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.

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There might be a bit more milk in a bowl of cheerios than in a bowl of raisin bran, but I think a more important factor is how efficiently the cereal transports milk into your mouth. This is more a function of the relative surface area/mass ratios. The milk that actually gets carried by the cereal is mostly a film covering the cereal surface. So, the high-surface-area raisin bran flakes carry a lot more milk per gram of cereal than the relatively low-surface-area cheerio toroids.

Plus, cheerios float, so the ones that get scooped up in the spoon tend to be only wet on the *bottom*.

As Tim indicated in his last sentence, I think you also need to consider the relative absorbency rates of the cereals. In my experience, raisin bran gets soggy much more rapidly than Cheerios.

Then again, it might be enlightening to experiment with equal weights of cereal and equal volumes of milk.

Here's a business opportunity for a cereal maker: Why not make cereal in various geometric shapes to assist kids in learning geometry?

Darn. Now I'm hungry for cubes, octahedrons, dodecahedrons, and icosahedrons! (Have to watch out for those tetrahedrons, though; they're kind of hard on the mouth.)


@Dave: lucky charms!!!!

I also second (or third I guess) the suggestion that absorbency rates for the different types of cereal are different. Thankfully that's easily testable (assuming you have a balance of some sort available).

While I would love to discuss the physics of packing in more detail because you have finally gotten to something I know a bit about, I think that is only a minor factor in the milk difference you notice. If you assume that you pack the bowl to the same line with either cereal, then add milk to the same or lower line, then your packing fraction analysis is relevant to the amount of milk currently in the bowl. However, it's not clear to me why one would assume that, as you eat the cereal, you take out the same volume of cereal but also the same volume of milk with each bite.

I suppose this depends entirely on how you pack your spoon. Let's first assume the milk fills in the intersticies between the cereal and nothing more (the cereal does not float and the milk is filled to the same volume as the bowl). In the deep spoon limit, without an extra pile of drained cereal above the fill line of the spoon, you take out milk and cereal at the same relative rate and there is no milk left over no matter what the packing fraction is. (I think that's closer to how I eat my cereal.) In the flat spoon limit, you take out drained, wet cereal and the key difference between cereals is how much milk they absorb.

Your analysis works under the assumption that you have a shallow spoon and take a constant volume of relatively dry cereal in a mound on top, but also a constant volume of milk in the bottom of the spoon. However, there is also cereal in the bottom of the spoon, and with the difference in packing fraction, you should take more milk along with the less densely packed cereal. (This leaves extra milk in the bowl unless you didn't fill up your bowl to the top of the cereal to begin with, but since we know your raisin bran comes with the proper milk:cereal ratio, we can assume that if the shallow spoon is the proper assumption, you do not fill the milk to the top of the cereal and instead fill it to the same proportion that you get in each spoonful. In this case again packing fraction doesn't matter.) If you are trying to say that you deliberately pack your spoon such that you get the desired amount of milk, which is constant regardless of cereal packing fraction, then that makes sense, but is not how I eat my cereal.

In reality, the shallow spoon with differentially absorbing cereal mounded on top is perhaps the most accurate, except for the floating thing (as also mentioned in the first comment). I think the fact that cheerios float, so you keep scooping them off the top with more milk underneath, is why you have extra milk in that case.

As Tim indicated in his last sentence, I think you also need to consider the relative absorbency rates of the cereals. In my experience, raisin bran gets soggy much more rapidly than Cheerios.

That's a factor as well, though I'm not convinced that the bran flakes actually absorb that much more milk than the Cheerios do. They get soggy, true, but they're considerably thinner, so it's easier to wet them through than it is to wet Cheerios.

A more direct measure would, of course, be to fill the bowl with cereal to the same line, and measure the amount of milk that goes into each. But that would've required me either to eat two breakfasts' worth of cereal this morning, or to be organized enough to measure milk amounts on two consecutive days.

Try goat milk. Works like a charm for kittens.

A glass of Carnation Instant Breakfast is more efficient than eating whole grain industrial waste. Supermarket price is excessive and the strawberry flavor is inexcusable. You can buy single flavor cases at discount from


We buy a year's supply at a whack, 15 cases. Including shipping, $4.92/box of 10 packets.

