Chenying Liu

The square root of two is infamous in the history of mathematics as Pythagoras’ Constant or the ‘number that got a guy killed’ (if you don’t know the story, I strongly encourage you to check it out here), but that isn’t the subject of this article. Instead, I’m going to convince you that the cube root of two has a double personality as a bringer of death and an angel, making it just as interesting as it’s better-known sibling…

Our story begins in the 5th Century BC, on a pearl of an island in the Mediterranean. The people of Delos were happily enjoying their slice of paradise until disaster struck. A deadly plague arrived on the island and within a few short weeks it mercilessly destroyed thousands of lives (unfortunately in 2022, we are experiencing a plague once again). Panic spread and people began to believe that the catastrophe had been sent by the sun god, Apollo, in an act of revenge. He intended to punish those who had offended him in his many and various roles as the god of the sun, light, truth, music, poetry, and healing (a rather versatile god shall we say…).

What were the residents to do? Their first idea was to go to the Temple of Apollo to pray for forgiveness. The high priestess Pythia, also known as the Oracle of Delphi, responded to their plea and proclaimed that to curb the plague, they must double the size of the god’s altar. The altar, which held mysterious power, stood 10 feet tall and had ornate carvings. It was in the shape of a regular cube, so doubling the size to be rid of the deadly disease sounded like a reasonably good deal and the Delians readily agreed. But alas, if only it were so easy!

First, they doubled each side of the altar, which resulted in a volume eight times larger. That just wouldn’t do. Next, they tried to extend the altar on one side only, but that changed its shape, distorting into a cuboid. No matter what they attempted, Apollo’s anger – and therefore the plague – raged on. Having done all they could, the Delians desperately turned to Plato. The famous philosopher was their last hope.

Initially, Plato and his students didn’t think the task was very difficult. As the nature of doubling the area of a square was to find the square root of two, they hypothesised that doubling the volume of a cube could be done in a similar way. (For instructions on how to solve the first problem of doubling a square, see the diagram below.) Plato and his students attempted to double the cube in a similar way, but to no avail.

The problem was that Plato was too optimistic in thinking he could solve the problem using only pure geometry. Indeed, he continued to study it for many years, but just like the residents of Delos, he could not find a solution. In fact, the problem of doubling the volume of a cube went on to become one of the three classical geometric problems of Ancient Greece (with the other two being angle trisection and the squaring of a circle). To quote Plato:

It must be supposed, not that the god specially wished the problem solved, but he would have the Greeks desist from war and wickedness and cultivate the Muses, so that, their passions being assuaged by philosophy and mathematics, they might live in innocent and mutually helpful intercourse with one another.

Plato was remarkably clever in understanding the intention of Apollo. The problem remained unsolved until 1873 when Pierre Wantzel proved that the cube root of two is not constructible using an unmarked straight-edge and compass – the tools with which Plato himself was working. However, this doesn’t mean that the problem cannot be solved geometrically via an alternative method. So my question to you is thus: how can you construct the cube root of two using only the tools of geometry? Have a think for yourself first and then check out the next article in the series to see how it’s done!