# Now Let Us Consider . . . a Herd of Spherical Cows?

Fall

2021

Physics Puzzler

# Now Let Us Consider . . . a Herd of Spherical Cows?

Brad R. Conrad, Director of SPS and Sigma Pi Sigma

During a planning session for our upcoming Centennial Physics Congress in Washington, DC (October 6–8, 2022), an immensely important puzzle came up:

**Theoretically, how many spherical cows could fit into Washington, DC?**

* A ridiculous question?* Definitely. Although you never know when you might need ridiculous information at your fingertips, and this definitely isn’t something you can look up.

* An answerable question?* Totally.

First things first—What would we need to know to make a good estimate?

How big is a spherical cow?

How do we pack them in DC?

Puzzle 1: How big is a spherical cow?

You probably won’t find this in many physics textbooks (but possibly in a theoretical animal science or quantum farming course...). I’ll assume we’re talking about dairy cows. I have seen them in person, and these cows are way bigger than me but also smaller than a standard US car. I’ll estimate they weigh 450 kg (which is about 1000 lb in the imperial system). THAT. IS. HUGE. We’re talking several football/soccer players big.

I’ll assume cows are mostly water (since humans are), and one of the few facts I’ve memorized is the density of water at STP: 1000 kg/m³. This is helpful because volume is mass divided by density, as you might remember from middle school. We also know that the volume of a sphere is 4πr^{3}/3, , where* r* is the radius of a sphere. Combining those equations for volume yields the following:

Solving for the radius r, we arrive at about 0.475 m as the radius of our spherical cow.

Now we just need to know how many spheres of radius 0.475 m fit into Washington, DC. From Fig. 1 we can see that Washington, DC, is roughly a diamond shape. I’m guessing it’s about 10 miles per side, or 100 square miles, minus the land given back to Virginia (on the west side of the Potomac River).¹ If we take out the water and now-Virginia part of the diamond, we can estimate the land to be about 67 square miles, or 173 km².

Puzzle 2: How do we pack them in DC?

If we really packed our* spherical* cows in there, they’d be touching each other. Just how tight could we pack them? Gauss and Lagrange determined the maximum packing of spheres in a 3D volume and on a flat 2D surface²—in fact, it’s a common calculation in condensed matter physics because it directly relates to crystal structures! If we put in as many cows as possible without stacking them, they’ll naturally form a hexagonal pattern, as shown in Fig. 2.

From this image we can see that for circles of radius r and a hexagon of side length 2r, the spheres fill this fraction of the space:

This is independent of sphere radius (which is super cool) and means that when we pack the cows in, we are left with about 9% empty space. If we ignored buildings, elevation changes, small rivers, and everything else already in Washington, DC, we’d be able to fit about 349 million spherical cows! That’s a lot of µ-ing (bad joke).

Next, I challenge you to think outside the two-dimensional world: What if we could stack the cows? **How many spherical cows could we stack inside Washington, DC?**

* Hints:* The final shape would probably be something like a pyramid. Check out the awesome Fall 2017 Puzzler written by Gary White for some useful insights.³

References:

1. It turns out this was a very good guess, as it originally was 10 miles per side. The actual area is about 68.3 mi2.

2. Zamponi, F., “Packings Close and Loose,” Nature 453 (2008): 606–607, https://doi.org/10.1038/453606a.

3. G. White, “Consider a Circular Cow,” The SPS Observer, Fall 2017, http://www.spsnational.org/the-sps-observer/fall/2017.

Willing to share your answer? Please email SPS-Programs [at] aip.org with a one-page PDF showing how you solved the above question. Submissions have a chance to be highlighted on SPS National social media pages, and participants may receive some SPS swag!