The Beauty of Fractals: Six Different Views


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Introduction

A good boundary is one made up of elements lying on the ragged edge of plausibility. They are outcomes that might conceivably happen but make one uncomfortable even to contemplate.

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The most commonly considered outliers are wild cards. My favorite example of a wild card, because its probability is so uncertain and its impact so great, is finding radio evidence of intelligent life somewhere else in the universe. Nobody knows if we will ever receive a message radio astronomers have been listening since the late s , but if we did, it would send a vast and unpredictable tremor through the zeitgeist. The tricky part about wild cards is that it is difficult to acknowledge sufficiently outlandish possibilities without losing your audience.

The problem—and the essence of what makes forecasting hard—is that human nature is hardwired to abhor uncertainty. We are fascinated by change, but in our effort to avoid uncertainty we either dismiss outliers entirely or attempt to turn them into certainties that they are not.

This is what happened with the Y2K problem in the final years before January 1, Opinions clustered at the extremes, with one group dismissing the predictions of calamity and another stocking up on survival supplies. The correct posture toward Y2K was that it was a wild card—an event with high potential impact but very low likelihood of occurrence, thanks to years of hard work by legions of programmers fixing old code. The result of the Y2K nonevent was that many people concluded they had been the victims of someone crying Y2K wolf, and they subsequently rejected the possibility of other wild cards ever coming to pass.

After all, airliners flown into monuments were the stuff of Tom Clancy novels in the s inspired by Clancy, I helped write a scenario for the U. Air Force in that opened with a plane being flown into the Pentagon , and it was widely known that the terrorists had a very personal antipathy toward the World Trade Center. Yet the few people who took this wild card seriously were all but dismissed by those who should have been paying close attention.

The Edge of Infinity - Mandelbrot Fractal Zoom (e2011) (4k 60fps)

The result of the Y2K nonevent was that many people subsequently rejected the possibility of other wild cards ever coming to pass. Human nature being what it is, we are just as likely to overreact to an unexpected wild card by seeing new wild cards everywhere. Above all, ask hard questions about whether a seeming wild card in fact deserves to be moved closer to the center.

Change rarely unfolds in a straight line. The most important developments typically follow the S-curve shape of a power law: Change starts slowly and incrementally, putters along quietly, and then suddenly explodes, eventually tapering off and even dropping back down.

But it will flatten eventually, certainly with regard to silicon circuit density. This distinction reveals another important feature of S curves, which is that they are fractal in nature. Very large, broadly defined curves are composed of small, precisely defined and linked S curves. For a forecaster, the discovery of an emergent S curve should lead you to suspect a larger, more important curve lurking in the background.

Miss the larger curve and your strategy may amount to standing on a whale, fishing for minnows. The art of forecasting is to identify an S-curve pattern as it begins to emerge, well ahead of the inflection point. The tricky part of S curves is that they inevitably invite us to focus on the inflection point, that dramatic moment of takeoff when fortunes are made and revolutions launched. His discovery falls at the inflection point of Western exploration.

Columbus was not the first fifteenth-century explorer to go to the New World—he was the first to make it back, and he did so at a moment when his discovery would land like a spark in the economic tinder of a newly emergent Europe and launch thousands upon thousands of voyages westward. Ironically, forecasters can do worse than ordinary observers when it comes to anticipating inflection points.

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Six Rules for Effective Forecasting

Ordinary folks are simply surprised when an inflection point arrives seemingly out of nowhere, but innovators and would-be forecasters who glimpse the flat-line beginnings of the S curve often miscalculate the speed at which the inflection point will arrive. As futurist Roy Amara pointed out to me three decades ago, there is a tendency to overestimate the short term and underestimate the long term. Our hopes cause us to conclude that the revolution will arrive overnight.

Then, when cold reality fails to conform to our inflated expectations, our disappointment leads us to conclude that the hoped-for revolution will never arrive at all—right before it does. One reason for the miscalculations is that the left-hand part of the S curve is much longer than most people imagine. Television took 20 years, plus time out for a war, to go from invention in the s to takeoff in the early s. Even in that hotbed of rapid change, Silicon Valley, most ideas take 20 years to become an overnight success. The Internet was almost 20 years old in , the year that it began its dramatic run-up to the s dot-com eruption.

So having identified the origins and shape of the left-hand side of the S curve, you are always safer betting that events will unfold slowly than concluding that a sudden shift is in the wind. Once an inflection point arrives, people commonly underestimate the speed with which change will occur.

The fact is, we are all by nature linear thinkers, and phenomena governed by the sudden, exponential growth of power laws catch us by surprise again and again. Even if we notice the beginning of a change, we instinctively draw a straight line diagonally through the S curve, and although we eventually arrive in the same spot, we miss both the lag at the start and the explosive growth in the middle.

