Thanks to Christopher Nolan — and Barbie, somehow — the Hollywood spotlight has been cast upon the Manhattan Project. While many are familiar with the large-scale and epoch-making dimensions of this project, some of its subtler aspects have been lost in the annals of history. One such fascinating episode was Enrico Fermi’s off-the-cuff estimation of the yield of the Trinity test, the world’s first atomic bomb detonation. His method, now known as “Fermi estimation,” is a brilliant testament to the power of guesswork.
During the Trinity test, Fermi, a key physicist on the Manhattan Project, was about 10 miles away from ground zero at Base Camp, which housed some of the few hundred witnesses of the world’s first atomic blast. While observing the explosion, he dropped small pieces of paper from his hand and measured how far they traveled due to the blast wave (it was about 5 feet). Based on this rudimentary observation, Fermi approximated that the bomb’s yield was about 10 kilotons of TNT. This informal estimate was shockingly close to the actual yield of around 20 kilotons.
The enormity of the context often overshadows the fact that Fermi’s guess during the Trinity test was not an isolated incident of his ingenious estimation method. Throughout his career, he became renowned for making accurate approximations that gave insight into complex problems. He used it in everything from education to guessing how many alien-inhabited worlds there ought to be.
The so-called Fermi paradox showcases his ability to pose profound questions based on simple reasoning. He famously asked, “Where is everybody?” in reference to the apparent contradiction between the high probability of extraterrestrial life and the lack of contact with such civilizations.
Fermi was known for posing questions to his students that encouraged them to think in terms of estimation. These “Fermi Questions” would typically involve making good guesses about quantities that seemed impossible to calculate. Examples include, “How many piano tuners are there in Chicago?” or “How many molecules are there in a glass of water?”
How to do Fermi approximation
When the steps of Fermi estimation are laid out, it is deceptively simple. That being the case, I’ll state the steps first before we look at some examples.
Step 1: Break Down the Problem. Break up the main question into smaller, more manageable components or questions that you can more easily estimate.
Step 2: Make Educated Guesses. For each component, provide an educated guess based on known or assumed information. Aim for the right order of magnitude, not exact accuracy.
Step 3: Calculate. Multiply or divide your estimates as needed to get an answer to the original question.
Now, let’s apply these steps to the aforementioned examples to get a feel for it.
Example 1: How many molecules are there in a glass of water?
First, we break the problem down into manageable chunks. The number of molecules in the entire glass is the same as (the number of molecules per gram) × (the number of grams per liter in water) × (the number of liters per glass). Each of these is easier to estimate than the original question.
In the second step, we need to come up with the estimates. This can come from existing knowledge or through accessing common knowledge (now on the internet!). Let’s say we have a big glass of water, so about 1 L. I don’t know the density of water off the top of my head, but a search engine provides a nice “rough” estimate for me! Google says it is 1,000 g/L. Last, the weight of a water molecule is, again, something we can refer to. Rounding to the nearest order of magnitude, a water molecule weighs a paltry 0.00000000000000000000001 grams. In other words, there are 100,000,000,000,000,000,000,000 water molecules in each gram of water.
Now we just multiply these numbers together to get our final “Fermi” guess of… 1 followed by 26 zeros. Or, a lot. The correct answer is a third of this number, by the way, which is pretty darn good!
Example 2: How many planets have intelligent life in the universe?
Step one. The number of planets with smarts is the same as (the number of galaxies in the universe) × (the number of stars per galaxy) × (the number of planets per star) × (the fraction of planets that are habitable) × (the fraction of habitable planets where life formed) × (the fraction of living things that have intelligence). Clearly, some of these are going to be wild guesses!
Oh, well — let’s go.
- NASA estimates there are about a trillion galaxies in the universe;
- There are about 100 billion stars in our galaxy;
- Maybe 1 in 100 stars have planets;
- Perhaps 1 in 1000 of those planets might support life;
- Generously (?), let’s say, even if all the conditions are right, life is a 1 in a million shot;
- Of the billion or so species on Earth, let’s round down to one being intelligent (cats, obviously).