Try this: Make oatmeal with water only - eat it with a little bit of cinnamon and sweetener (I use sucralose, but pick your own poison ;-) ) Packing fraction is 1.0!

BTW, I started eating this to keep my weight down (no milk - fewer calories) and bring down my cholesterol (it works). Anyway, cheerios is basically just toroidized oats.

P.S. Get the *thick flakes* and stay await from that deflavorized instant crap.

Don't forget to account for surface tension effects, which will affect the amount of milk adhering to the cereal -- for instance, affecting how much milk is found inside the hole in a Cheerio, as well as affecting how milk adheres between multiple bran flakes.

Of course, the surface tension will vary depending on the type of milk you use, starting with milkfat percentage but also including variation between species. See for example this paper, which is admittedly 60+ years old.

I've never been one for dry cereal myself, but my parents were always big Cheerios eaters. Their dogs always viewed the milk left over when the Cheerios were eaten as theirs by rights. I wouldn't be surprised if Emmy had an opinion on the matter, too.

By marygriff (not verified) on 25 Sep 2009 #permalink

In my family we always layered our cereal. Bottom to top, it usually went: granola / All-Bran or Raisin Bran (depending if anyone had bought raisin bran) / Rice Krispies / cornflakes or Cheerios (the top two layers varied depending on what was in the house).

In adulthood I do a stripped-down version of this: I've recently rediscovered my grandmother's recipe for granola, so I now eat that with Cheerios. This messes still further with the milk packing ratio, but it is quite tasty and makes me feel all virtuous.

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

Wouldn't it depend on the radius of the spheres. Seems to me it would be an asymptotic with the limit approaching 100% as the radius approached zero. But I digress, we have the intertubes now I'll get back to you.

By Doug Little (not verified) on 25 Sep 2009 #permalink

With my bowls of Cheerios it always turns out that I eat the last Cheerio with the last drop of milk. That's because I have learned from experience how much milk to put on the cereal, plus I make minor adjustments in the cereal to milk ratio in the spoonfuls if I see things aren't going to work out evenly. This isn't rocket science. :D

Wha? That breakfast bowl has a handle. Is it still a breakfast bowl? How is this possible?

I grabbed that image from a Google search. I'm not quite dorky enough to take and post pictures of my actual breakfast.

Another factor in packing fraction is whether the shapes are completely uniform in size as well as shape, or only uniform in shape and not in size. Shapes that come in a modest range of sizes pack a little better than uniformly-sized shapes. Shapes that come in a broad distribution of sizes can pack dramatically better than same-size shapes.

Imagine packing spheres: small enough spheres can pack in the interstices of the bigger ones that are touching each other, and smaller spheres yet can fit in the remaining spaces. With the right mix, you can get a packing fraction arbitrarily close to 1.0.

By Del Cotter (not verified) on 25 Sep 2009 #permalink

Doug, no, if you ignore the surface of the bowl for now, and just concentrate on the packing fraction in the middle of the mass, away from the outsides, then sphere size has no effect. Because although each of the littler spheres has less space around it, there are now many more spheres. The two effects cancel out and all that counts is how much space each sphere is occupying or blocking another sphere from occupying, compared to that sphere.

By Del Cotter (not verified) on 25 Sep 2009 #permalink

Without googling, you can estimate the volume of a torus as that of a cylinder of diameter equal to the difference between radius of the outside and radius of the hole, and of length equal to the mean of the circumferences of the outside and the hole. The cylinder is bent around and its two ends joined to make the torus.

You might think some complicated calculus would be needed to explain how the compression on the inside matches the expansion on the outside to make the cylinder keep its volume when you bend it into a circle, but a much less complicated thought experiment gets the idea across intuitively: cut the torus in half and rejoin the halves into an S shape. Clearly no volume change in that transformation.

Now cut the two semitori each in half and rejoin to make a wiggly shape that's beginning the look like a wavy cylinder. Do this an infinite number of times and the original cylinder is reconstituted! with no squeezing or stretching involved, only slicing and rotating. This also works with a rectangular box to make a cylinder-with-a-hole, and with any prism with a 180° rotational symmetry to make the equivalent toroid shape.

