Archytas and the Delian Problem

Ancient and later writers preserved several versions of how it arose, and today it is impossible to say precisely where history ends and legend begins. According to the best-known tradition, during an epidemic people sought guidance from the oracle of Apollo. The answer seemed simple: the god’s altar was to be doubled in size.

But there was a problem. Apollo’s altar was in the shape of a cube. If the length of each side is doubled, its volume does not double — it becomes eight times larger.

How, then, could one construct a cube whose volume was exactly twice that of the original?

This question became one of the three famous classical problems of ancient Greek mathematics — alongside the trisection of an angle and the squaring of the circle. Perhaps we will return to the other two problems on future postcards.

A Sacred Island

Delos is a tiny island in the Aegean Sea, yet to the ancient Greeks it was one of the most sacred places in the world. According to myth, it was here that Leto gave birth to the twins Apollo and Artemis, and the island became a great centre of the cult of Apollo.

Among its most recognizable monuments is the Terrace of the Lions. The people of Naxos dedicated a row of marble lions to Apollo around the end of the seventh century BCE. Scholars estimate that originally there were about nine to twelve. That is why the lions appear on our postcard: around 400 BCE, an ancient visitor to Delos would have seen far more of these stone guardians than survive today.

The Trap Hidden in the Cube

Imagine that the side of the original cubic altar is 1. Its volume is therefore:

Now double each side. The side of the new cube is 2, and its volume becomes:

We wanted twice the volume. Instead, we got eight times the volume.

Today we write the required length of the side of the new cube as:

But knowing this number is not the same as solving the ancient problem.

The central difficulty lay in the restrictions of classical geometric construction. The problem had to be solved on a plane using only a compass and an unmarked straightedge — a ruler with no graduations.

A compass could be used to draw circles and transfer distances, while the straightedge could only draw straight lines through given points. You could not simply measure the required length from a scale, mark it on the ruler, or use a ready-made numerical value.

And under these restrictions, doubling the cube is impossible.

Today we know why: using only a compass and an unmarked straightedge, it is impossible to construct a segment of length ∛2 from a unit segment. The ancient Greeks had no way to prove this. A rigorous proof of the impossibility appeared only in the nineteenth century, when algebra made it possible to determine the precise limits of classical geometric constructions.

But Greek mathematicians did not know that the problem before them was impossible under the given conditions. They continued to search for a solution.

And this is where Archytas enters the story.

Archytas of Tarentum

Archytas lived approximately from 428 to 347 BCE. He was a mathematician, philosopher, statesman, military commander, student of music, and one of the outstanding figures of the Pythagorean tradition.

His solution to the Delian Problem was remarkable precisely because Archytas stepped beyond the restrictions of classical construction.

He carried the problem from the plane into space. His construction involved the intersection of several geometric surfaces. In modern interpretations of his method, these are described as surfaces related to a cylinder, a cone, and a torus.

Their intersection produced the geometric quantity needed to double the cube.

It was an extraordinarily daring idea. Archytas did not simply propose a different method of solution — he changed the very space in which he searched for the answer. Geometry was no longer only straight lines and circles drawn on a plane. It became geometry in space.

Archytas’ own description of this construction has not survived. We know of it through later mathematicians, above all Eutocius of Ascalon, who centuries later preserved an account of the method in his commentary on Archimedes’ On the Sphere and Cylinder.

What has reached us, then, is not Archytas’ own voice, but an echo of his thought carried across many centuries.

Can You Double the Cube?

Look again at the two cubes on the back of the postcard. The side of the first is 1. The task is to find the side of a second cube whose volume is exactly 2.

Today the answer can be written in a single line:

So the side of the new cube must be about 26% longer than the side of the original — not twice as long.

But try constructing that length on a plane using only a compass and an unmarked straightedge.

You cannot.

That is one of the most remarkable features of the Delian Problem: for more than two thousand years, mathematicians searched for a solution under conditions in which no solution exists.

Archytas found a way forward not by forcing compass and straightedge to do the impossible. He changed the conditions of the search itself — and carried geometry into space.

A simple command — “Double the altar” — opened the door to an entirely new way of thinking about geometry.