Introduction to Exponentials

V2.1 September 2016

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1 Paenza, A. (n.d.). How folding paper can get you to the moon - Adrian Paenza. Retrieved June 24, 2016, fromhttp://ed.ted.com/lessons/how-folding-paper-can-get-you-to-the-moon

When you fold a common piece of paper roughly 0.005 cm thick (5/1000th of 1 centimeter), the paper will be just over 1 cm thick after the 8th folding. Yet after only 42 foldings, that piece of paper is thick enough to reach the moon! And after just one more fold, the 43rd doubling, it reaches all the way to the moon and back.1 The doublings beyond that start to become difficult for us to conceptualize – in only 94 foldings, the paper would be the length of the observable universe. Exponential growth is both deceptive and astonishing, as the doubling of small numbers will inevitably lead to numbers that outpace our brain’s intuitive sense.

This is meaningful because we as humans tend to overestimate what can be achieved in the short term, but vastly underestimate what can be achieved in the long term.

Exponential Thinking

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The Law of Accelerating Returns:

The rate of progress in any evolutionary learning environment (a system that learns via trial and error over time) increases exponentially.

The more advanced a system that improves through iterative learning becomes, the faster it can progress.

Advances breed faster advances, which we describe as the Law of Accelerating Returns.

Law of Accelerating Returns Fast Facts2

by Ray Kurzweil

● The rate of progress of an evolutionary process increases exponentially over time.

● Technological evolution is one such evolutionary process.

● A specific paradigm (a method or approach to solving a problem, such as shrinking transistors on an integrated circuit as an approach to making more powerful computers) provides exponential growth until the method exhausts its potential.

● When this happens, a paradigm shift (a fundamental change in the approach) occurs, which enables exponential growth to continue.

2 http://www.kurzweilai.net/the-law-of-accelerating-returns

Contrast this with the following:

Exponential: If we picture 30 doubling steps (1, 2, 4, 8, 16, 32, etc.), our brains have a difficult time understanding the magnitude of that trip. It’s more than 1 billion meters (1,073,741,824 meters) or roughly 26 times around planet Earth.

It’s the gap between linear and exponential that is at the heart of why the future is so uncertain and the present is so unbelievably surprising.

Humans are not equipped to process exponential growth. Our intuition is to use our assessment of how much change we’ve seen in the past to predict how much change we’ll see going forward. We tend to assume a constant rate of change.

Linear: If we picture 30 linear steps (measured in meters) in our minds (1, 2, 3, 4, etc.), our brains have an easy time understanding that we’d arrive 30 meters away at the end of that trip.

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The Law of Accelerating Returns:

2 http://www.kurzweilai.net/the-law-of-accelerating-returns

Our computing power has exponentially increased in price performance for more than a century.

Moore’s Law is the observation that the number of transistors per square inch on integrated circuits has doubled every 18 months since they were invented

in 1958 and the prediction that this trend would continue into the foreseeable future. Moore’s Law is only applicable to the most recent paradigm of computing, however, the exponential increase of price performance holds true across all five of the paradigms of computing depicted in this chart.

Exponential Growth of Computing Power

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3 Smart, John. "Singularity Watch." Singularity Watch. N.p., 2002. Web. 18 June 2016.

“This is the most important graph in all of technology and business” - Steve Jurvetson, Partner, Draper Fisher Jurvetson

include quantum computing (leveraging the very effects that are bringing an end to Moore’s Law) and 3D self-organizing molecular circuits.

Each version of computing is less human-dependent than the last. Consider that the first paradigm, electromechanical, included human labor. Rooms of people, usually women, called “calculators” were involved in processing information. And since 1978, software (and thus computing) has aided in the design of the next generation of computing.3 Faster computers have and will continue to bring about faster, cheaper, more powerful, and less power-hungry computers.

The feature size of transistors is reaching its practical limit. Effects such as quantum tunneling begin to play a non-negligible role when each layer of the transistor is less than ~10 atoms across. Because of this, Moore's Law will come to an end, but this doesn’t mean the end of the Law of Accelerating Returns. The significance of Kurzweil’s chart is that after individual paradigms experience an exponential rate of advancement, they eventually level off. This creates a series of sigmoids (S-curves) that, when stacked on top of each other, continue an accelerated pace of development. No one is certain what the next paradigm will be, though candidates

Exponential Growth of Computing Power

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3 Smart, John. "Singularity Watch." Singularity Watch. N.p., 2002. Web. 18 June 2016.

Why Does Technology Accelerate?

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Virtual/augmented reality: Experiences themselves are now being turned into software with profound implications for the future of education and communication, among other fields.

Software enables a shift from a physical trial-and-error experimentation science, to a digital one. The time it takes to advance an area through testing and experimentation is dramatically reduced when it becomes an information-based domain.

Physical experimentation: In the past, NASA personnel would sketch thousands of pen-and-paper drawings to test the fluid dynamics of their aircraft, and then test for hours in physical wind tunnels.

Software experimentation: Today, NASA uses computational fluid dynamics to simulate their aircraft on computers, iterating hundreds or even thousands of designs at a fraction of the cost.

Computing is the engine driving the world forward today. As we digitize the world around us, we turn entire domains (areas like biotechnology, communication, health and medicine, energy systems, etc.) into information sciences.

When something becomes an information science, it becomes subject to exponential advances. When something analog becomes digital, it can leverage and ride on the back of the exponential curves of computing.

Biotechnology/bioinformatics: With the sequencing of the human genome, genomics and bioinformatics have turned biology into a computer science developing at an exponential rate.

Neuroscience: The increasing resolution of CT and MRI/functional MRI scans are allowing us to begin mapping the connectome, the 86 billion neurons and their connections within the brain. We’re moving into the age of computational neuroscience, and most of the innovation here is still to come.

