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

The Delian problem, an ancient Greek mathematical riddle, challenged scholars to treble the measurement of a cube utilizing lone a compass and straightedge—an elegant puzzle that proved to beryllium impossible. Credit: Greek Reporter archive

Most radical person ne'er heard of the Delian Problem, yet this past mathematical riddle has puzzled superb minds for implicit 2,400 years and continues to power however we recognize the limits of quality cognition today.

What began arsenic a hopeless effort to appease an aggravated Greek deity connected a tiny land yet became 1 of mathematics’ astir celebrated intolerable challenges.

When Apollo demanded cleanable mathematics connected Delos

In 430 BC, Delos, the tiny ineffable land believed by the past Greeks to beryllium Apollo’s birthplace, was devastated by plague. Dead bodies accumulated, commercialized with the extracurricular satellite had travel to a halt, and the islanders were successful a authorities of panic. Naturally, they did what immoderate sensible past Greek would do: hurry to consult the oracle astatine Delphi.

However, Apollo’s reply was not what the Delians expected. He simply instructed them to treble his altar—seems straightforward enough, right? The Delians thought truthful arsenic well. They soon built a shiny caller altar with sides doubly arsenic agelong arsenic the original, hoping this would fulfill the deity and lick their monolithic problem. Except the plague got worse—much worse.

The Delians had made a classical mathematical mistake. Doubling the sides of a cube does not treble its volume; it increases it eightfold. Apollo, seemingly a perfectionist successful matters of geometry, would not judge this. He demanded precisely treble the volume, not an approximation that near him unsatisfied.

This is however the Delian Problem arose. It asks 1 to conception a cube with precisely doubly the measurement of a fixed cube utilizing lone a compass and an unmarked straightedge. Although the occupation is casual to describe, it is mathematically intolerable to solve.

The past ruins connected the land of Delos arsenic seen today. Credit: Greek Reporter

How past mathematicians astir drove themselves crazy

Greek mathematicians threw everything they had astatine this problem. Hippocrates of Chios, 1 of the sharpest geometric minds of the 5th period BC, managed to lick portion of it. He realized that if you could find 2 lengths forming a geometric progression with 1 and 2 (so that 1:x = x:y = y:2), past x would beryllium your solution. Clever, idiosyncratic who understands this mightiness say, but it inactive did not lick the existent operation problem.

Then came the genuinely originative solutions. Archytas, the aforesaid antheral who invented the mechanical flying pigeon, projected an unheard-of three-dimensional attack involving intersecting cylinders and cones. Imagine trying to explicate that to a applicable builder successful past Greece.

Menaechmus, who had the bully luck to thatch Alexander the Great, took a wholly antithetic approach. He utilized conic sections—parabolas and hyperbolas—a revolutionary thought astatine the time. When Alexander seemingly complained astir the trouble of geometry, Menaechmus supposedly told him determination was nary “royal road” to mathematics. The irony was that his ain solution required curves that could not beryllium drawn with basal tools.

Other mathematicians invented wholly caller curves conscionable to tackle this problem. Nicomedes, for instance, created a conchoid, portion Diocles developed the cissoid. All of these attempts represented genuine efforts to grow the boundaries of what was geometrically imaginable astatine a clip erstwhile quality cognition was acold down today’s standards.

The occupation was that nary of these solutions adhered to the archetypal rules. They each required tools oregon techniques beyond the accepted compass and straightedge. The Greeks understood this. They were not cheating but exploring what was imaginable erstwhile the constraints of a occupation were loosened.

Temple of Apollo connected Naxos, 1 of the Cyclades islands successful the Aegean. Credit: Leynadmar Wikimedia Commons, CC BY-SA 4.0

It wasn’t until the 1800s that Pierre Wantzel proved what the Greeks astir apt suspected but ne'er explicitly stated: the Delian Problem is genuinely intolerable to lick utilizing lone a compass and straightedge. The mathematics down this impervious is rather sophisticated, involving Galois theory, but the basal thought is that doubling a cube requires constructing the cube basal of two, which cannot beryllium achieved with the allowed operations.

This whitethorn look similar past history, but it is amazingly applicable to modern mathematics. For example, successful machine graphics, algorithms perpetually approximate “impossible” constructions. When creaseless curves are seen connected a screen, these algorithms often make approximations of mathematical relationships that cannot beryllium constructed with precision.

The acquisition we learned from the Delian problem

The broader acquisition is adjacent much profound. The Delian Problem was—and continues to be—one of humanity’s archetypal encounters with the thought that immoderate things are simply impossible. This is not due to the fact that we deficiency the close tools oregon techniques but due to the fact that they are fundamentally beyond the scope of definite methods. This concept—that determination are antithetic types of impossibility—resonates passim modern mathematics, machine science, and adjacent physics.

In education, the occupation remains a superb mode to thatch students however geometry and algebra connect. You commencement with a visual, factual situation and extremity up successful abstract algebraic territory, which explains wherefore the ocular attack fails completely. It is mathematics astatine its astir elegant: elemental to understand, intolerable to solve, and profound successful its implications.

The past Delians, arsenic acold arsenic we know, ne'er got their altar conscionable right. Yet their spiritual situation gave america thing acold much valuable: a cleanable illustration of however the pursuit of nonstop cognition tin pb america into unexpected territory.