Our capacities to think and rationalize are inseparable from our bodies that extend in space and act dynamically through time. To understand how to navigate space, as all animals do, we must possess the seeds of geometry in our consciousness. “Geometry is the area of mathematics that involves shape, size, position, direction, and movement and describes and classifies the [spatial] world we live in” [1]. As civilizations developed, these intuitions were put through the forge of categorization and empiricism which produced constellations of facts for purposes both practical and religious. We should note that geometry literally comes from geo – Earth, meter – measure. One such place that Earth measurement was required was in the farmlands of ancient Egypt. The floods of the Nile were necessary for the flourishing of crops, but the Greek historian Proclus noted this overflow would “obliterate the boundary lines between properties” [2]. Geometers arose to settle land disputes and redraw the boundaries using agreed upon truths. Egyptian geometers were called “rope-stretchers” as rope was their primary tool. Rope can be pulled taught to create straight lines. Circles can be constructed by staking one end of the rope down and then walking around said stake. Egyptians used their knowledge of the Pythagorean 3, 4, 5 triangle to create perfect right angles.

Over time, those who studied the collection of empirical facts constituting geometry began to understand that some of these facts depend on others. These realizations began a transition from trial and error to logical deduction that came to fruition in ancient Greece. This transition was motivated in part by the knowledge gained in breaking something down to its elements. People didn’t just want the facts that constitute geometry, or the fruit, they wanted to understand how the tree produces such fruit. When we understand things from first principles, from the roots, we can better innovate, optimize, and explain our understanding. In the words of Stephen Hawking, the geometer Euclid is arguably the “greatest mathematical encyclopedist of all time” [3]. He earned this title in consolidating the known geometry of his time (roughly 300 BCE), and thoroughly demonstrating that one could logically derive virtually all known theorems by tarting from a surprisingly small set of assumptions [4]. Euclid followed the branches of geometry down to the roots. The bedrock assumptions of Euclid’s Elements of Geometry consist in 23 axioms to be taken as self-evidently true (including his definitions of points, lines, etc.), and five postulates “which were assumptions whose validity seemed less certain, yet which appeared to be true of the geometry of our world” [5]. Here is one modern way of presenting Euclid’s Five Postulates:

1. If you have two points, you can draw a unique line segment between them.
2. If you draw a line segment, you could keep drawing it forever, and make a line
3. If you have a line segment, you can use it to make a circle with a radius equal to the segment length, where one of the endpoints is the middle.
4. All right angles are congruent, which means they have the same measure (90 deg).
5. If a straight line crosses two other lines, and the two interior angles it forms on one side with those two lines add up to less than two right angles, then those two other lines will eventually cross (if you extend them long enough).

Notice how postulate five stands apart from the others, which seem more accessible and sensible. There are equivalent ways of stating the fifth postulate using theorems that depend on it like, “The angles of all triangles always sum to 180 degrees.” John Playfair’s 18th century version helps to clarify why it is often called the “parallel postulate” stating: Given a line and a point not on said line, there is but a single unique line parallel to the first that passes through that point [5]. The parallel postulate is necessary for many important theorems of geometry like the ubiquitous Pythagorean Theorem, and all common formulas for volume and are likewise dependent on it. Herein lies a key point: When we use or prove theorems that are ultimately dependent on the root of the fifth postulate, we are operating in Euclidean Geometry. From one perspective, the parallel postulate feels natural to our conception of space. We can easily imagine two parallel lines like train tracks extending forever into space without intersecting, and if any other straight line intersected one of the two tracks it would have to intersect both… or is it really so simple?

