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, will fit together snugly in a packing carton. As will the hexagonal prisms of Droste. (2nd time I’m mentioning Droste—see my earlier post about the Droste effect). 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.

(more pictures & more to read after the jump)


For shipping and display purposes, how well your packaged products fit together is good thing to keep in mind. Some shapes—(like spheres)—will pack pretty close together, but still leave gaps in between. (Tetrahedrons and dodecahedrons, for example.)

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 [see: my 9/13/11 retraction about this one!]


5. Truncated octahedrons

So I ask you, “where are all the rhombic-dodecahedral and truncated-octahedral, close-packing packages?”

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