Euclid
In ancient Greece, the mathematician, Euclid, set out to define the laws that governed the perceivable universe. In his book, “The Elements,” Euclid would ultimately assembled 5 postulates from whom he derived all of the laws of geometric space.
Euclid’s Postulates:
1. A straight line segment can be constructed to run through 2 points
2. Any straight line segment can be extended infinitely in a straight line
3. Given any straight line segment, a circle can be constructed having that segment as its radius
4. All right angles are congruent
5. If two lines intersect a third, such that the sum of the inner angles on one side is less that two right angles, then the two lines inevitably must intersect each other on that side if extended far enough
These postulates are rather straightforward, and a simple sketch would illustrate the intuition at there core. However, it is that 5th postulate that is critical here: many throughout history, including Euclid himself, noticed how complicated it is relative to his other postulates. In fact, Euclid uses his first 4 postulates to prove his first 28 propositions, or the rules he derives based on these laws, and only involves his 5th postulate for his 29th proposition (proving corresponding and alternating angles are congruent). Further, Euclid never proved this 5th postulate.
The Search To Prove It
In attempts to simplify the 5th postulate, mathematician formulated equivalent statements, like “Playfair’s axiom:”
In a plane, given a line, and a point not on it, at most one line parallel to the given line can be drawn through the point
Still, the 5th postulate continued to remain unproven. Rene Descartes, the 17th century French mathematician and philosopher, approached the problem from a different tack; rather than attempt to prove the 5th postulate, Descartes replaced Euclid’s laws with the laws of algebra.
Analytic Geometry
Descartes’ “Analytic Geometry” turns the geometry of Euclid with algebra. Specifically, Descartes created coordinate system to represent geometry: the Cartesian coordinate system. Analytic geometry includes the representation of geometry with a variety of coordinate systems like Cartesian coordinates (x,y), Polar coordinates (r, theta), and Parametric coordinate systems (x = f(t), y = g(t)). Descartes’ analytic geometry is Euclidean in that it behaves like Euclidean geometry, and the same conclusions are drawn, however the postulates are substituted with the laws of algebra.
Other mathematicians created their own geometric systems, like Affine Geometry and Perspective Geometry, but these still followed Euclid’s postulates. At the same time, the efforts to prove the 5th postulate were unsuccessful.
Then mathematicians wondered, “what if Euclid’s 5th postulate does not have to be true?”
Enter: Non-Euclidean Geometry
Janos Bolyai
Bolyai recognizes that most of Euclidean geometry can be proven without the 5th postulate, and in the early 19th century, Bolyai removed it. His resulting system was dubbed, “Absolute Geometry.” Absolute Geometry maintains the first 4 of Euclid’s postulates, and in and of itself, Absolute Geometry is incomplete, as it no longer has any proof regarding the relationship between corresponding and alternating angles (Euclid’s proposition 29). As a result, it is open to allow the addition of different axioms. Thus, Absolute Geometry can be used to build spaces that appear very different that Euclidean geometry. Moreover, because it maintains the first 4 postulates, anything proven in Absolute Geometry, holds true in any geometric system that can be built off absolute geometry.
Nikolai Ivanovich Lobachevsky
Instead of removing the 5th postulate, Lobachevsky alters Playfair’s axiom. Instead of:
In a plane, given a line, and a point not on it, at most one line parallel to the given line can be drawn through the point
Lobachevsky writes:
In a plane, given a line, and a point not on it, more than one line parallel to the given line can be drawn through the point
But how can this work? How could there be multiple lines running through a point, with different slopes, without them all intersecting with the original line?
Gaussian Curvature
Imagine a surface, at any point on that surface, the vector perpendicular to the point is called the “normal vector.” There exists two planes for every normal vector that the normal vector lies upon, these planes are called “normal planes.” The intersection of a normal plane with the surface is a “normal curvature;” the “principal curvatures” are the maximum and minimum curvatures at a point. Finally, with all of this terminology out of the way, the Gaussian Curvature is the product of the maximum and minimum curvatures. If the Gaussian Curvature is positive, then the surface is curving in the same direction at the point; if the Gaussian Curvature is negative, then the surface is curving in opposite directions at that point; if the Gaussian Curvature is 0, then the surface is straight at one or more of the normal sections.
Notice that negative curvature means the normal sections are moving in opposite directions, away from one-another, further and further apart; in a word: diverging. Thus, on a part of a surface with negative curvature, lines will diverge from one another. This can be seen in the hyperboloid, a surface with global negative curvature.
Back to Lobachevsky
If one were to draw a line on a surface with negative curvature, a hyperboloid for example, they could draw multiple lines through a point not on that line, and they would diverge from the original line. In this way, a surface with negative curvature fulfills Lobachevsky’s version of Playfair’s axiom, and his axioms thereby must create a space with negative curvature.