Uday I/O thoughts on computers and things

Predicting the Future and Exponential Growth

“How many times would I have to fold a sheet of paper for the height of the folded paper to reach the moon?”

Human beings have terrible intuition for exponential growth. If I asked you how many times you would have to fold a single sheet of US Letter paper to reach the moon, it would be difficult to intuitively comprehend that it only takes twenty folds to reach Mount Everest, forty-two folds to the moon, and fifty to reach the sun.

Compound interest is the eighth wonder of the world. He who understands it, earns it … he who doesn’t … pays it.” –Albert Einstein

Often times when looking at the future, we use the projection of the past to predict the outcome of the future. This suggestion is generally not understood properly as due to the natural linearity of time, people are more likely to look at the next five years assuming the growth will march linearly like the past five years.

The truth is that even from the last five years to now, growth particularly technological growth is never linear and is always exponential.

The Apollo Guidance Computer, for example, was built of 4100 integrated circuits, each of which was a 3 input gate for a total of approximately 12,300 transistors – having its performance clocking in at 41.6 instructions per second.

An iPhone 6 has 3.36 billion instructions per second meaning an iPhone 6 is 80 million times faster than Apollo on just instructions per second and around 120 million times faster than the guidance computer that put Neil Armstrong and Buzz Aldrin on the moon. A single iPhone 6 could theoretically guide 120 million Apollo rockets at the same time (an iPhone 6S is 70% faster than an iPhone 6). This also doesn’t include the fact that a 64 bit processor can use less operations to process more complex computations - so 120 million times faster is definitely an understatement.

So how can we explain this type of growth?

This level of exponential growth can be broadly defined by stating that the future that we build in N years will have the tools of N - 1 years where N is years in the future. More importantly, the rate of development is compounded, so the future we build on the Nth year is a result of exponentially growing by multiplying 2(N-1)/1.5 by 2(N-1)/1.5 to reach 2N/1.5.

This value assumes that the technological output doubles every eigtheen months. This is true in observed Moore’s Law which shows the exponential increase of the number of transistors on integrated circuits.

Moore's Law

A similar growth rate also applied to the increase in computational capacity per second per computer in a given year.

Computational Capacity

On a small scale this does not seem like much - moving from the value 21/1.5 (1.58) to 22/1.5 (2.52) in technological output seems minor, but similar to folding a piece of paper to reach the moon - the compounded growth is where the magic happens.

Exponential Future Graph

We usually don’t notice exponential growth and expect that exponential growth in the future is something we will be surprised to see, but if we plot ourselves along an exponential growth curve and we zoom out, it is easy to see that there is exponential growth even though there appears to be a knee to the curve. By simply zooming out, the knee minimizes and we can physically remain in the same place despite when we zoom in being higher on the curve.

So in the future, the past will feel as natural as it does now, but growth will continue becoming more and more drastic relative to the present. Additionally, you can place yourself at the tip of the curve, and by simply adjusting the y-axis again, it is easy to see how incredibly fast we move from the top of one curve, to the bottom of the same curve by modifying our perspective. So although often people believe that growth is slowing down an easy way to prove this is to imagine how far in the past you would have to go to get someone to react strongly to the present – this in itself proves exponentiality.

Perhaps someone from 1990 coming to today would be shocked looking at the iPhone; to do the same in 1990, we would have to go exponentially far back to give someone the same level of awe that someone from 1990 would have here in 2015. This becomes more difficult and requires you to reach further back in time, eventually asking people from early civilizations to reach thousands of years back to shock and awe people from the past to achieve the same level of shock that someone from 1990 would get in 2015.

In simpler English, this rapid construction of the future happens because the tools that are built today for tomorrow make it easier to build the future the next day.

Unfortunately we are so bad at even conceiving what the tools of tomorrow are that more often than not, we make terrible attempts to predict the future today thinking in terms of the tools today.

Jean-Marc Côté in 1899 was, for example, hired to create a series of picture cards as inserts, according to Matt Noval from the Smithsonian magazine. The images were to depict how life in France would look in a century’s time. He probably didn’t see the computer vision revolution coming (or the computer for that matter).

France in 2000s Drawn from 1900s

The most useful heuristic I find to clarifying predictions about the so-called immediate future is what I call the “fifteen to twenty year fallacy”. Most people who cannot fathom exponential growth tend to use linear projections (y=x) on a future roughly “fifteen to twenty years” away. It is a safe enough distance that they can guess without feeling the pressure of having to be right, but close enough that they tend to project the tools of the present on this particular future. This is generally done by people who are unaware of the tools and the growth of a particular domain. Generally when I hear something like this I know that any prediction fifteen to twenty years away can generally be reevaluated as the 1.5 * log2 of number of years.

So when I read a report showing that autonomous vehicles will the hit the roads by 2030, I just do a little math.

if (15 < (futureYear - currentYear) < 20):
return actualYears = 1.5 * (math.log((futureYear - currentYear), 2))

So fully autonomous vehicles in a little under six years? I’d bet it will be even sooner.


  1. I do not believe that rate of growth of technology definitely doubles every eighteen months as stated above. Although my base could be wrong, I am sure that growth happens as f(x)=nx versus f(x) = nx.
  2. I do believe that despite the fact that Moore’s Law may be slowing down, the growth rate for general technology does hold.

Images From:

  1. Max Roser, Technological Progress
  2. Isaac Asimov, Futuredays: A Nineteenth Century Vision of the Year 2000
  3. The Internet of Things and Transportation