Artificial intelligence: Many of today’s breakthroughs in machine learning are powered by exponential increases in processing power.

*NASA, Public Domain

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Why is it that we’re witnessing such a rapid explosion of innovation all over the planet today?

Throughout all of human history, the rate of innovation has been a function of how connected people are, and how easy it is for ideas to spread.

Urbanization: The first cities appeared between 9,000-10,000 years ago, and ever since, cities have produced more innovation per capita than anywhere else on earth. There are more patents and inventions per capita in cities because people can come together to share ideas. Today, for the first time in human history, more people live in cities than anywhere else.

The Internet: We tend to forget how recently the Internet was created. It was developed in only the last few decades and has forever transformed human history. Ideas are no longer separated by great distance, while the cost of replicating, copying, and sharing them is falling to near zero

Three billion new minds are expected to come online over the next five years, representing the largest potential innovation explosion humanity may ever see.

We’re only in the beginning stages of the Internet age. Most of the innovation we’re going to witness is still ahead of us.

Today’s Innovation Explosion

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Predicting Exponential Growth

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4 "Cutting the Cord." The Economist, 7 Oct. 1999. Web. 18 June 2016.5 Straubel, JB. Tesla Motors, 15 June 2015. Web. 18 June 2016.

This same plotline is taking place today with battery and renewable energy technology. Late in 2015, the Energy Information Alliance federal agency predicted that the total number of electric vehicles with a range of 200 miles per charge would not exceed 1,000 - by the year 2040. A few months later, Tesla announced its Model 3, a car which has a battery range of 215 miles per charge, and it received more than 300,000 pre-orders in a week! Ouch.5

In biotechnology, the Human Genome Project is a great example of exponential growth. Seven years into the 10-year project, only 1% of the genome had been sequenced. Mainstream critics complained that finishing the project could take 700 more years. However, the Human Genome Project and Celera Genomics ultimately published a draft of the human genome two years ahead of schedule, thanks in large part to DNA sequencing technology entering an exponential curve in 2008.

Internalizing the implications of exponential growth can present enormous business opportunities. Alvy Ray Smith co-founded Pixar in 1986. He realized that computers would eventually be able to produce feature-length animated films, but he also recognized that the company would have to bide its time building hardware instead of making movies until the capabilities came to fruition. As soon as the time was ripe, Pixar struck a deal with Disney to make Toy Story, which went on to earn more than $373 million worldwide.6 Disney would go on to buy Pixar in 2006 in a $7.4 billion all-stock deal.7

Human beings are notoriously bad at making forecasts with exponential growth in mind. No one knows for sure why our intuition for exponentials is so poor, but Kurzweil suggests that since our biological systems have evolved to perceive in a linear and sequential fashion, we better understand movement occurring at a fixed rather than an accelerating rate, and we thus apply this innate understanding to our predictions about growth. Exponentials only become dramatic over long timescales, and the initial doublings can be almost imperceptible.

In the early 1980s, AT&T asked McKinsey to predict how many mobile phones would be in use by the year 2000. Aware of the many challenges the industry faced (bulky devices, high cost of data usage, poor battery life, etc.), McKinsey predicted a total market size of 900,000 devices. By 1999, there were 900,000 people signing up for a mobile phone service every 3 days.4

Note that when McKinsey made this prediction, mobile phones were big, clunky things. It was the exponentially improving price-performance of the components inside the mobile phones that allowed for an explosive penetration and rate of user adoption.

6 "Toy Story (1995)". Box Office Mojo. Retrieved September 12th, 2016.7 La Monica, Paul R. (January 24, 2006). "Disney buys Pixar". CNN.

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Peter Diamandis constructed a framework to help navigate the exponential journey that a digital technology takes, known as Peter Diamandis’ 6 D’s. This framework explains what happens as an area becomes information-enabled:

Digitized – Once a technology is digitized it becomes an information science, and so we can use computers to manage it.

Deceptive – Exponential growth is hard to spot. At the beginning of most exponentially advancing environments, the early stages of development are almost imperceptible.

Disruptive – After the initial deceptive growth, the development of an exponentially advancing technology can make the previous paradigm effectively obsolete, out-performing it in both effectiveness and cost.

The 6 D’s

The 6 D’s

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Dematerialized – Items that were once large and unwieldy can now fit easily into our pockets. The miniaturization of sensors paired with digitization allows for the elimination of dedicated single-use physical devices. (Harvard’s Innovation Lab has illustrated this point with a beautiful video.)

Demonetized – GPS systems and high resolution video cameras were prohibitively expensive in the past, but almost anyone can access them now that they are applications and sensors included on and in your phone. The cost of producing and replicating software is dramatically cheaper than creating the physical version of it, and the economies of scale associated with the sensors allow them to become eminently affordable.

Democratized – Products, services, and information that were once only available to wealthy nations, research labs, or companies, are now becoming accessible by an ever-increasing percentage of the global population. If you can buy a cheap smartphone with an Internet connection, you have the same communications capabilities and access to the same exact platforms as a billionaire.

So What?!

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The future is notoriously difficult to predict. Our shortcomings in seeing and extrapolating the exponential trends that will shape the coming century sets the stage for us to experience perpetual futureshock.

The deceptive and then explosive growth of exponentials often leads us to believe that science fiction is far from being science fact. However, it is hard to overstate the staggering implications of the disruptive advances in genetic modification, augmented and virtual reality, artificial intelligence, and other exponentially advancing domains that will unfold in our lifetimes. Businesses, governments, and we as individuals will need to drastically modify our predictive models to adapt to this changing landscape.

The rate of change is only increasing.

Did we miss a key insight?

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