Friday was pi day. I don't know when people started celebrating the fact that the American notion for March 14th looks something like the first three digits of the irrational number pi, but it has grown into something of an event in schools. Along with being a good excuse for having a piece of pie, it also happens to be Albert Einstein's birthday.
Apparently Einstein had a fondness for sweets - ice cream on a hot day, but also pie. Apparently a favorite was cherry pie, which happens to be my favorite, but there were good apples and a homemade vanilla streaked with a swirl of caramel and pecans would do the trick.
Enjoying apple pie à la mode, I found myself thinking about the harsh Winter in much of the US and wondering how cold it was historically. It turns out the season wasn't that harsh - in 114 years of record keeping the season that just ended only ranks as the 34rd coldest.1 The record was set in the Winter of 1978/79 with many of the other near records set in the first half of the century. I remember 78/79 being brutal on Long Island, but the most dramatic I've seen came in 68/69 when I was growing up in Montana. A few weeks where the high temperatures were below zero and a few nights that fell on the cold side of minus forty.
Montana Winters can be cause some mechanical systems to act a bit differently. A car starting out on a -30°F morning gives a bumpy ride even on a dry road. The tires have deformed into a D shape giving a thump every time they go around. After awhile friction warms the tire enough restore a bit of flexibility and the ride gets smoother, but this is really entertaining if you're a kid.
As it happens even on a warm day tires are far from round - they have to have a contact area with the road so a car can "grab the road" - a perfectly round tire would only have the narrowest contact with the road and the car would not be able to start, stop or turn. There is quite a bit of physics involved in how a car moves.
Racing has been a laboratory for ideas that lead to automotive innovations. In the 50s and 60s serious work was done on the aerodynamics of cars and the physics of their suspension, tires, brakes and power trains. Some of this work came from rather curious corners rather than famous race car designers. People like Gordon Murray and, before him, Jim Hall and Jim Clark were pioneers stealing some of the limelight from the intuitive and artistic designers like Enzo Ferrari.
Jim Hall was particularly interesting with ideas that came directly from simple physics. His secret weapon was his degree in mechanical engineering from Caltech combined with a rather creative mind.. Inverted wings appeared on his Chaparrals to keep the tires in better contact with the road making his cars accelerate and corner better than much of the competition. He not only worried about the flow of air over the car, but also under it. The model 2J had a "skirt" around the car and a 45 hp auxiliary engine driving twin 17 inch fans that sucked air from under the car creating a partial vacuum between the car on the track. The 2J had enough extra contact with the ground that it was seriously faster than the competition even with the weight penalty of the engine. Of course it was quickly banned with a rule change.
My experience with thinking about things is a lot of play is involved. When starting with an idea a good trick is to simplify it as much as possible and create a thought experiment - what Einstein called a gedanken experiment. These usually happen in your head or on a blackboard and are supported by quick mostly mental calculations that are sometimes called Fermi calculations.2 Gedanken experiments and Fermi calculations usually aren't accurate - they don't have to be. Rather they are often accurate enough to sort useful approaches from those that are destined to fail - if an idea is wrong you don't need to get it wrong to five significant digits. They also happen to be delightfully fun and are central to the playfulness of the sport. A great way to find an upper or lower bound it is a powerful tool for thinking about technologies. I'm guessing other fields play similar games.
As with anything, experience makes you better at Fermi calculations. You intuitively combine formulae - often very simple approximations - with "good enough" numbers, fuel yourself with chocolate or coffee and see where your mind takes you. There are a number of "rules" you devise along with numbers you tend to remember. Light travels about a foot in a billionth of a second, there are pi * 107 seconds in a year, water is about 800 times as dense as air at sea level, a person requires about 100 watts of power from their metabolism, a square meter of surface receives about a kilowatt of power from sunlight on a clear day with the Sun directly overhead, pi is about 3 and so on...
Pi is 3?
Sometimes that's good enough. There have been clueless attempts to define pi as something other than - well, pi - but this is completely daft unless you are in a world with very primitive engineering capabilities. The trick is knowing how much pi is enough for the purpose.3
Most of us approximate pi as 3.14 and sometimes as 22/7. It turns out 22/7 is a bit closer than 3.14 which gives much of the world claim to the 22nd of July as pi day. Of course you can break up the day to find the proper hour, minute and second to celebrate a bit more accuracy.
Estimating pi as 3 puts you about 4.5% from the real value - fine for Fermi calculations where the error is a small part of the overall acceptable error, but if you are making a liner for something 10 feet in diameter, you'll be off by about 17 inches. 3.14 gets you to about a half percent of the real value, 3.14159 gets you within 0.000084 percent. For most construction this is overkill.
The Committee on Data for Science and Technology is the source of accepted values for fundamental constants. It turns out the fundamental constants of the universe only require 32 digits of pi. NASA "only" uses 16 for its space flights. So while the number is extremely useful and mathematically interesting, you don't need very much of it to make sense of the Universe or to build anything.4
In honor of pi and Einstein I was going to continue with a post on a beautiful connection. That is going to take at least and hour to write and I'm near the end of my time for this post so, as they say, to be continued ...
1 NOAA has a great site with historical climate information from 1985. The Winter "season" is usually expressed as December through February rather than the astronomical season, which squares more with how most of us view the season.
2 Enrico Fermi may have been the last physicist who excelled at both theory and experiment. He was exceptionally clever at quickly finding the core of a problem and testing it with a simple calculation. Physicists had been doing this long before, but his name became attached to the process.
3 There is also the question of how much pie is enough. For me that depends on the type of pie and if ice cream is available. In moments of weakness two slices of a tart cherry pie are required.
4 While there is very serious math in irrational numbers like pi (are the digits really random for example - it is believed they are, but no proof exists), there are any number of silly games you can construct. To celebrate the day I sent an email with this note.
There isn’t a limit on where the behavior of being a kid ends, so there are games you can play. It isn’t known if the digits of irrationals are random, but pi gives the appearance. With a computer i is easy to find strings of digits. One of the first interesting strings comes at the 762nd decimal place where there are six 9s in a row. Richard Feynman noted if you are memorizing pi, to it to that level and then you can end by saying "... two, one, one, three, four, nine, nine, nine, nine, nine, and so on ..." suggesting pi is rational:)
For nerdish kids a game is to calculate pi to a few million digits and then search for short strings. Represent your initials by numbers and find where you first show up in the number. My initials are ESC. If you use an ascii representation my string is 698367. This first occurs at the 1,092,663rd decimal position. Now if I forget my initials all I have to do is remember my position in pi:-)
Another short observation. The question came up can you cook an avocado? I had no idea, but this worked well and might be the basis for your own experiments.
Cut an avocado in half, remove the pit, pour on a bit of olive oil and balsamic vinegar and season with a sea salt. Place on a grill cut side up and let it cook away. "Done" is a relative term and I don't know what is best yet. Letting it go to where the skin chars gives a lovely flavor. Serve with a bit of freshly ground pepper.