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non-Euclidian Geometry – When an Idea’s Time Had Come

The odds are pretty good that you’ve had at least one moment in your life where you were able to “read the mind” of someone close to you.

Maybe it’s a friend you’ve known since early childhood, or someone you’ve been in a relationship with for decades – or just a family member that you have a particular closeness with. There came a moment when you both seemed to blurt out the exact same thing in the exact same way at the exact same time, laughing and marveling at how the universe “syncs up”.

But imagine if you were able to “read the mind” of someone thousands of miles away from you, a complete and total stranger, never realizing that not only were you both seized by the exact same passion but also following the EXACT same process that would inevitably result in one of mathematics most important discoveries. Sounds crazy, right?

That’s the story of János Bolyai and Nikolai Ivanovich Lobachevsky. We know very little of the early life of Bolyai, aside from the fact that he was born in Hungary in 1802 and a bit of a backwater stretch of Transylvania in what is now today known as Romania. His father reports that he had a very strong aptitude for mathematics from a young age, something that was no doubt inspired by his father’s work as a mathematician as well as the close working relationship the family had with legendary mathematician Carl Friedrich Gauss.

After his father failed to get Gauss to take Janos on as an apprentice, Bolyai decided that the world of mathematics had little place for him. Instead, he prepared for life in the military and was getting set to undertake his training at the Royal Engineering College in Vienna to become a military engineer.

It was here that his passion for mathematics – particularly the field of geometry – really took off, and where the seeds of his now legendary approach to “inventing” the non-Euclidian geometry of curved spaces would be sewn.

At the exact same time that Bolyai was in Vienna exploring the potential of non-Euclidian geometry, a young man by the name of Lobachevsky was also captivated by the potential of non-Euclidian geometry. Born in 1792 to a family that couldn’t have been more different than the Bolyai family, Lobachevsky worked his way up from his blue-collar, working-class roots in Russia to not only attending university but later on becoming a professor and a rector. He would learn mathematics under the close tutelage of a teacher and close friend of Gauss, which is likely how he was able to germinate the same kind of conceptions about Euclidian geometry that Bolyai was able to – putting them both on the same path to this incredible discovery within just a few short months of one another.

Amazingly, the approach that both Bolyai and Lobachevsky took to discover or “invent” the concept of curved spaces in non-Euclidian geometry was almost identical. Neither man had met or even heard of one another,and they conducted their research independently separated by thousands of miles.

Both men worked to redefine Euclid’s conception of parallel lines and space.  Euclid stated that there was only one line, A, on a plane, that went through a point, B, not on another line C on the same plane, that never met line C, no matter how far it extended. For them, there were more than one line, even infinite lines like A that never touched C on the same plane. In their geometry, space though was curved, the shortest distance between two points a curve, not a line and triangles’ interior angles added up to less than 180 degrees. Interestingly, this is actually reality in outer space near large bodies exerting gravitational force.

The progress of both mathematicians later led both of them to prove a number of results having to do with the manipulation of trigonometric formulae. They were both able to independently prove that a unique set of formulae regarding triangles on the surface of their non-Euclidian spheres would work independent of the parallel postulate. They were also able to analytically come up with a number of different equations that describe the way that “ordinary triangles” worked in non-Euclidian space – again following the same path of research and achieving the same results without ever becoming aware of the other’s work, progress, or results.

At the end of the day, Lobachevsky was the first to publish his findings. His work on hyperbolic geometry was first recorded in 1826 and later published in 1830, with Bolyai working on his projects throughout the same timeline but not actually publishing his findings until 1832. Historians later discovered that both men had been working along parallel lines while researching parallel lines. And if that isn’t an example of the universe syncing up or at least being ironic, what is?