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May 2017

Change: The beguiling nature of exponential curves

By | Startup general interest | No Comments

Consider the three charts above. They are all representations of the same exponential function where the Y value is equal to two times it’s previous value. The first chart shows the data series for the first twenty values, the second chart shows the data series for values on through ten, and the third chart shows the data series for values eleven through twenty. Notice that all the charts look similar and that the second and third charts are virtually identical.

The takeaway: when you are on an exponential curve the trajectory looking forward is the same at any point on the curve.

For me, at least, this is highly counter-intuitive. I think that’s because the mind sees change in absolute rather than relative terms. I know that things like processing power, storage, bandwidth (fixed and wireless), solar, and genome sequencing have been improving exponentially for some time so I expect to feel the change to a much greater extent today than I used to, and by extension my natural inclination is to expect that change will get bewilderingly fast in the next decade or two.

However, when you think it through properly our experience of change will remain the same. There will be a doubling each year (or halving, or whatever the exponential function is).

This is all very abstract. Let me try and make it real. When I think about Moore’s law and the acceleration in computer power, it feels that the change should be faster than it was when I was a kid. I remember when I was 8 the Sinclair ZX81 was released and then the big news a year later was when the Sinclair ZX Spectrum came out. Memory went from 1k to 16k and there was colour! More importantly for me at the time, the games were much better :). That was a notable advance, however, for the next few years after that there were no really major steps forward. When I compare that to progress in computing over the last few years or so it seems to me we have seen a similar rate of change, although we have to look to the cloud services we use rather than our personal devices to see the change. I will call out Uber and the Amazon Echo as two new things that are changing the way we go about our lives in a way of similar significance to what those Sinclair computers did in the 1980s.

I should say at this point that in the real world exponential curves don’t continue for ever. We get S-curves which closely mimic exponential curves in the beginning, but then tail off after a while often as new technologies hit physical limits which prevent further progress. What seems to happen in practice is that some new technology emerges on its own S-curve which allows overall progress to stay on an something approximating an exponential curve.

The chart above shows interlocking S-curves for change in society over the last 6,000 years. That’s as macro as it gets, but if you break down each of those S-curves they will in turn be comprised of their own interlocking S-curves. The industrial age, for example, was kicked off by the spinning jenny and other simple machines to automate elements of the textile industry, but was then kicked on by canals, steam power, trains, the internal combustion engine, and electricity. Each of these had it’s own S-curve, starting slowly, accelerating fast and then slowing down again. And to the people at the time the change would have seemed as rapid as change seems to us now. It’s only from our perspective looking back that change seems to have been slower in the past. Once again, that’s only because we make the mistake of thinking in absolute rather than relative terms.

I’m writing this now because I only just created the charts at the top of the page. The mathematical side of my brain has known for some time now that when you are on an exponential curve the trajectory going forward is always the same, but there was some other part of my mind that didn’t quite believe it. If you’ve reached this far in the post you have seen my mind in action getting to the bottom of this piece of inner conflict. I think I see the world a little more clearly now. I hope you do too!