Volume Packing of Breakfast Cereal

i-d2c04b4d75071250965940da4774255c-25950_cheerios.jpgWe’re working on moving SteelyKid from formula to milk (which isn’t going all that well– dairy seems to make her gassy). This has led me to switch over to cereal in the mornings, since we’re buying milk anyway, which frees up the time otherwise spent waiting for the toaster.

Cereal-wise, I tend to alternate between Cheerios (which we also buy for SteelyKid) and Raisin Bran– my parents never bought sugary breakfast cereal, so I never developed a taste for any of those things. Being the ridiculous geek that I am, I’ve noticed something about the relative amounts of milk and cereal I use for the two different brands.

With Raisin Bran, I tend to fill the bowl with cereal, then add milk, and when I finish the cereal, there’s only a small amount of milk left. With Cheerios, on the other hand, after I finish all the cereal from a full bowl plus milk, there’s still rather a lot of milk left. I generally put in another half-bowl (maybe two-thirds) worth of cereal, and finish that, too.

Being a physicist (and, as noted earlier, a gigantic dork), it occurs to me that this can probably be explained by the different volume packing factors for the different shapes. Raisin Bran is mostly flat flakes, which Cheerios are little toroids. Those two shapes will fill space very differently.

The packing fraction of different shapes is a well-known problem in science– see, for example, this write-up in Science (PDF). If you take some container, and pack in as many spheres as you can, you find that those spheres only occupy about 64% of the volume. So, if you fill a 10-liter container with as many marbles as you can stuff in, you can still pour in a bit less than 4 liters of water.

The packing fraction for flat flakes is a less famous number, but this PNAS article mentions in passing that the packing fraction for cylinders is in the neighborhood of 0.9 (90% of the available space). A really physicist-like way of looking at the flat flakes in Raisin Bran would be as really short, wide cylinders, so you could take that as an upper limit on the packing fraction.

Cheerios, on the other hand, are shaped kind of like M&M’s which, as noted in the PDF link above, have a packing fraction around 0.7. They’re toroids, though, not ellipsoids, and as a result have some missing volume. The hole in the center of the Cheerio is about the same diameter as the width of the Cheerio, and if you estimate the effect on the volume, you find that this takes out about 1/8th of the volume of the full shape (estimating the volume as a cylinder with a hole in, because I can do that without Googling anything). 7/8ths of 0.7 is a bit more than 0.6.

And, hey, look at that. The packing fraction for Raisin Bran is half again that of Cheerios. Now, this is almost certainly dramatically overestimating the real packing fractions for the two different types of cereal– Raisin Bran is not optimally packed into a bowl, and of course the raisins throw things off even more– but the general conclusion is probably sounds: Raisin Bran fills more space than Cheerios. Which would mean that there’s significantly more milk in the bowl with the Cheerios, consistent with my observation that there’s enough left for most of a second bowl.

So there’s my Randy Waterhouse moment for the month: breakfast considered as a problem in random packing of small solid shapes. While I do own both a computer and a katana, this is as close as I ever hope to get to being a Neal Stephenson character.