Mar 10

How Enrico Fermi Takes Integrals

You know those interview questions? The ones where you’re supposed to make spurious assumptions and use them to compute something like: “How many piano tuners are there in Chicago?”
The correct term for one of these is a “Fermi Problem” after Enrico Fermi the physicist, who apparently was spectacularly good at such problems.
Legend has it he made a stunningly accurate guess at the power of a Nuclear Explosion based on how far some nearby scraps of paper moved.

The point of Fermi Problems—and the reason they’re so popular in interviews—is to find the quickest possible path to an answer.
Even if the answer itself isn’t close enough to be useful the intuition it gives you is.
And Fermi had a great love for finding this intuition in a problem.

Square Root

Integration is one of the most deceptively intuitive problems out there, in the picture above for example the answer is just the red.
This easy to digest definition is what you get in the first 5 minutes of a calculus class, and then the other 1795 minutes everything is complicated and unrelated to the red.
And integrals aren’t just hard for you, it really doesn’t take much to make an integral impossible for everyone.
For example \int \sin(x)^{\sin(x)} \ dx gets you a forlorn “no result found in terms of standard mathematical functions” from Wolfram Alpha. Math just doesn’t have a good answer for it.

But this shouldn’t sit well with any of us because you know, and I know that that function has some red under it.
And it really didn’t sit well with Fermi so here’s what he’d do: he’d draw the function out with pencil and paper, cut it out and then weigh it.
Divide that weight by that of a 1 by 1 square of the same paper and that’s the answer.
All he needed was the first 5 minutes.


Jul 09

How Physicists Build a Bridge

Imagine the following, you have a room, twenty stories high and 50 feet across. You want to hang a Foucault Pendulum in this room. Here’s the catch: the ceiling is made of glass and can’t support the pendulum. The only way to hang it is with a cable that traverses the gap between the walls. Now we do have a little something to work with, on the other side of the walls are banks of offices from floors 1 to 20. Furthermore these offices have windows opening on to the room.

This is the situation physicists at Fermilab found themselves in when they wanted to hang a pendulum in Wilson Hall [ picture ]. Now they could call in a construction crew, build scaffolding up twenty stories and just connect the cable like that. Not pretty but it would get the job done. However they wanted something pretty. Firstly, things are a little easier. We don’t actually need to get the heavy duty cable across to begin with, we can start with a lightweight string, and use that to pull across increasingly heavy cables. The Niagara Falls Bridge for example started when a young boy flew a kite across the falls.

Unfortunately  kites don’t work inside. Well maybe we could have someone throw a baseball across the gap with the string pinned to it. Maybe but the physicists were more inspired by the 0 brawn approach: two people go up to 20th story offices, each with a ball of string. They drop the balls (holding on to an end) all the way down to the ground 20 stories below.  A third guy ties the ends together. Wind the string up. voila, we have our bridge.

Jul 09

What You Learned About How Asparagus Works Was Probably Wrong

It’s a small quirk of human metabolism that Asparagus makes our urine smell. Due to some research in the 80s we’re now pretty darn sure that the smell is caused by a triage of sulfur containing alkyl compounds citation. But a few decades before scientists pulled out the big guns and nailed down exactly what was causing this phenomenon, they looked at the much simpler question: “Who makes smelly urine?” Observations of the day held that many people met tales of Asparagus Urine with vacant stares, and a few studies revealed that the percentage of people making smelly urine was in the low 40s citation . (This study was done in The UK and would probably yield very different results in other countries.)

So the scientists concluded, this is a polymorphism, different phenotypes existing in the same population. Some people’s digestive tracks produce the malodorous compounds, and some don’t. Simple enough right? I’ve been told they even managed to find the corresponding genotypes, and the correlation they had was great. I heard it got as far as children’s textbooks, before someone pointed out a tiny flaw in the experiment.

The experimenters had been a bit shy in their work. They’d fed people asparagus and asked: “Now does your pee smell?” when they should have asked the more risque: “Does his pee smell?” Had they asked the latter they would have found that ≈40% of people said yes all the time, and  ≈60% said no all the time. Because it turns out that everyone digests asparagus the same way and outputs the same urine, it’s just that not everyone can smell it.

…transforms my chamber-pot into a flask of perfume -Marcel Proust

This post was inspired by an old Daniel Miessler post which surfaced yesterday on Hacker News entitled “ Why Planes Fly: What They Taught You In School Was Wrong .”