I was struck the other day by the following: The cooking instructions on my Bob’s tri-colored quinoa package said to combine 2 cups of water with 1 cup of dried quinoa, which would ultimately create 4 cups of cooked quinoa. See image above.

My first reaction was to believe that some error had been made. However, I then realized that the explanation was packing: When one packs spheres or other awkward solid geometric shapes into a container, they cannot fill the space completely. Little pockets of air sit between the spheres. A quick google search for the packing fraction of spheres gives a value of \(0.75\) for a crystalline structure and about \(0.64\) for random packings — apparently a universal law.

We can get a similar number out from my quinoa instructions: Suppose that before the quinoa is cooked, the water fills its volume completely. However, after cooking, the water is absorbed into the quinoa and forced to share its packing fraction. The quinoa stays at the same packing fraction before and after cooking, so the water must be responsible for the volume growth. This implies it went from 2 cups to 3, or

\begin{eqnarray} \tag{1} \label{1} 2 = \rho \times 3, \end{eqnarray}

where \(\rho\) is the packing fraction of the quinoa “spheres”. We conclude that the packing fraction is \(\rho = 2/3\), very close to the googled value of \(\rho = 0.64\).

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Jonathan Landy Avatar Jonathan Landy Jonathan grew up in the midwest and then went to school at Caltech and UCLA. Following this, he did two postdocs, one at UCSB and one at UC Berkeley.  His academic research focused primarily on applications of statistical mechanics, but his professional passion has always been in the mastering, development, and practical application of slick math methods/tools. He worked as a data-scientist at Square for four years and is now working on a quantitative investing startup.

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Cooking math