On Episode 150 of their podcast Dear Hank and John, Hank and John Green were pondering this, haunting questions:

How many visitors to the Eiffel Tower would have to poop there when they arrive before the tower is covered in poop? How long would that take?

Well, let’s have at it!

I’m a physicist so I feel compelled to rephrase the question in a way that better accommodates my math–at least at first. So the question I will actually tackle is:

If every visitor to the Eiffel Tower pooped there when they arrive, how long would it take for the pile to become as tall as the Tower?

To calculate the answer, then we need to do the following:

1. Figure out the volume of poop involved
2. Calculate how much mass of poop that corresponds to
3. Calculate how many people will take to produce that much poop, ie divide by the average amount each of us drops every day
4. Divide by the average number of yearly visitors to the Eiffel Tower to find how many years the process takes

Let’s start then by figuring out how big a pile of sh*t this actually is. This turned out to be sneaky hard to do.

The fantastic monument that will come to existence in the process will be called The Paris Poop Pile–the Pile for short–and will unavoidably be a cone. That’s the natural form piles of dropped materials take.

The volume V of the cone is given by the formula V=(1/3)πhr²: h is the height of the cone (324 meters, that’s how tall the tower is), π is everyone’s favorite mathematical constant, r the radius at the base of the cone. That, with a little trigonometry turns out to be just h/tan(α), or the height divided by the tangent of the steepness angle of the side.

As a matter of fact, every time we just drop materials and let them pile up, they form a cone with a specific steepness, that depend on the properties of the material. For example, the sand falling through an hourglass.

This steepness angle is called angle of repose and its value is known for a lot of materials. Not poop, though. But I’ll guesstimate it to be about 30 degrees: like slimy earth, stony gravel, or mid-solid clay. That number also comes in handy because it makes the radius of the Pile be exactly √3 times the height, and thus gives us a simple formula for the volume: V=πh³.

We have all numbers we need for that calculation: the Pile’s volume is around 106 million cubic meters.

(*)

Now, apparently, the average density of poop is about the same as water. I’m not quite convinced of this but people seem to agree with that, so let’s go with it. Because of that, the Pile weighs 106 million metric tons.

According to a 2015 paper, people produce an average of 128 grams (around 1/4 of a pound) of feces per day. On the internet you can find any sort of estimate, up to 500g. This number seems reasonable and well sourced. Also, if it’s good enough for Wikipedia is good enough for us. Dividing 106 million metric tons of poop by 128g of poop per person, we get that the Pile takes the work of 834.77 billion people.

Every year, 7 million people visit the Eiffel Tower. 7 million tributes to the Pile. That’s plenty, but it still means we need around 119.3 thousand years to finally, completely eclipse the Tower using poop.

It’s a very long time. But don’t forget just how big the Pile is. The Eiffel Tower is really tall, and the Pile is quite shallow, so a lot of material will just slide down the side.

Of course, I took a couple of shortcuts: how much stability does the tower itself or the surrounding buildings provide to the Pile? The poop in the Pile is not inert: how will it change? How do these and other things factor in the construction times? Will visitors diminish because there’s a ton of crap in front of the Tower or will they increase because they want to participate? This is just a starting point, a first approximation. Science is made by building upon each other’s work: be a scientist!

Bonus: condensing to a single formula

If you want to play around with this or check my numbers (please do: that’s how science progresses), here’s a formula that puts all calculations in a single step:

time=π*height³/[3*YearlyVisitors*DailyPoopedMass*(tan(AngleRepose))²]

Putting in your own numbers or estimates you can also calculate how long it would take visitors to cover other monuments in poop.

 

(*) Looking at this fragment from Jurassic Park, this pile of triceratops poop seems to have a much steeper angle of repose. Few things are worth noting: first of all, that’s triceratops poop, not human poop–triceratops eat tons of vegetation, so their feces are probably less slimy than ours. Second of all, that’s a movie, not real life (might be real poop, I don’t know). Finally, even with a much steeper angle of repose it’d take thousands of years to complete the Pile. With a 45° angle, we’d still need about 40 thousand years.