Enrico Fermi, the face of a genius

 

Enrico Fermi and his problems

Enrico Fermi might have had problems, like most people, but that's not what this post is about.  I'm gonna talk about the type of problem that bears his name: (cue hero music) THE FERMI PROBLEM!  Fermi problems involve math, but they don't usually have a specific number solution, only a close estimate based on logic, deduction, math, physics, estimation, and even guessing.  These are the kind of problems that drive the average person to a "What?" and a shoulder shrug before they return to their pizza and beer and football game, but a curious sort might be driven to hours of distraction as he or she attempts to explore all the nuances that can lead one closer and closer to the right answer without ever achieving a FINAL answer!  Madness!!!

How did this quiet man of Italy come to have his name associated with these mathematical torture devices?  Well first of all, Enrico Fermi was a smart man.  I mean, like, really smart.  The kind of smart that makes a Mensa candidate look like the village idiot.  During his teen years he conducted experiments to test the density of the water supply of his hometown of Rome.  Just for fun!  He was so smart... (all together now, "how smart was he?")... he was so smart, that when he applied to university, the admissions panel admitted him not as a freshman, but as a doctoral candidate!  Yup.  Here, friends, was the Doogie Howser of physics and mathematics.  Enrico Fermi also had a nickname: (cue ominous music) "The Father of the Atomic Bomb".  Before the Manhattan Project became reality, it was Enrico Fermi that proved sustained nuclear fission was possible.  By nearly blowing up the city of Chicago! 

"The Italian Navigator has just landed in the New World"

Chicago Pile-1; from such humble beginnings...

 Enrico Fermi and his wife immigrated to America in 1938 to escape fascism in Mussolini's Italy.  Just a few years later, on December 2nd, 1942, Fermi and his team of scientists, in order to test his theories of sustainable nuclear fission, set up a rather large, orderly pile of wood and graphite and uranium, dubbed "Chicago Pile-1",  on a squash court underneath the football stadium of the University of Chicago.  The idea was to use large cadmium rods to absorb uranium neutrons to control the speed of the fission reaction.  However, if Fermi's calculations were wrong and the fission reaction raced ahead uncontrolled, there would have been a nuclear blast in downtown Chicago.  So, no pressure.  The test was a success, and the nuclear age was born.  To inform the White House of their success, a cable was sent bearing the following cryptic message: "The Italian Navigator has just landed in the New World".  Fermi's reward was being appointed one of the overlords of the Manhattan Project, which was tasked with creating an atomic bomb.

The Fermi Estimate

Enrico Fermi was well known for his ability to estimate answers to complex problems by using logic and deduction in lieu of hard and fast mathematical calculation.  The legend goes that he estimated the strength of the first atomic bomb, codenamed "Trinity", by dropping pieces of paper during the explosion and seeing how far they blew away from him.  His estimate was 10 kilotons, which was remarkably close to the accepted value of 20 kilotons, especially considering he came up with his estimate using no hard number calculations.  Below is a typical Fermi estimate attributed to Fermi himself and taken from Wikipedia:

"How many piano tuners are there in Chicago?" A typical solution to this problem involves multiplying a series of estimates that yield the correct answer if the estimates are correct. For example, we might make the following assumptions:

1. There are approximately 9,000,000 people living in Chicago.
2. On average, there are two persons in each household in Chicago.
3. Roughly one household in twenty has a piano that is tuned regularly.
4. Pianos that are tuned regularly are tuned on average about once per year.
5. It takes a piano tuner about two hours to tune a piano, including travel time.
6. Each piano tuner works eight hours in a day, five days in a week, and 50 weeks in a year.

From these assumptions, we can compute that the number of piano tunings in a single year in Chicago is

(9,000,000 persons in Chicago) / (2 persons/household) × (1 piano/20 households) × (1 piano tuning per piano per year) = 225,000 piano tunings per year in Chicago.

We can similarly calculate that the average piano tuner performs

(50 weeks/year)×(5 days/week)×(8 hours/day)/(2 hours to tune a piano) = 1000 piano tunings per year per piano tuner.

Dividing gives

(225,000 piano tunings per year in Chicago) / (1000 piano tunings per year per piano tuner) = 225 piano tuners in Chicago.

The actual number of piano tuners in Chicago is about 290.

