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| Nobel Laureate Enrico Fermi, c. 1943 |
Dr. Enrico Fermi was one of the most brilliant physicists of the 20th century, winning the Nobel Prize in 1938, and producing the first self-sustaining power-generating fission reaction in 1942. His theoretical and practical contributions to atomic physics can scarcely be overstated. In 1938, Fermi and his family left their hometown of Rome as a consequence of Mussolini's anti-Semitic laws. Fermi himself was not a Jew, but his wife was. They settled in Chicago and Fermi became a naturalized American citizen in 1944.
Fermi was as close as modern scientists have come to a completely apolitical technocrat. Neither did he help draft the 1939 Einstein letter, nor did he provide an active role in the postwar order as his confederates Einstein, Szilard, and Teller did. Fermi's role in the Manhattan Project was crucial, but he was basically anti-war. He was also an intensely humble scientist who could be found machining his own parts or helping graduate students move conference tables around the university. Tragically, Fermi died in 1954 as a result of stomach cancer.
One of the stories about Fermi is that he did the first back-of-the-envelope guess about the yield of the first atomic explosion (Trinity test in 1945) by standing at a known distance from the explosion holding strips of cut paper. He threw the pile into the air periodically. When the shock wave hit, the airborne paper was displaced a certain distance that he estimated. Then, in his head very quickly, Fermi estimated the yield of the bomb at 10 kilotons of TNT. The actual yield as determined by measurement instruments was 19 kT. Granted, he wasn't precisely accurate, but it's remarkable how "in the ballpark" a scientist can be simply by using estimation.
To posit a question which requires estimation and layers of assumption is known as creating a "Fermi problem". The "solution" to a Fermi problem is sometimes indeterminate and should be carefully quantified in terms of the uncertainty of each assumption, which can be substantial.
The original question that Dr. Enrico Fermi asked while teaching at the University of Chicago was this: How many piano tuners are there in the city of Chi
cago?
According to Wikipedia, Fermi's assumptions are given as follows:
1. There are approximately 5 million people in Chicago (c. 1940).
2. The average household size in Chicago is roughly 2 persons.
3. Roughly one household in 20 has a piano that is kept in a good state of repair.
4. A piano in good state of repair requires tuning once per year.
5. A piano tuner can perform a complete tuning, plus travel time, in 2 hours.
6. Piano tuners work standard 8-hour days, 5 days per week, and take off 2 weeks per year.
Using the above logic, we estimate 125,000 piano-owning households in Chicago, and the 125,000 piano tunings per year can be done by a total of 125 piano tuners.
When a guest lecturer in my extremely large freshman physics class posed this seemingly insipid question, some of the very large student body promptly went to sleep, and others took it as some kind of joke being wound up. Only a moderate portion of the students actually sought to do a dimensional analysis and helped walk the professor through our class-consensus estimates. I do not believe that many of us present had been familiar with the concept of Fermi problems before. Perhaps in recent years it has become (and will continue to become) more important to teach physicists, engineers, mathematicians, and students of all hard sciences to learn the art of estimation in research.
Actually, our guest lecturer planned to devote as much time as he felt fruitful to the development of the point of this Fermi problem. To that end, the students and he hammered out a list of not merely guesses, but upper and lower bounds for those guesses. Wow, sophisticated! But don't give us too much credit, since the lecturer did get us to answer the problem using more or less the same approach.
1. Chicago's population was estimated at between 3 and 8 million by the students, some of whom had some personal knowledge of the city, and the uncertainty largely rested on whether the city proper or the metro area was being considered. Fermi's estimate of 5 million was probably more consistent with a metro area. Calculations were done with both 3 and 8 million.
2. We were not aware of Fermi's guess of 2, but we estimated that single workers would be pretty commonplace in a highly urbanized area like Chicago, and that households of 1 would be equal in number to childless couples, which together would outnumber those with kids two to one. Of those with more than 2 residents, one of the students proposed doing some kind of summation up to 12 (10 kids and 2 parents being very uncommon) with the total probability from 3 to 12 being 0.33. We arrived at an average household size of about 2.5.
3. Pianos seem to be less common, since they remain expensive while families have found other ways to encourage gathering such as television and home computers. We reckoned that 2-4% of households had pianos, of which most (75%) were regularly tuned, since a piano way out of tune is of little musical benefit. This amounts to 1.5% to 3% of Chicago households.
