Butterfly Effect

TC_ButterflyEffect

The Butterfly Effect is a poetic name to say that in some situations, small changes in initial conditions can lead to major variations of final outcome. It was first observed while working on computer models for weather predictions in the 1960s.
An American mathematician and meteorologist named Edward Lorenz, while running a weather computer simulation, decided to truncate decimals entering data only to obtain a completely different weather scenario. He used 0.506 instead of 0.506127.

There is a few things to consider:

  1. Computers were very slow back then. A weather scenario could easily take several hours, if not days to complete. Scientists and computer users had a major incentive to use any possible means to shave computing steps and creating smarter algorithms was highly rewarded. Each decimal could eat a lot of memory just for storage and exceed the capacity of the computer during intermediary computations. In 2015, poor programming delays output by the microseconds but until the 90s it would have your machine crash.
  2. Truncating decimals is extremely common in science, especially in physics because a variation beyond the first 3 or 4 digits is not easily measurable. These digits are called the Significant Figures. Sometimes, calculations will produce a trail of decimals but the resulting data cannot be more precise than the original data, hence the practice.
  3. Scientists often do not have more than 3 or 4 significant figures at their disposal to begin calculation simply because the instruments could not provide measurements with such precision. If 0.506127 was, say, the height of mercury in a barometer measured in meters, the third digit (6) already represents millimeters (less than 1/32 of an inch). Try to read that on a yardstick. The last digit (7) is expressing microns, one thousand time smaller a unit, invisible to the naked eye. Modern instruments using electronics are capable of delivering 5 to 7 significant figures in measurements.
  4. Gas weights 6lbs/gal
    we considered 25 MPG for this example
    Average gas tank is 17gal. 17X6lbs = 102lbs
  5. Such precision is quickly irrelevant in real life. For example, the average car in the US weights 4000 pounds, that is 64000 ounces. Driving one mile could shave 4 ounces of gas and bring that weight down to 63996 ounces. The gas tank being full or empty can make the car 100 pounds heavier or lighter . knowing the weight of the car to the ounce, that is with 5 significant figures is totally useless and that is true for almost everything.

All these considerations explain why truncating decimals beyond the third digit did not seem a big deal at first.

So why can it have consequences?
There are instances, when calculations are repeated over and over and each calculation is starting from its previous result (this is called an iteration) then a small change in initial conditions can lead to major variations in final outcome. In a weather model, everything is recalculated by the hour so 24 times per day of prediction.

Below is a chart of a sample calculation made with the same two numbers, one of them constant. We ran it with 4 different degrees of precision. The calculation is repeated 20 times, each time using the result of the previous round. The calculation is very simple, we take a number, calculate its square, add the constant. The result of this calculation is used to do the same thing again (take the result, calculate its square, add the constant) 20 times. This is called a Quadratic Polynomial. The original number is 1, and the constant is respectively -1.618033988, -1.618034, -1.618 and -1.6 (the same number, rounded to 9, 6, 3 or 1 decimals)

ButterflyEffect

As you can see using 3 or 6 decimals has an impact after 25 iterations, the error accumulates and make the results diverge (rounding to 1 decimal has an impact way earlier than that). This with only two numbers, and two basic operators (square and addition). Imagine the consequences if it were a few dozens formulas and data entries, like weather models have.

Compare two mortgages: same amount, same duration, slightly different rates:
$400,000 borrowed for 30 years at respectively 6% and 5.5%. What that zero point five percent can do? Over the life of the loan, it will turn into a 5% variation between the two initial situations, that is $45,800.

That is what the butterfly effect is all about. An tiny difference that in itself seems to have zero or little impact, can have dramatic consequences after several iterations of a repetitive process. This phenomena is observable in many domains in real life. Like how bad a document can become after successive copies or how being consistently late a few minutes makes someone fired one day.

Same is true with good habits and how having higher standards and demanding a little bit more of oneself on a regular basis can lead to a higher life overall.