If you think the fifth postulate feels a little more convoluted than the others, you are not alone. No postulate is as controversial as the parallel postulate. Even Euclid would refrain from using it until he absolutely had to in his proofs [6]. While known and important theorems appeared to depend on it, it unsettled Greek mathematicians. Writing in 5th century AD Proclus explains, “The postulate makes a positive statement about a region beyond the reach of possible observation or geometric intuition” [7]. We can never exhaustively imagine something extending to infinity, so how would we ever really know whether or not parallel lines intersected? Proclus declared the parallel postulate “plausible, but not necessary.” Mathematicians believed in the fifth postulated but held that it ought to be proved. Early attempts at proof typically ended up importing the fifth postulate in a concealed form. Communities of mathematicians from India, the Middle East, and Eurasia would attempt to replace the parallel postulate or prove it from the other four to no avail for nearly 2000 years. The conversation seemed stuck. It began to move forward in 1733 thanks to the work of Girolamo Saccheri.

Girolamo Saccheri was convinced that his life’s work was a failure. He spent a great amount of effort attempting to prove the fifth postulate from the other four. His attempt consisted in proof by contradiction. By assuming an alternative to the fifth postulate that opposes it, he would demonstrate that such an assumption necessarily leads to a contraction, and that fifth postulate must therefore be true. He started by assuming the existence of triangle whose angles sum to less than 180 degrees and began working out all manner of theorems using this as a starting point rather than the fifth postulate (which demands that all triangles have internal angles that sum to exactly 180 degrees). Despite immense struggle, none of his results contain contradictions. Instead, he created an entirely new tree of geometry by swapping only the fifth postulate, changing only one the roots. He could not see that he was one of the first to witness the bizarre fruits of hyperbolic geometry. Saccheri was not the only one to feel besieged by the necessity of parallel lines. In 1775 Farks Bolyai was scrambling to make ends meet through a multitude of professions. He was particularly concerned with the fifth postulate, and at one point thought he finally found a way to prove it from the other four, but other mathematicians found it flawed [6]. Bolyai turned his back on the study of the fifth postulate, not wanting to waste another second on it. His son Janos excelled in math from adolescence and was drawn to the parallel postulate precisely when his father warned him, “I have measured that bottomless night, and all the light and all the joy of my life went out there” [4]. Janos’s method was like Saccheri’s, but he could see that this new world of geometry could stand on its own. Janos proudly wrote to his father, “I have created a new and different world” [4]. At the same time, Nikolai Lobachevsky published work on the same subject by likewise supplanting the fifth postulate by not putting any restrictions on the number of lines parallel to a given line that pass through a point (remove the word “unique” from Playfair’s axiom) [6]. Mathematicians began to understand that another tree of geometry could exist independently of Euclidean geometry, a non-Euclidean geometry, and this tree is called Hyperbolic Geometry. The only thing separating it from Euclidean geometry is the alternative to the fifth postulate, a single different root. The question naturally shifted from trying to prove the parallel postulate to entertaining “what could be like if it was not the case?”

The parallel postulate cannot be proved from the others postulates. It is not necessary as there are alternative geometries. The mathematician Poincare, reflecting on the efforts of past geometers, stated, “...by natural selection our mind has adapted itself to the conditions of the external world. It has adopted the [Euclidean] geometry most advantageous to the species or, in other words, the most convenient. Geometry is not true, it is advantageous” [8]. Moreover, there are multiple non-Euclidean geometries. While the tree of hyperbolic geometry can be constructed by assuming the existence of triangles with angles less than 180 degrees, elliptic geometry can be constructed by assuming the existence of triangles with angles greater than 180 degrees. Carl Friedrich Gauss was one of the first to truly grasp spherical geometry, a subset of elliptic geometry which is another non-Euclidean geometry. He too could see the possibility of an internally consistent alternative to Euclid. He did not publish his results as he was fearful of the potential ensuing controversy. He began developing this perspective while helping surveyors map large plots of land and paying close attention to the ways that maps warp details in accordance with the curvature of the Earth [5]. This is how mathematicians established the two primary gates through which we entered the world of non-Euclidean geometry: elliptic and hyperbolic geometry, both objecting to the fifth postulate in different but complementary ways.