Wednesday, August 6, 2014

Sixty

Aviation for me has always come with an intrinsic conflict, namely that I seem to enjoy numbers and formulas more than most other pilots.  Mathematics has always come with an intrinsic conflict, too, since I seem to enjoy flying and airplanes more than most other mathematicians.

I think pilots would fly better if they calculated more, and I think mathematicians would calculate better if they would fly.  (That's not quite right, since the purpose of mathematics is to get the right answer without calculating.  But you get the idea.)

So I have been writing an essay about the number 60.  I think it's the most important number in aviation, with the possible exception of the price of self-service Avgas at my local airport.  I won't reproduce the essay here, but will share some thoughts.

The mathematician likes 60 because it has so many factors: 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30.  This makes it easy to divide by 60.  (Someday I will get to teach a semester-length course in long division, which is an important technique in coding and cryptography, but that's the subject of a different essay.)

I am unsure of how this came to be, and don't want to replay the "Babylonian mathematics" game, but we divide a lot of things into 60 pieces: hours are divided into 60 minutes, each minute is 60 seconds; each degree of arc is 60 minutes, and each of these minutes is 60 seconds; and, what is the same thing, each degree of latitude is divided into 60 nautical miles (in theory). The circle is divided into 360 degrees, that is, 6 times 60, too.

All of these divisions are part of aviation.  Remember the "Rule of 60" that appeared on every FAA knowledge test that you ever took?  That one degree of error is one mile of error after 60 miles?

To focus on error is a mistake; focus on desired performance instead.  Lots of turbine pilots use the 3-to-1 rule for descent planning: 3 miles for every 1000'.  A little fooling around with 60 shows that's remarkably close to 3 degrees.

That's fine if your pressurized jet can sustain 2,000 fpm without busting eardrums, but what if the airplane isn't pressurized?  Then you're looking at a descent rate of 500 to 1,000 fpm.  To lose, say, 4,000 feet, takes 8 minutes at 500 fpm. A groundspeed of around 120 knots is about 2 miles per minute (see the role of 60?), so the descent takes 16 miles.  The same idea works at 110, or 130.  A groundspeed of 180 knots is 3 miles per minute (we divided by 60 again), so the same descent takes 24 miles.  This is still close enough at 150 knots, or at 210.  Oh, add a couple of miles to slow down to traffic pattern airspeed.  That's experience, not math.

Using a base of 60 means that you can work with whole numbers, which are a whole lot easier than fractions.

Now if only coming up with money to pay for Avgas were as simple...

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