Why Ancient Greece’s Smartest Minds Couldn’t Solve the Delian Problem

Most people have never heard of the Delian Problem, yet this ancient mathematical riddle has puzzled brilliant minds for over 2,400 years and continues to influence how we understand the limits of human knowledge today.

What began as a desperate attempt to appease an angry Greek god on a small island eventually became one of mathematics’ most famous impossible challenges.

When Apollo demanded perfect mathematics on Delos

In 430 BC, Delos, the small sacred island believed by the ancient Greeks to be Apollo’s birthplace, was devastated by plague. Dead bodies accumulated, trade with the outside world had come to a halt, and the islanders were in a state of panic. Naturally, they did what any sensible ancient Greek would do: hurry to consult the oracle at Delphi.

However, Apollo’s answer was not what the Delians expected. He simply instructed them to double his altar—seems straightforward enough, right? The Delians thought so as well. They soon built a shiny new altar with sides twice as long as the original, hoping this would satisfy the god and solve their massive problem. Except the plague got worse—much worse.

The Delians had made a classic mathematical mistake. Doubling the sides of a cube does not double its volume; it increases it eightfold. Apollo, apparently a perfectionist in matters of geometry, would not accept this. He demanded exactly double the volume, not an approximation that left him unsatisfied.

This is how the Delian Problem arose. It asks one to construct a cube with exactly twice the volume of a given cube using only a compass and an unmarked straightedge. Although the problem is easy to describe, it is mathematically impossible to solve.

How ancient mathematicians nearly drove themselves crazy

Greek mathematicians threw everything they had at this problem. Hippocrates of Chios, one of the sharpest geometric minds of the 5th century BC, managed to solve part of it. He realized that if you could find two lengths forming a geometric progression with 1 and 2 (so that 1:x = x:y = y:2), then x would be your solution. Clever, someone who understands this might say, but it still did not solve the actual construction problem.

Then came the truly creative solutions. Archytas, the same man who invented the mechanical flying pigeon, proposed an unheard-of three-dimensional approach involving intersecting cylinders and cones. Imagine trying to explain that to a practical builder in ancient Greece.

Menaechmus, who had the good fortune to teach Alexander the Great, took a completely different approach. He used conic sections—parabolas and hyperbolas—a revolutionary idea at the time. When Alexander apparently complained about the difficulty of geometry, Menaechmus supposedly told him there was no “royal road” to mathematics. The irony was that his own solution required curves that could not be drawn with basic tools.

Other mathematicians invented entirely new curves just to tackle this problem. Nicomedes, for instance, created a conchoid, while Diocles developed the cissoid. All of these attempts represented genuine efforts to expand the boundaries of what was geometrically possible at a time when human knowledge was far behind today’s standards.

The problem was that none of these solutions adhered to the original rules. They all required tools or techniques beyond the traditional compass and straightedge. The Greeks understood this. They were not cheating but exploring what was possible when the constraints of a problem were loosened.

It wasn’t until the 1800s that Pierre Wantzel proved what the Greeks probably suspected but never explicitly stated: the Delian Problem is truly impossible to solve using only a compass and straightedge. The mathematics behind this proof is quite sophisticated, involving Galois theory, but the basic idea is that doubling a cube requires constructing the cube root of two, which cannot be achieved with the allowed operations.

This may seem like ancient history, but it is surprisingly relevant to modern mathematics. For example, in computer graphics, algorithms constantly approximate “impossible” constructions. When smooth curves are seen on a screen, these algorithms often create approximations of mathematical relationships that cannot be constructed with precision.

The lesson we learned from the Delian problem

The broader lesson is even more profound. The Delian Problem was—and continues to be—one of humanity’s first encounters with the idea that some things are simply impossible. This is not because we lack the right tools or techniques but because they are fundamentally beyond the reach of certain methods. This concept—that there are different types of impossibility—resonates throughout modern mathematics, computer science, and even physics.

In education, the problem remains a brilliant way to teach students how geometry and algebra connect. You start with a visual, concrete challenge and end up in abstract algebraic territory, which explains why the visual approach fails completely. It is mathematics at its most elegant: simple to understand, impossible to solve, and profound in its implications.

The ancient Delians, as far as we know, never got their altar just right. Yet their religious crisis gave us something far more valuable: a perfect example of how the pursuit of exact knowledge can lead us into unexpected territory.