Multiplying these numbers together, we get…*drumroll*… 1,000. Neat, though that still sounds pretty lonely. Of course, we don’t have the correct answer to compare it to. (Nor would we want to, but that’s a story for another post.)
Using this type of Fermi estimation, with different estimates, Frank Drake famously estimated that the number of civilizations we could communicate to (or from) in our own galaxy is 20. Others have come up with still different answers. But the beauty of Fermi estimation is that we easily see where the disagreement lies.
Why does it work?
Naively, we might expect that every error we make in estimating the component problems will compound into a final estimate that is wildly off. However, when rounding, we are just as likely to underestimate as overestimate. While we wouldn’t expect these effects to perfectly cancel one another, they provide remarkable mitigation.
Physicists have created various arguments trying to “prove” it works. These require several assumptions or approximations of their own. But, more than that, they are also very complicated. I prefer a more straightforward way: simulation.
In the context of science, a simulation is a model of a process, often run on a computer nowadays, used to predict real-world behavior. Simulations recreate specific scenarios or conditions, allowing scientists to test theories or experiment with variables without directly interacting with the real world, which may be messy, expensive, or simply impossible. Examples of simulation include predicting weather patterns, modeling the spread of diseases, and recreating the birth of galaxies and stars. I’m more modest — I just simulated Fermi approximations. Here’s how I did it.
In broad strokes, a Fermi approximation is the multiplication of a bunch of numbers that have been rounded to the near order of magnitude. That is very easy to model. The steps are as follows: (1) pick several random numbers; (2) multiply those together to get “the answer” — save that for later; (3) for each random number, round to the near order of magnitude; (4) multiply the rounded numbers together to get “the estimate”; (5) compare “the estimate” to “the answer.”
Here’s an example — four random numbers: 8, 1201, 98, and 223. Multiply them together to get the answer 209,973,232. If we round those numbers, we get 10, 1000, 100, and 100. Multiplying these together (just add the zeros!) gives 100,000,000, and voila! We are only off by a factor of 2. Amazing.
Ok, but that’s just one example. The key to successful simulation — especially when randomness is involved — is to repeat many times. So, that is what I did.
In the above figure, I’ve plotted a histogram of a million random numbers between 1 and 10,000, along with the values that result after they are rounded. By eye, you can see that the distributions of numbers are quite different indeed. But what happens when we multiply them together? To test this, I split all these numbers up into 20,000 groups of 5. For each group of 5, I multiplied the numbers and their rounded versions and then compared them to see the difference. The results are summarized below.
On average, the Fermi approximation is roughly the same as the correct value. That’s to be expected. But what about the other cases that aren’t so average? The simulation shows us that 90% of the time, the Fermi approximation is within an order of magnitude of the correct value. While this is not a proof that will convince a mathematician, I’m satisfied. I’ve tested it — Fermi approximation works. Proof by simulation!
Cute, but who cares?
The charm of Fermi’s approximation isn’t just in its historical significance. It highlights a crucial skill for scientists, students, and, indeed, anyone looking to tackle large problems — namely, the ability to make educated guesses based on limited data. In an era where information overload is common, the ability to distill vast amounts of data into actionable insights is invaluable.
Moreover, it underscores the human element in science. In a landscape dominated by supercomputers, sophisticated equipment, and the looming threat of AI, Fermi’s simple approach reminds us that innovation and insights can come from the most basic observations.
As moviegoers appreciate the larger narrative of the Manhattan Project — and the thousands of ways to interpret the meaning of Barbie — it’s worth pausing to remember these smaller moments as well. They remind us of the ingenuity, creativity, and curiosity that drive scientific exploration and of the individuals like Fermi who, even in the face of the unknown, sought to understand and quantify the world around them.