Powers of Two

Place one gold piece in the first square, double the number in the second square, double again in the next square, and so on, filling all 64 squares of the board. How many gold pieces is that in total?

In real life, you should not try this indoors, since the stacks of coins will soon increase in height beyond any roof made by man. The board should also be sturdy, to support the tons of weight placed on it. Due to shearing winds at higher altitudes, the coins would need to be fused together pretty soon, turning from stacks into solid rods of gold.

Barely halfway through the board, at square 35, all gold ever refined by humankind so far, 180 thousand metric tons, would have been cast into coin-rods and placed on the same chessboard. Around the same time, the rods are no longer pressing all their weight against it, since their center of mass would be orbiting the Earth at ever higher altitudes. Eventually the rods would become tethers, their rotating inertia pulling away from the board game with more force than Earth’s gravity pressing them against it.

In the end, assuming enough gold is refined, the stack of coins in the final square would reach well into the Kuiper belt, beyond the orbit of Neptune. Even if the gold is welded into a solid rod, it would not stay straight, or even in one piece, for very long. Light would take over four hours to travel from one end to the other, and local pulling forces would not be able to balance out along the whole rod.

The whole exercise is of course just an educational story, meant to reveal the poor grasp of numbers and magnitude that human intuition is cursed with. It is not known where or by whom it was originally told, but most versions of it are told with grains of wheat or rice, or salt, all things that come in small sizes.

But what if instead of doubling the amount in each square, you were to halve the amount of gold in each square? Start with a single gold coin, cut it in half, move the half into the next square, then a quarter, and so on? Would it be possible to have a piece of gold in each 64 squares of the chessboard?

Gold is soft and malleable enough to cut, at least when pure (this is why real gold coins are usually not pure gold; but let’s assume purity here, for the sake of the argument). In fact, with stone-age tools it is possible to beat gold to such thinness that its edge becomes invisible: it is thinner than the shortest wavelength of visible light. Already on the second row of the chessboard it becomes necessary to use gold leaf instead of nuggets. This is good for the visibility of the remaining gold; it also helps that gold foil naturally attaches itself to the underlying surface, making it less likely that a light breeze blows away the invisible gold dust in the lower squares.

Gold was considered the noblest substance by the ancients, embodying the idea of material permanence. If gold can be beaten so thin that sunlight is visible through it, why not beat it even thinner, until it becomes as thin and light as the Emperor’s new clothes? What is the internal force that makes thinning the foil ever harder, the thinner it gets? And could we, in theory, continue dividing the gold forever, if we had the means to see it and the power to thin it?

In the middle of 19th century, the great experimentalist Michael Faraday attached gold leaf to glass plates, and studied it under the most powerful microscopes available. He was looking for, among other things, hints of any fine structure, such as the existence of atoms or molecules, strongly suggested by the works of Dalton, Avogadro, Berzelius and others. In his Bakerian lecture from 1856 he writes

“Yet in the best microscope, and with the highest power, the leaf seemed to be continuous, the occurrence of the smallest sensible hole making that continuity at other parts apparent, and every part possessing its proper green colour. How such a film can act as a plate on polarized light in the manner it does, is one of the queries suggested by the phenomena which requires solution.”

Faraday had a knack, and was already famous, for making unusual experiments and finding strange natural phenomena for other, more theoretical-minded scientists to then try to explain. It was only a few years later, in 1865, that Jacob Loschmidt referred to Faraday’s experiment, in a paper that, for the first time in history, made a reasonable estimation for the mass of a single atom: a trillionth of a milligram [he used trillion in the long scale, meaning 1018].

Applying Loschmidt’s estimation to the chessboard, assuming the coin is a solid 8 grams of Au, it could easily be divided 63 times, with still hundreds of gold atoms left even in the last square (8×1021 / 263). There is indeed plenty of room at the bottom, as Dr. Feynman said. Since then, we have of course made more accurate measurements of the atomic mass, but Loschmidt’s estimate was very close to the mark [the actual number of atoms in 8 grams of Au is about 2.4×1022, just three times higher].

We have had the knowledge about the approximate size of atoms for more than 150 years, and have made steady progress in the precision and accuracy of our instruments. But the sight of the apparently empty lower half of the chessboard really demonstrates how much about the physical world is hidden from the senses we were born with.

The gold leafs in squares 38-50 are of the diameter of typical living cells, and visible with a microscope. Optical microscopes become useless pretty soon after square 50, because the diameters become smaller than the wavelengths of visible light. All the complex biochemistry of life, with practically infinite variations of form, happens at a scale too small for us to see. But there is plenty of room for the variations; every view in a microscope is like choosing one asteroid in a galaxy cluster of star systems to look at.

Taking the hint from Feynman, the world-wide electronics industry has proceeded down the ladder, doubling the transistor count of production chips about every 18 months since the mid nineteen-sixties, by miniaturizing components. After fifty years, this so-called Moore’s Law has an unknown future, but the incredible impact to human society of affordable computing has been achieved by traversing just half-way down the chessboard of magnitude.

The problem of accuracy is not just with the instruments used, it is also in the amount of raw data needed to process the information. Every new bit of information doubles the number of possible combinations. To represent measurements of 22 digits of accuracy, for example the recent discovery of gravitational waves, more than 64 bits of precision is needed. For comparison, all the pictures in this post were created with blender, which uses single-precision floats internally, with just 24 bits of precision. This is enough for human vision, and requires less hardware.

Even with trillions of pixels in quadruple-precision accuracy, the human brain would not have the internal bandwidth to grasp both ends of the chessboard of magnitude at the same time. The most common way to display physics magnitudes is to use the zoom effect, as famously done in the Powers of Ten film in 1977. The zoom is a compromise, with the apparent motion as if traveling to other worlds, and no intuitive way to gauge the logarithmic speed of movement; but it seems no better way to convey differences in magnitude has been invented, so far.