The golden ratio has spawned a beautiful new curve: the Harriss spiral

Timing, of course, is everything, and Silicon Valley is littered with the corpses of companies who mistook a clear view for a short distance and others who misjudged the magnitude of the S curve they happened upon. Also expect the opportunities to be very different from those the majority predicts, for even the most expected futures tend to arrive in utterly unexpected ways.

In the early s, for example, PC makers predicted that every home would shortly have a PC on which people would do word processing and use spreadsheets or, later, read encyclopedias on CDs. But when home PC use did finally come about, it was driven by entertainment, not work, and when people finally consulted encyclopedias on-screen a decade after the PC makers said they would, the encyclopedias were online. The established companies selling their encyclopedias only on CD quickly went out of business.

The entire portion of the S curve to the left of the inflection point is paved with indicators—subtle pointers that when aggregated become powerful hints of things to come. A classic example is the first sales of characters and in-game objects from the online game EverQuest on eBay in the late s.

Through the avatars, members engage in social activities, including the creation and sale of in-world objects in a currency Linden dollars that can be exchanged for real dollars through various means. Where it ends is still uncertain, but it is unquestionably a very large S curve.

Terms indexed in Chaos and Fractals

More often than not, indicators look like mere oddball curiosities or, worse, failures, and just as we dislike uncertainty, we shy away from failures and anomalies. Its earliest graphical antecedent was Habitat, an online environment developed by Lucasfilm Games in Though nongraphical MUDs multiple user dimensions were a cultish niche success at the time, Habitat quickly disappeared, as did a string of other graphical MUDs developed in the s and s. Then the tide turned in the late s, when multiplayer online games like EverQuest and Ultima started to take off.

It was just a matter of time before the S curve that had begun with Habitat would spike for social environments as well as for games. So although the explosive success of Second Life came as a considerable surprise to many people, from a forecasting perspective it arrived just about on time, almost 20 years after Habitat briefly appeared and expired. As the Second Life example illustrates, indicators come in clusters.

Department of Defense to design robots that could compete in a mile-plus race across the Mojave Desert. Most of the robots died in sight of the starting line, and only one robot got more than seven miles into the course. But just 19 months later, at the second Grand Challenge, five robots completed the course. Around the same time I noticed a sudden new robot minicraze popping up that many people dismissed as just another passing fad. What was odd was that my friends with Roombas were as wildly enthusiastic about these machines as they had been about their original K Macs—and being engineers, they had never before shown any interest in owning, much less been excited by, a vacuum cleaner.

Alone, this is just a curious story, but considered with the Grand Challenge success, it is another compelling indicator that a robotics inflection point lies in the not-too-distant future. One indicator: Roomba owners today can even buy costumes for their robots! One of the biggest mistakes a forecaster—or a decision maker—can make is to overrely on one piece of seemingly strong information because it happens to reinforce the conclusion he or she has already reached.


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This lesson was tragically underscored when nine U. The bearing placed his ship, the Delphy , north of its dead reckoning position.

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Convinced that his dead reckoning was accurate, the commander reinterpreted the bearing data in a way that confirmed his erroneous position and ordered a sharp course change towards the rapidly approaching coast. Nine ships followed the disastrous course. Meanwhile, the deck officers on the Kennedy , the 11th boat in the formation, had concluded from their dead reckoning that they in fact were farther north and closer to shore than the position given by the Delphy. The skipper was skeptical, but the doubt the deck officers raised was sufficient for him to hedge his bets; an hour before the fateful turn he ordered a course change that placed his ship several hundred yards to the west of the ships in front of them, allowing the Kennedy and the three trailing destroyers to avert disaster.

He hedged their bets and, therefore, saved the ship. In forecasting, as in navigation, lots of interlocking weak information is vastly more trustworthy than a point or two of strong information. The problem is that traditional research habits are based on collecting strong information. And once researchers have gone through the long process of developing a beautiful hypothesis, they have a tendency to ignore any evidence that contradicts their conclusion.

This inevitable resistance to contradictory information is responsible in no small part for the nonlinear process of paradigm shifts identified by Thomas Kuhn in his classic The Structure of Scientific Revolutions. Once a theory gains wide acceptance, there follows a long stable period in which the theory remains accepted wisdom.

All the while, however, contradictory evidence is quietly building that eventually results in a sudden shift. Good forecasting is the reverse: It is a process of strong opinions, weakly held.

The Beauty of Fractals: Six Different Views The Beauty of Fractals: Six Different Views
The Beauty of Fractals: Six Different Views The Beauty of Fractals: Six Different Views
The Beauty of Fractals: Six Different Views The Beauty of Fractals: Six Different Views
The Beauty of Fractals: Six Different Views The Beauty of Fractals: Six Different Views
The Beauty of Fractals: Six Different Views The Beauty of Fractals: Six Different Views

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