By Del Cotter (not verified) on 25 Sep 2009 #permalink


Yeah on thinking about it that makes sense. The scale doesn't matter. From what I could dig up the limit is 63.4%. It's kinda counter intuitive though if you don't think about it too hard.

By Doug Little (not verified) on 25 Sep 2009 #permalink

i'm loving this discussion, but i'd like to add that your daughter is getting gassy because the milk is pasteurized and so lacks the active enzymes needed for proper digestion. raw milk is much easier on the system if you can get it.

Raw milk is hard to find and milk fever is nasty, nasty stuff. The best source of "active ingredients" is yogurt. A week or so of having yogurt twice a day is more than enough to allow most lactose intolerant people to drink milk without the associated gas. If you do get yogurt instead of milk for a bit I recommend that you invest in a good granola mix too. Cheerios don't go that well with yogurt.

Then there is Grape Nuts, the ultimate in dense cereal. Much smaller package, smaller bowl full, soaks up milk, and fills you up. I think I'll go fix a bowl.

Gas suggests that SteelyKid might be lactose intolerant. She's probably a bit young for Lactaid pills, and they don't work for everyone.

I agree generally with Cheem #22 about yogurt self-inoculation, but effects vary. I didn't know I was genetically lactose intolerant (both parents) till a 23andMe spit test because I can have a scoop of ice cream or a 16oz cafe-made hot chocolate without trouble. (I do like yogurt, and no doubt it's helped.) Meanwhile, my spouse needs Lactaid even for the "good" yogurt from the health food store, despite concerted attempts.

Real granola goes pretty well with yogurt....

By thistleingrey (not verified) on 26 Sep 2009 #permalink

A scientist should not forget the outliers. Puffed rice, the most buoyant and flavorless of breakfast cereals absorbs virtually no milk, and therefore the bowl of milk can be refrigerated for the next day's breakfast. Grape-nuts cannot be overwhelmed by milk. The dense little nuggets generally require a further splash of moo-juice before the bowl can be finished. Unfortunately for any experimentation puffed rice is only sold in pillow sized bags and you will end up with a package of the nasty stuff nobody will eat taking up room in your pantry.

This is a worthwhile article and blog discussion about a common but yet intriguing phenomenon. To let you real scientists know that it's not only an exercise in your realm of pure research, the issues of volume packing have been well-known to us chemical engineers since the dawn of the Industrial Age.

In one aspect of our profession, known as Unit Operations, we use huge vessels called packed column chemical reactors to control the chemistry of processes. For example, scrubbing toxic effluents from smokestacks and extracting coffee from beans employ this technique. In these vessels, we packed catalytic beads or coffee beans and pump all manners of fluids through them. Our concerns match a lot our yours plus others: packing fraction, fluid absorption and adsorption, flow rates as well as bead buoyancy and geometry. In striving for maximum economic efficiency of the packed column chemical reactors, the calculations do tend to become like rocket science.

So, from the breakfast tables of dorks springs the mighty cauldrons of industry.

FYI: Generally at this age it's not lactose intolerance, it's intolerance of a protein in cow's milk. (SteelyKid did fine on human milk, after all.)

As an engineer who mixes his cheerios with his raisin bran (note - no caps as I buy generic) - the key is the flotation of the cheerios. For mostly cheerios - add milk only until you see the cereal 'lift' as it starts to float. If mostly raisin bran, it takes proportionally more milk as it doesn't float as easily.

fran e is right. And I think you are overthinking this.

Cheerios float.

Raisin Bran doesn't, so much.

More milk underneath cheerios.

By Luna_the_cat (not verified) on 28 Sep 2009 #permalink

You can answer your critics with a simple experiment: try various kinds of Cheerios cereals. They all have the same shape, but different densities and milk absorbancies. If it is simply a matter of packing, your results will be Cheerios-invariant.

Thanks for the distinction, Kate. FWIW, however, lactose-intolerant individuals do fine with human milk when they're still in the young age range during which it's usually consumed. The designations might be somewhat orthogonal.

By thistleingrey (not verified) on 29 Sep 2009 #permalink