4. My old friend Dan plays the piano as well as the pipe organ and violin, so he dutifully told the class in boring detail about how often tuning can become necessary, depending on how often the instrument was used. Eventually the lecturer simply suggested once per year might be accurate, and we accepted it.
5. Dan pointed out that the job of tuning a piano could be done in 1 hour. When we were asked to think of how the schedule of the piano tuner might be padded with real-life constraints such as driving his vehicle to the owner's home, and waiting for the phone to ring in the first place, we reasoned that only maybe half of the time on the job was spent doing tuning. Therefore, in a roundabout way, we also reckoned 2 hours per job to be fairly accurate. However, we took "piano tuner" in the most literal sense, and only included those who actually performed that kind of work, and not support staff for the businesses that do this work.
6. We did agree that piano tuning is neither a common nor extreme vocation, and that the practitioners are likely to set themselves very reasonable 8-hour days. The stereotypical 2-week vacation and 5-day workweek were thrown in as well. It was assumed that tuners would not have to work weekends. Here we agreed completely with Fermi again.
Using our estimates, taking just Chicago proper resulted in a range of piano tuners from 18 to 36, or if the whole metropolitan area was considered, from 48 to 96.
The real advantage of an estimation analysis like a Fermi problem is that it teaches a number of fundamental skills that help in the life of a scientist:
1. Ability to get a "reality check" on figures to within an order of magnitude, if not higher precision. If we check Fermi's guess, and try to do research on exactly how many piano tuners exist, is it reasonably accurate? If an estimator feels it is exceedingly likely that there are more than 12 piano tuners but fewer than 1,250, then he or she has a certain amount of confidence that Fermi's guess is within an order of magnitude of the "real" value.
2. Proper dimensional analysis skills. Multiplying jobs per day by days per year equals jobs per year.
3. Comfort in dealing with ranges of numbers, or in uncertainty values. We might have said that Chicago's population was 5.5 million, plus or minus 2.5 million.
4. Familiarity with maintaining proper significant figures. For example, if Chicago had 8 million or 8.001 million people, what difference is implied? If you add a 100 kg man to a 40-tonne boat, is the weight now exactly 40,100 kg?
5. Crossover fields of knowledge and "common sense". If asked for the population of Chicago, a common-sense adult should know that it is in the millions (certainly not 100,000) but not greater than the population of the whole region (certainly not 30 million). Asking engineers and scientists to familiarize themselves with pianos seems silly, but it's merely an example of how arcane knowledge could be called upon. Here is some more motivation (not that any was needed by geeky undergrads in the first place) to have a very diverse base of knowledge.
6. Intellectual flexibility in quite an intangible way. The scientist who is comfortable with relying on estimates while carefully examining the limitations of those estimates, is the one who will actually get a job. A scientist who is obsessed with Ivory-Tower exactness and does not like the real world, will have a hard time finding a job, and will also find it difficult to make substantial contributions to his or her field.
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| Brig. Gen. Leslie Groves |
Order-of-magnitude estimates are occasionally called for when approaching theoretical physics topics, but they tend to infuriate the uninitiated! Perhaps no man was so plagued by imprecision as General Leslie Groves, who was responsible for managing the Manhattan Project. Although he had almost unlimited resources (given the time period), Groves was initially unsure if the atomic bomb could even exist at all.
General Leslie Groves, the military officer in charge of the Manhattan Project, viewed the Los Alamos scientists as in need of constant reminders of reality. Their initial estimates for a critical mass of fission material were only precise to within an order of magnitude, which would have made it either easy (10 times less than the estimate), possible (close to the estimate) or impossible given the uranium-refining capability of the day (10 times greater than the estimate). He likened order-of-magnitude accuracy in the following anecdote: "The wedding party is planned to have 100 people. However, maybe 10 people will show up, and maybe 1000 people will show up." Clearly effective planning is difficult in such a situation, and his anger is in fact clearly understandable when explaining the awesome responsibility of planning the atomic bomb project.
But an estimation problem is either a thought experiment for learning purposes, or the start of a more solid investigation with greater accuracy. It should be taken for what it is. The Fermi problem continues to be an important lesson in physics and related disciplines.
Edouard Benedictus was a historical French badass whose name I did not know until recently. Given that he was skilled as a chemist and painter as well as a bookbinder and fabric designer (interesting), I think he qualifies as a polymath. If your method of discovering the world is through Google, you would have likely seen no association with M. Benedictus apart from his paintings and design of patterns.
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