“Euclid alone has looked on Beauty bare,” Edna St. Vincent Millay wrote in her lovely ode to how the father of geometry transformed the way we see and comprehend the world. But although the ancient Alexandrian mathematician provided humanity’s only framework for understanding space for centuries to come, shaping both science and art, his beautiful system was wormed by one ineluctable flaw: Euclid’s famous fifth postulate, known as the parallel postulate — which states that through any one point not belonging to a particular line, only one other line can be drawn that would be parallel to the first, and the two lines, however infinitely they may be extended into space, will remain parallel forever — is not a logical consequence of his other axioms.

This troubled Euclid. He spent the remainder of his life trying to prove the fifth postulate mathematically, and failing. Generations of mathematicians did the same for the next two thousand years. It even stumped Gauss, considered by many the greatest mathematician of all time. It took a Hungarian teenager to solve the ancient quandary.

In 1820, more than two millennia after Euclid’s death, the seventeen-year-old János Bolyai (December 15, 1802–January 27, 1860) told his father — the mathematician Wolfgang Bolyai, who had introduced his son to the enchantment of mathematics four years earlier — about his obsession with the parallel postulate.

In the exchange that followed, recounted in George E. Martin’s classic 1975 primer The Foundations of Geometry and the Non-Euclidean Plane (public library), Bolyai Senior responded with the opposite of encouragement, writing to his son:

Don’t waste an hour on that problem. Instead of reward, it will poison your whole life. The world’s greatest geometers have pondered the problem for hundreds of years and not proved the parallel postulate without a new axiom. I believe that I myself have investigated all the possible ideas.

But the young man persisted. On November 3, 1823, the twenty-one-year-old mathematical maverick wrote to his father while serving as an artillery officer in the Hungarian army:

I have resolved to publish a work on the theory of parallels as soon as I have arranged the material and my circumstances allow it. I have almost been overwhelmed by them, and it would be the cause of constant regret if they were lost. When you see them, my dear father, you too will understand. At present I can say nothing except this: I have created a new universe from nothing. All that I have sent to you till now is but a house of cards in comparison with a tower. I’m fully persuaded that this will bring me honor, as if I had already completed the discovery.

The discovery in which he exults is one of humanity’s most groundbreaking insights into the nature of reality: Bolyai had laid the foundation of non-Euclidean geometry — a wholly novel way of apprehending space, which describes everything from the shape of a calla lily blossom to the growth pattern of a coral reef, and which would become a centerpiece of relativity; without it, Einstein couldn’t have revolutionized our understanding of the universe with his notion of spacetime, the curvature of which is a supreme embodiment of non-Euclidean geometry.

Impressed by his son’s tenacity and swayed by the significance of the breakthrough, Wolfgang pivoted 180 degrees and now urged his son to publish his findings as soon as possible in order to ensure priority of discovery:

If you have really succeeded in the question, it is right that no time be lost in making it public, for two reasons: first, because ideas pass easily from one to another, who can anticipate its publication, and secondly, [because] there is some truth in the fact that many things have an epoch, in which they are discovered at the same time in several places, just as the violets appear on every side in spring.

These were words of remarkable prescience. When János’s paper, completed in 1829 and published as an appendix to a book of his father’s in 1832, reached Gauss — an old friend of Wolfgang Bolyai’s — the great German mathematician was astonished. He responded that he couldn’t praise János’s work, for it would mean praising himself — the young mathematician’s breakthrough, from the central questions he had tackled to the path he had pursued in answering them to the results he had obtained, coincided “almost entirely” with what had been occupying Gauss’s own mind for more than thirty years, though he had resolved never to publish these meditations in his lifetime. With the selfless graciousness of a true scientist, who sets aside all personal ego and celebrates any triumph of knowledge, Gauss wrote to János’s father:

So far as my own work is concerned, of which up till now I have put little on paper, my intention was not to let it be published during my lifetime… On the other hand it was my idea to write down all this later so that at least it should not perish with me. It is therefore a pleasant surprise for me that I am spared this trouble, and I am very glad that it is just the son of my old friend who anticipates me in such a remarkable manner.

But the young Bolyai’s elation at having “created a new universe from nothing” was swiftly grounded when he realized that a third mathematician — Nikolai Lobachevsky in Russia — had preceded both him and Gauss in publishing a paper outlining the selfsame ideas. Lest we forget how information traveled in the pre-Internet era, it took Bolyai sixteen years to learn of Lobachevsky’s book. Once he read it, he reconciled himself to the loss of priority by rooting his ego in the animating principle of science, which he recorded in an uncommonly poetic and profound meditation in his notebook:

The nature of real truth of course cannot be but one and the same [in Hungary] as in Kamchatka and on the Moon, or, to be brief, anywhere in the world; and what one finite, sensible being discovers, can also not impossibly be discovered by another.

The discovery at which these three finite, sensible beings had arrived simultaneously and independently forever changed not only mathematics but our fundamental grasp of nature. For a fine complement, see mathematician Lillian Lieber’s 1961 masterpiece drawing on the non-Euclidean revolution to illustrate the building blocks of moral values like democracy and social justice, then revisit physicist Alan Lightman on the shared psychology of creative breakthrough in art and science.