It is not clear whether it was rice or wheat. We're also not sure of the origin of the story, for there are many versions. But it goes something like this: A king was presented with a beautiful game of chess by its inventor. So pleased was the king that he asked the inventor to name his own reward. The inventor modestly asked for some rice (or wheat). The exact quantity would be calculated through the simplest formula: put a grain on the first square, two on the second, four on the third, and so on doubling the number of grains until the sixty-fourth and last square. The king readily agreed, before realizing that he had been deceived. By mid-chessboard, his castle was barely big enough to contain the grains, and just the first square of the other half would again double that.

The story has been used by anyone from 13th century Islamic scholars to scientist/author Carl Sagan to social media videographers to explain the power of exponential sequences, where things begin small, very small, but then, once they start growing, they grow faster and faster—to paraphrase Ernest Hemingway: they grow slowly, then suddenly.

The idea of "exponential" and its ramifications ought to be better known and understood by everyone (and the chessboard fable is a useful metaphor) because we live in an exponential world. It has been the case for a while, actually. However, so far we have been in the first half of the board—and things are radically accelerating now that we're entering its second half.

The "second half of the chessboard" is a notion put forth by Ray Kurzweil in 1999 in his book The Age of Spiritual Machines. He suggests that while exponentiality is significant in the first half of the board, it is when we approach the second half that its impacts become massive, things get crazy, and the acceleration starts to elude most humans' imagination and grasp.

Fifteen years later, in The Second Machine Age, Andrew McAfee and Erik Brynjolfsson looked at Kurzweil's suggestion in relation to Moore's law. Gordon Moore is a cofounder of Fairchild and Intel, two of the pioneering companies of Silicon Valley. Reflecting on the first few years of development of silicon transistors, in 1965 Moore made a prediction that computing power would double roughly every eighteen- to twenty-four months for a given cost (I'm simplifying here). In other words, that it would grow exponentially. His prediction held for decades, with huge technological and business impacts, although the pace has been slowing down a bit in recent years—it's worth mentioning that Moore's was an insight, not a physical law, and that we're likely moving away from transistors to a world of quantum computing, which relies on particles instead to perform calculations.

McAfee and Brynjolfsson reckon that if we put the starting point of Moore's law in 1958, when the first silicon transistors were commercialized, and we follow the exponential curve, in digital technology terms we entered the second half of the chessboard sometime around 2006 (for context, consider that the first mapping of the human genome was completed in 2003, the operating systems for our smartphones of today were launched in 2007, the same year IBM's Deep Blue beat Garry Kasparov in chess, while scientists at Yale created the first solid-state quantum processor in 2009).

Hence we find ourselves right now somewhere in the first, maybe in the second square of the second half of the chessboard. This helps make sense of the dramatically fast advances that we see happen in science and technology, from smartphones and language translation and the blockchain to big analytics and self-driving vehicles and artificial intelligence, from robotics to sensors, from solar cells to biotech to genomics and neuroscience and more.

While each of these fields is by itself growing exponentially, their combinatorial effect—the accelerating influence that each has on others is prodigious. Add to that the capacity for self-improvement of artificial intelligence systems, and we are talking about almost incomprehensible rates of change.

To stay with the original metaphor, this is what entering the second half of the chessboard means: Until now we were accumulating rice grains at an increasingly fast pace, but we were still within the confines of the king's castle. The next squares will inundate the city, and then the land and the world. And there are still thirty-two squares to go, so this is not going to be a brief period of transformation. It is going to be a long, deep, unprecedented upheaval. These developments may lead us to an age of abundance and a tech-driven renaissance, as many claim and/or hope, or down an uncontrollable dark hole, as others fear.

Yet we still live for the most part in a world that doesn't understand, and is not made for, exponentiality. Almost every structure and method we have developed to run our societies—governments, democracy, education and healthcare systems, legal and regulatory frameworks, the press, companies, security and safety arrangements, even science management itself—is designed to function in a predictable, linear world, and it interprets sudden spikes or downturns as crisis. It is therefore not surprising that the exponential pace of change is causing all sorts of disquiet and stresses, from political to social to psychological, as we are witnessing almost daily.

How do we learn to think exponentially without losing depth, careful consideration and nuance? How does a society function in a second-half-of-the-chessboard reality? What do governance and democracy mean in an exponential world? How do we rethink everything from education to legal frameworks to notions of ethics and morals?

It starts with a better understanding of exponents and of the metaphor of the "second half of the chessboard." And with applying "second-half thinking" to pretty much everything.