Of Euclid’s five postulates, which form the basis of Euclidean geometry, the fifth is the most controversial. Called the Parallel Postulate, it’s a pretty basic idea. Take a flat piece of paper, draw a line segment and then draw two lines that are perpendicular to it. The Parallel Postulate simply says that the lines will remain forever parallel, staying precisely the same distance apart even if we extend them to infinity on either side of the line segment.

For a long time mathematicians strived to prove the Parallel Postulate using the first four (more) basic postulates, and so elevate it from a postulated axiom to a proven theorem. It turns out that you can’t, and for good reason—the postulate isn’t true for all geometries. Euclidean geometry, represented by the flat piece of paper, and on which lines that start off parallel never intersect, is just one kind of geometry, but there are others. You can imagine replacing the Parallel Postulate with some alternative: maybe lines that are parallel initially draw closer together and touch eventually, or maybe they always move apart. The first option is the case with spherical geometry, which describes shapes drawn on the surface of a sphere (think of lines of longitude that start of parallel at the equator but converge at the poles), while the second defines hyperbolic geometry, which is more like the surface of a saddle or a Pringle’s potato chip.

These non-Euclidean alternatives weren’t noticed for millennia. Euclid himself flourished around 300 BCE, and the first alternatives to the Parallel Postulate were proposed in the early 1800s by Russian mathematician Nikolai Lobachevsky and Hungarian mathematician János Bolyai. Interestingly, both studied the hyperbolic alternative to flat Euclidean geometry.

This raises two questions: First, why did it take so long? And second, why did the mathematicians study hyperbolic geometry before spherical? Everyone knew about spheres. How hard could it have been to imagine drawing some lines on them, and exploring the geometric consequences of doing so?

The questions share a common answer. Yes, spheres were commonplace, but that’s exactly the point. Nobody thought that you needed an entirely new kind of geometry to deal with them. The two-dimensional surface of a sphere might be curved, but it’s easy enough to construct one in ordinary three-dimensional space: Pick some origin point, and consider the set of all points some fixed distance from that origin; that set is a sphere. You could draw lines on a sphere’s 2D surface, but mathematicians thought “real” lines live in 3D space, and so ignored the curved lines on the sphere.

You cannot construct hyperbolic geometry, however, in ordinary 3D space. One can glibly say things like “saddle shaped,” but every saddle has a special point in the middle where the curvature is highest, and the curvature fades as you move away from the middle. So, in regular 3D space, unlike a sphere, all points on a saddle are not created equal. The new hyperbolic geometry of Lobachevsky and Bolyai is special: there, too, every point is treated the same.

The reason mathematicians took so long to invent hyperbolic geometry is that the perfect form of such a geometry exists only in our minds. You can’t build it, the way you can build a sphere or a saddle, or even imagine building it. It’s an example of an imaginary space that human beings can postulate and study mathematically, without ever being able to reach out and touch it.

You can, nevertheless, try to visualize hyperbolic space as honestly as possible, keeping in mind that some distortions are inevitable when going from hyperbolic to Euclidean space. One of the most interesting visualizations is the hyperbolic disk (shown above), also known as the Poincaré disk. It was first used by Italian mathematician Eugenio Beltrami (1835-1900) and only later by French mathematician Henri Poincaré (1854-1912), but Poincaré is more famous, and famous people often get their names attached to things.

The image highlighted above is a hyperbolic disk tessellated by triangles. It represents an infinitely big two-dimensional hyperbolic geometry, but clever mathematical techniques have been used to compress the infinite space down to a finite disk-shaped region. That manipulation introduces a difference between what we see in 2D Euclidean space and the “true” hyperbolic geometry being represented. In particular, distances are compressed as we approach the boundary of the disk; in reality, both the radius of the disk and the circumference of its perimeter should be thought of as infinitely long.

In the 2D image, there are straight lines running at various angles through the center of the disk. But in the true geometry, all of the lines depicted are “straight,” although they appear to us as arcs stretching from one point on the edge to another one. And all of the triangles thus described have precisely identical geometries in hyperbolic space, despite appearances to the contrary. It’s just that our compression technique has squeezed and distorted them, especially near the boundary, analogous to how flat maps of Earth’s curved surface distort straight lines and measures of lengths.

This representation of the hyperbolic disk famously made an impression on artist M.C. Escher. Inspired by a figure in a paper by British mathematician H. S. M. Coxeter, Escher used it as the basis for a series of woodcuts titled Circle Limit, employing fish and angels and demons to divide up the disk.

Hyperbolic space played a crucial role in the development of non-Euclidean geometry. It was the first explicit example of a non-Euclidean space that mathematicians could only conceive of in the abstract, freeing their minds to imagine other kinds of wild geometries. Soon thereafter, German mathematician Bernhard Riemann developed a more general theory of non-Euclidean spaces, ones where arbitrary kinds of curvature could exist in spaces of arbitrary dimensionality. It is this Riemannian geometry that serves as the basis for general relativity, Einstein’s formulation of gravitation as a manifestation of the curvature of spacetime.

The hyperbolic disk continues to astound us. As a simple and concrete example of a non-Euclidean space, it pops up in different contexts within modern physics. Today, a great deal of effort is being expended studying something called the “AdS/CFT correspondence,” a relationship between a physical theory without gravity in D dimensions and another theory with gravity in (D+1) dimensions. In particular, the gravitational theory lives in a spacetime with hyperbolic geometry.

No wonder then that theoretical physicists spruce up their lecture slides with images of Escher prints, to help their audience appreciate the allure of hyperbolic spaces. We might not be able to build one, but they are no less beautiful for that. ♦