With the recent release of “Terminator: Salvation” I took some time to re-familiarize myself with the franchise by watching the first three Terminator films. While watching Terminator 2 I was enlightened to something I had not noticed before. The T-800 says to John Connor “Skynet began to learn at a geometric rate”. Of course I’d heard this line before, but I didn’t really understand it, until now that I have a background in mathematics.

Certainly “geometric rate” sounds really cool, but several meanings could be inferred. The movie seems to imply that in this case a geometric rate is an exponential rate. Meaning that Skynet learned a little at first, then a lot. Of course a geometric rate may also mean something totally different, in fact it could be quite the opposite.

In calculus a geometric series is an example of an infinite series with a finite sum. To visualize what that means, imagine a 1×1 square, it’s surface area is 1. Cut the square in half, you now have two surfaces areas of 1/2 which sum up to 1. Cut one of those halves in half, you then have a rectangle with area 1/2 and two squares with area 1/4. Proceed cutting one of the smaller rectangles in half. You can do so forever. You’ll essential have a series of rectangles and squares with the areas as follows 1/2, 1/4, 1/8, 1/16, 1/32, … 1/(2^n). Even though you have an infinite series, the sum of all those elements is still the area of the square, which is 1. The example below demonstrates this with a ruler.

With this understanding of a geometric series it is possible that the geometric rate at which Skynet was learning, was a geometric series with a finite sum. In which case Skynet learned a whole lot right off the bat, and then continued to learn, but not nearly as much as it learned at it’s onset, and the total sum of all knowledge that Skynet will learn is limited to a finite sum.

Indeed this is not inconsistent with the film. Skynet did destroy most of mankind right away with the technology it had. Later on it developed some more knowledge about destroying humanity, but not nearly at the rate at which it had when it first went online.