Atoms or gunk? The chessboard rice-halving challenge

We know today that every physical thing consists of atoms, particles so small that we cannot see them with our eyes. The idea of atoms was invented thousands of years before the particles named after them were discovered by scientists. For most people, seeing is believing, so how did the early atomists argue about the existence of the practically invisible?

The numbers of atoms in small everyday objects, like drops of water, are so ridiculously large that the hunter-gatherer mind has no intuitive grasp of them. For example, if I had a gold coin for every atom in one gold coin, they would fill so many olympic-size swimming pools that it would be impossible to walk, or even run, past all of them in a single human lifetime.

Start from the first square, cut the rice grain in half, take the smaller half and move it to the next square. Continue until you have a piece of rice in every square.

An ancient story about grains of rice doubling each step on a chessboard can be changed slightly, so that it is about splitting things in halves, instead of doubling them. All that is needed is a small object, and a sharp knife. Start from the first square, cut the rice grain in half, take the smaller half and move it to the next square. Continue through the squares until you have a piece of rice in every square.

The halving version of the chessboard challenge is much more practical than the doubling challenge, since less rice is needed. It is also quick and easy to perform in practice, and if all the rules are followed (“always take the smaller half”), quite impossible to perform for all 64 squares.

Here is a quick attempt with some modeling clay. After 16 steps, the pieces become smaller than the flecks caught in the knife blade.chessboard1

In theory, it is possible to cut a 1 oz gold coin in half 78 times before reaching individual gold atoms. In practice, filling even the 64 squares of a chessboard would still require more than a steady hand and sharp eye (by square 50 the nuggets become smaller than the wavelength of visible light, making optical microscopes useless). Halfway down the board, the remaining pieces would probably need to be attached to some larger object, to keep them from floating away with other dust. (But when an object is attached to another, do they not then together form a new object? According to atomism, yes they do.)

It is of course much easier to dilute the gold coin with another substance than to cut it into individual atoms. Metals can be worked together by stretching and folding repeatedly, without actually melting them into liquid form. Applying pressure, such as with a smith’s hammer, stretches the material nicely when it is at the right temperature.

This picture demonstrates the principle, again using colored modeling clay. It has a doughy consistency, always somewhere between fluid and solid, and easy to mold with hands.folditIf the component pieces were not consisting of atoms that attach to other atoms, then the process could be reversed, by separating and unfolding, even after being stretched and folded together 16 times, or even 64.

This number of foldings is only possible with a material that stretches, folding a paper more than 7-8 times requires a very large piece of paper. (Baking also provides practical examples of repeated folding and stretching. There is a natural limit to how many times dough can be folded to make the finest noodles.)

The patterns caused by repeated folding and stretching are similar to what can be seen in Damascus or Wootz steel. Although the original methods were kept secret and lost, ingots are usually folded a number of times when worked into a blade. Mixing iron with impurities, mainly carbon, then folding repeatedly produces a material that is stronger than any of its component materials. How can that be explained without atoms?

The historical origin of the game of chess happened in the same place where atomism was first mentioned, somewhere in India. The same region was also first to produce these steel blades, and I wouldn’t be surprised if they also thought of the rice-halving challenge, and presented it as an argument for atomism.

I have no knowledge of any reference to this kind of challenge actually being presented back then, and if anyone reading this can provide links I would be grateful. The closest thing that I know are the various paradoxes of Zeno, where distance or time is repeatedly divided. Zeno does not present his paradoxes in the context of atomism, either for or against. Then again some atomists apply the principle to matter only, and therefore would accept empty space/time as being infinitely divisible “gunk“.