

People will happily tell you that the growth of their website is exponential - but really they mean sigmoidal. Puzzled? We'll explain..

Those graphs beloved of people showing how fast their sales, website users or collections of foil-wrapped biscuits are growing keep being described as "exponential". But the truth is that nothing in real life can keep growing exponentially forever - not even the universe.

Instead, as Tim O'Reilly points out, we ought to be saying that this growth is sigmoidal. That's S-shaped to you, sir with the dyscalculia:

Over Thanksgiving dinner, Saul Griffith was complaining about the lack of mathematical literacy among people who should know better. "Take all that talk about the exponential growth of various web sites. Don't people realize that those curves are actually sigmoidal?"

We've, um, borrowed Tim's graphs below, themselves borrowed from Wikipedia, which show the difference:
exponential vs. linear and cubic curves Exponential, linear and cubic curves

sigmoidal curve A sigmoidal curve.


Which means that the more important question to ask wehn someone proudly starts their presentation and points to their "exponential" growth is to put your hand up and ask "when do you think the inflexion will come? What factors might cause an earlier inflexion?" (In the past the answers used to be "As soon as Microsoft enters the market" and "I think I just answered that", but now it's become "As soon as Google enters the market.")

O'Reilly also notes that the importance of correct modelling to making accurate predictions about pretty much everything:

"...as Saul noted a few moments later, most of these curves aren't even sigmoidal, they are sinusoidal. (This is, incidentally, why Ray Kurzweil is most likely wrong about the singularity.)"

The "singularity" being the point when, Kurzweil says,


And that certainly sounds bad, especially if you imagine Doctor Who saying it. It would take more than a sonic screwdriver and Kylie Minogue to fix that, you know.