The Black Swan: The Impact of the Highly Improbable

Visual arts: Most computer-generated objects are now based on some version of the Mandelbrotian fractal. We can also see fractals in architecture, paintings, and many works of visual art – of course, not consciously incorporated by the work’s creator.

Music: Slowly hum the four-note opening of Beethoven’s Fifth Symphony: ta-ta-ta-ta. Then replace each individual note with the same four-note opening, so that you end up with a measure of sixteen notes. You will see (or, rather, hear) that each smaller wave resembles the original larger one. Bach and Mahler, for instance, wrote submovements that resemble the larger movements of which they are a part.

Poetry: Emily Dickinson’s poetry, for instance, is fractal: the large resembles the small. It has, according to a commentator, “a consciously made assemblage of dictions, metres, rhetorics, gestures, and tones”.

Fractals initially made Benoit M. a pariah in the mathematical establishment. French mathematicians were horrified. What? Images? Mon dieu! It was like showing a porno movie to an assembly of devout Eastern Orthodox grandmothers in my ancestral village of Amioun. So Mandelbrot spent time as an intellectual refugee at an IBM research center in upstate New York. It was a f*** you money situation, as IBM let him do whatever he felt like doing.

But the general public (mostly computer geeks) got the point. Mandelbrot’s book The Fractal Geometry of Nature made a splash when it came out a quarter century ago. It spread through artistic circles and led to studies in aesthetics, architectural design, even large industrial applications. Benoit M. was even offered a position as a professor of medicine! Supposedly the lungs are self-similar. His talks were invaded by all sorts of artists, earning him the nickname the Rock Star of Mathematics. The computer age helped him become one of the most influential mathematicians in history, in terms of the applications of his work, way before his acceptance by the ivory tower. We will see that, in addition to its universality, his work offers an unusual attribute: it is remarkably easy to understand.

A few words on his biography. Mandelbrot came to France from Warsaw in 1936, at the age of twelve. Owing to the vicissitudes of a clandestine life during Nazi-occupied France, he was spared some of the conventional Gallic education with its uninspiring algebraic drills, becoming largely self-taught. He was later deeply influenced by his uncle Szolem, a prominent member of the French mathematical establishment and holder of a chair at the College de France. Benoit M. later settled in the United States, working most of his life as an industrial scientist, with a few transitory and varied academic appointments.

The computer played two roles in the new science Mandelbrot helped conceive. First, fractal objects, as we have seen, can be generated with a simple rule applied to itself, which makes them ideal for the automatic activity of a computer (or Mother Nature). Second, in the generation of visual intuitions lies a dialectic between the mathematician and the objects generated.

Now let us see how this takes us to randomness. In fact, it is with probability that Mandelbrot started his career.

A Visual Approach to Extremistan/Mediocristan

I am looking at the rug in my study. If I examine it with a microscope, I will see a very rugged terrain. If I look at it with a magnifying glass, the terrain will be smoother but still highly uneven. But when I look at it from a standing position, it appears uniform – it is almost as smooth as a sheet of paper. The rug at eye level corresponds to Mediocristan and the law of large numbers: I am seeing the sum of undulations, and these iron out. This is like Gaussian randomness: the reason my cup of coffee does not jump is that the sum of all of its moving particles becomes smooth. Likewise, you reach certainties by adding up small Gaussian uncertainties: this is the law of large numbers.

The Gaussian is not self-similar, and that is why my coffee cup does not jump on my desk.

Now, consider a trip up a mountain. No matter how high you go on the surface of the earth, it will remain jagged. This is even true at a height of 30,000 feet. When you are flying above the Alps, you will still see jagged mountains in place of small stones. So some surfaces are not from Mediocristan, and changing the resolution does not make them much smoother. (Note that this effect only disappears when you go up to more extreme heights. Our planet looks smooth to an observer from space, but this is because it is too small. If it were a bigger planet, then it would have mountains that would dwarf the Himalayas, and it would require observation from a greater distance for it to look smooth. Likewise, if the planet had a larger population, even maintaining the same average wealth, we would be likely to find someone whose net worth would vastly surpass that of Bill Gates.)

FIGURE 11

Apparently, a lens cap has been dropped on the ground. Now turn the page.

Figures 11 and 12 illustrate the above point: an observer looking at the first picture might think that a lens cap has fallen on the ground.

Recall our brief discussion of the coast of Britain. If you look at it from an airplane, its contours are not so different from the contours you see on the shore. The change in scaling does not alter the shapes or their degree of smoothness.

Pearls to Swine

What does fractal geometry have to do with the distribution of wealth, the size of cities, returns in the financial markets, the number of casualties in war, or the size of planets? Let us connect the dots.

FIGURE 12

The object is not in fact a lens cap. These two photos illustrate scale invariance: the terrain is fractal. Compare it to man-made objects such as a car or a house. Source: Professor Stephen W. Wheatcraft, University of Nevada Reno.

The key here is that the fractal has numerical or statistical measures that are (somewhat) preserved across scales – the ratio is the same, unlike the Gaussian. Another view of such self- similarity is presented in Figure 13. As we saw in Chapter 15, the superrich are similar to the rich, only richer – wealth is scale independent, or, more precisely, of unknown scale dependence.

FIGURE 13: THE PURE FRACTAL STATISTICAL MOUNTAIN

The degree of inequality will be the same in all sixteen subsections of the graph. In the Gaussian world, disparities in wealth (or any other quantity) decrease when you look at the upper end – so billionaires should be more equal in relation to one another than millionaires are, and millionaires more equal in relation to one another than the middle class. This lack of equality at all wealth levels, in a nutshell, is statistical self-similarity.

In the 1960s Mandelbrot presented his ideas on the prices of commodities and financial securities to the economics establishment, and the financial economists got all excited. In 1963 the then dean of the University of Chicago Graduate School of Business, George Shultz, offered him a professorship. This is the same George Shultz who later became Ronald Reagan’s secretary of state.

Shultz called him one evening to rescind the offer.

At the time of writing, forty-four years later, nothing has happened in economics and social science statistics – except for some cosmetic fiddling that treats the world as if we were subject only to mild randomness – and yet Nobel medals were being distributed. Some papers were written offering “evidence” that Mandelbrot was wrong by people who do not get the central argument of this book – you can always produce data “corroborating” that the underlying process is Gaussian by finding periods that do not have rare events, just like you can find an afternoon during which no one killed anyone and use it as “evidence” of honest behavior. I will repeat that, because of the asymmetry with induction, just as it is easier to reject innocence than accept it, it is easier to reject a bell curve than accept it; conversely, it is more difficult to reject a fractal than to accept it. Why? Because a single event can destroy the argument that we face a Gaussian bell curve.