produce department: rectangular arrangement = big gaps; triangular/hexagonal arrangement = small gaps.

For shipping and display purposes, how well your packaged products fit together is good thing to keep in mind. Some shapes — like spheres and tetrahedrons (the familiar triangular tetrapak below) — will pack pretty close together, but still leave gaps in between.
What are the space-filling shapes that leave no gaps in between? Of the polyhedral shapes with regular faces there are only five that can do this:

1. Cubes (obviously)
2. Triangular prisms (like the Toblerone boxes above)
3. Hexagonal prisms (like the Droste boxes above)
4. Rhombic dodecahedrons

5. Truncated octahedrons
The truncated octahedron is the basis of an innovative packaging solution that protects and maintains the purity of its content  –  Luso natural mineral water  –  while seeking to optimize and ease storage, transportation and display, as well as handling and consumption.  What's next?