The more than 2,000‑year search for impossible geometries

If your child asks you for help with their math homework and you see that, in one exercise, the angles of a triangle add up to only 90 degrees, you’d immediately tell them they’re wrong. But what if there were a geometry where their answer could be correct? Mathematicians’ long-standing obsession with understanding the formal foundations of geometry has led to the discovery of unexpected universes where these kinds of phenomena are possible. But far from settling the question, it has opened another: how can we imagine them?

The geometry we studied in school, known as Euclidean geometry, was formalized over 2,000 years ago in The Elements. Although, from today’s perspective, this work contains formal errors from the very first page (which have been corrected over time, so there’s no need to worry), it changed the way mathematics is done. Its author, Euclid, assumed five statements (or postulates) to be true in order to, simply by applying logic, obtain all the theoretical results of plane geometry known at the time, such as the Pythagorean theorem, and to develop new theorems.

Euclid’s first four postulates were very simple and obvious—for example, that a unique straight line passes through any two distinct points—and were accepted without question. However, the fifth was more complex. It stated, in a rather convoluted way, that through a point outside a given line, only one line parallel to it can be drawn. This prompted the question of whether it could be deduced from the first four, which would therefore be the only necessary postulates.

Throughout the centuries, mathematicians from all over the world tried, in vain, to formally prove it. The first attempts came from Greece. In the 5th century, Proclus believed he had done so, but he assumed, without justification from the postulates, that any two straight lines that do not intersect are always equidistant. From the 9th century onward, mathematicians of the Arab empire took up the problem. Ibn al-Haytham “proved” it by assuming, again without justification, that two straight lines cannot delimit an area. At the beginning of the 12th century, The Elements was translated from Arabic into Latin. It was done by Adelard of Bath who, according to legend, learned Arabic while traveling through Anatolia and Egypt and visited Córdoba, where he posed as a Muslim scholar in order to obtain a copy. From that moment on, numerous Western mathematicians joined the effort. In the 17th century, the English mathematician John Wallis offered a “proof” in which he unjustifiably assumed that any triangle can be rescaled.

It wasn’t until the first half of the 19th century that the young Hungarian military officer János Bolyai and the Russian professor Nikolai Lobachevsky independently developed a geometry in which more than one parallel line passes through a point outside a given line. Although Lobachevsky called it imaginary geometry, it is what we know today as hyperbolic geometry and it plays a central role in mathematics.

Bolyai’s and Lobachevski’s ideas needed a model that could prove their geometry was actually realizable. That model only arrived formally almost half a century later. Mathematicians such as the German Bernhard Riemann were beginning to understand that there are other spaces on which geometry can also be done, where the notion of a straight line becomes the shortest distance between two points and can take different forms depending on the shape of the space itself. Using these ideas, in 1868 the Italian mathematician Eugenio Beltrami introduced a mathematical model of hyperbolic geometry, now known as the Poincaré disk. Here, Euclid’s fifth postulate does not hold, but the rest do.

We can see the Poincaré disk in several works by the Dutch artist M. C. Escher. It is a circle in the plane where, from our perspective, objects become infinitely small as they approach the edge. Lines are now either diameters of the circle or arcs of circles perpendicular to the boundary. Because of this, given a line and a point outside it, there are infinitely many hyperbolic lines passing through that point without intersecting the given line. Another consequence is that the sum of the angles of a triangle is not 180 degrees, but strictly smaller and variable.

The mathematical community was also interested in finding physical models. Beltrami himself produced some in paper, which are still preserved in Pavia, and the German student Walther von Dyck made others in plaster. Both were partial or local models. In fact, in 1901 the German mathematician David Hilbert proved that it was impossible to visualize hyperbolic geometry in its entirety in three dimensions. In everyday life we can find local models of hyperbolic geometry: in a kale leaf, in a saddle, or in a well‑known (and perfectly wavy) potato crisp. We can also build them ourselves, for example using the colorful hyperbolic crochet developed by the Lithuanian mathematician Daina Taimina in 1997.

The arrival of new technologies has multiplied the ways of visualizing this hyperbolic world: from interactive websites that let us draw with a hyperbolic ruler and compass to videos that help us understand the relationship between different models through light. Recently, the video game Hyperbolica (which allows an immersive experience with virtual‑reality headsets) uses this landscape — slow‑cooked over more than two millennia of mathematics — as its setting, and gives it undeniable prominence.

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