Notice the number of different estimates involved in the above problem.  Assuming that each estimate is reasonably correct, the final answer will be a reasonable approximation to the actual value because any overestimating of one individual value will be cancelled out by underestimating on a different value.  Accuracy in a Fermi problem is usually measured in orders of magnitude, as some Fermi problems are extremely complex and have many components.  Take, for example, a formula called the Drake Equation, which is used to estimate the number of intelligent civilizations in the universe.  The equation looks like this:

where:

N = the number of civilizations in our galaxy with which radio-communication might be possible (i.e. which are on our current past light cone);

and

R_* = the average rate of star formation in our galaxy

f_p = the fraction of those stars that have planets

n_e = the average number of planets that can potentially support life per star that has planets

f_l = the fraction of planets that could support life that actually develop life at some point

f_i = the fraction of planets with life that actually go on to develop intelligent life (civilizations)

f_c = the fraction of civilizations that develop a technology that releases detectable signs of their existence into space

L = the length of time for which such civilizations release detectable signals into space

Clearly there are values in this equation that are unknown and will remain so for the foreseeable future, such as the number of planets with life or with intelligent civilizations.  We can only guess.  As guesses become more refined, the final answer, in theory, becomes more accurate.  Values for N in this equation range from 2 to 28x10^7, so clearly this is NOT a very good equation.  Plus, the lack of any contact with all of these supposed 280 MILLION civilizations has led to something called, get this, The Fermi Paradox!!

So, how is this idea of the Fermi problem useful?  Well, physicists and mathematicians and engineers will often use Fermi estimates before making actual calculations in order to not only provide context in which to do their actual calculations, but also to pre-test the validity of their final results.  How important is Fermi estimation?  Believe it or not, entire courses are taught on this at such prestigious places as MIT and the California Institute of Technology.  Often Fermi problems lead to more rigorous calculation, but the Fermi estimate at least gives one a starting point by figuring out parameters to refine.  This sort of logical problem solving has practical application in any number of fields of endeavor, such as engineering or city planning, architecture or chemistry.  Or just plain old mental torture frustration exercise.  So, try out this typical Fermi problem:

Archimedes of Syracuse was a philosopher, inventor, and scientist who lived and died in Sicily more than 2,000 years ago.  Assuming that the earth's atmosphere has been thoroughly mixed since then, how many molecules from his final breath are in your lungs right now?

Before you shrug, say "What?" and return to your pizza and beer, take a moment to think about not only the different values involved, but also what they might be.  Approximately.  When you have your Fermi estimate, or when you have simply given up, check the solution generally accepted below (to see the answer, highlight the dark text boxes below):

22.4 liters = 1 mole = 6x1023 molecules. Typical breath = 1 liter = 3x1022 molecules in

Archimedes’ last breath.

Volume of earth’s atmosphere = 8x1014 m2 (area of earth, see last question) * 30 km

(mean height of atmosphere) = 2.4x1018 m3 = 2.4x1021 liters. So 10 of Archimedes’ last

molecules are in each liter of the atmosphere.

Your lungs now hold roughly one liter. So there should be (on average) 10 of

Archimedes’ last molecules in your lungs now.

 Were you close?  Were you creeped out?  How many zeros did you have in your final answer?  Subtract your number of zeros from those of the final solution, and the absolute value of the resulting number shows you the order of magnitude you were in disagreement with the solution.  If you dare to try some more Fermi problems, click on the link below:

 

http://www.fermiquestions.com/play

or this one:

http://www.nytimes.com/interactive/2009/03/31/science/20090331-angier-quiz.html

Some sick, twisted person created a kind of Fermi estimate GAME!!!  You keep score by keeping a running total of your differences of order of magnitude.  Lowest score wins.  It's the new family game that will be sweeping the nation!

 

Final thoughts, or how I learned to stop worrying and love the bomb, or "If everybody is thinking alike, then somebody isn't thinking" -George S. Patton

When it comes to Fermi problems, for me personally it was love at first fight frustration *!@&#! noble attempt.  I deeply love the challenge of trying to come up with all the various components of a problem and estimate values for each one.  As an educator I put high value on demonstrating to math students that sometimes it is not all about coming up with the correct answer or regurgitating formulas and methods learned by rote. In the real world there are some problems that don't have set solutions or clearly mapped out paths to conclusion.  Sometimes we simply have to make our best guess.  A great deal of pressure is removed in such calculations, as it is not the answer but the process that has the greatest value.  You compare your answer with others, you re-evaluate, you live with the uncertainty of your final answer.  This, my friends, (cue cheesy music) is what makes the Fermi problem such a beautiful metaphor for life itself.