Sami was a student in the Fall 2016 course “Geometry of Surfaces” taught by Scott Taylor at Colby College. The essay has been lightly edited before being published here.
This essay is an introduction to the history of hyperbolic geometry. Euclid, Gauss, Felix Klein and Henri Poincare all made major contribution to the field. We will discuss some of their influences in the following sections, starting with Euclid and his postulates that defined geometry. Then we will look at the effect of Gauss’s thoughts on Euclid’s parallel postulate through noneuclidean geometry. Later, Klein settled any doubt of noneuclidean consistency. Lastly, Poincare makes some notable contributions to solidifying hyperbolic geometry as an area of academic study.
Euclidean geometry came from Euclid’s five postulates. It is the most intuitive geometry in that it is the way humans naturally think about the world. Nonetheless, there are a few other lesser-known, but equally important, geometries that also have many applications in the world and the universe. These “other” geometries come from Euclid’s fifth postulate: “If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles the two straight lines if produced indefinitely meet on that side on which the angles [are] less than two right angles” . See Figure 1(A) below for an illustration of this. An equivalent way to express this is that the angle sum of a triangle is two right angles.
This is also called the parallel postulate because it is equivalent to the following statement: “if one draws a straight line and a point not on that line, there is only one unique line that is parallel to given line through the point” . See Figure 1(B) below.
Unlike the other postulates, the fifth is not obvious (to satisfy your curiosity, the rest of Euclid’s postulates are attached at the end of this essay). For centuries, mathematicians and amateurs alike attempted to prove that the fifth postulate is a consequence of the first four postulates and other established theorems . However, it turns out this postulate determines whether we are in euclidean or noneuclidean geometry .
The Birth of Hyperbolic Geometry
Over 2,000 years after Euclid, three mathematicians finally answered the question of the parallel postulate. Carl F. Gauss, Janos Bolyai, and N.I. Lobachevsky are considered the fathers of hyperbolic geometry. Gauss published little of his work on it because he, allegedly, feared disrespecting Euclid by disproving the parallel postulate. Yet, Gauss extended geometry to include what he coined “non-euclidean,” or the contradiction of the parallel postulate . Bolyai and Lobachevsky published their works on noneuclidean geometry independently but around the same time; however, their findings were not popularized until after 1862, when Gauss’s private letter to Schumacher about his “meditations” on hyperbolic geometry was published. With the work of these three mathematicians, the controversy of the parallel postulate was put to rest. It was definitive that without the parallel postulate, the remaining four postulates created a geometry that is equally consistent .
What is hyperbolic geometry?
Now that a brief history of the sources of hyperbolic geometry has been provided, we will define hyperbolic geometry. It has constant negative Gaussian curvature, which resembles a hyperboloid (See Figure 2). Note, that spherical geometry has constant positive curvature .
Visualization of Hyperbolic Geometry
A more natural way to think about hyperbolic geometry is through a crochet model as shown in Figure 3 below. This discovery by Daina Taimina in 1997 was a huge breakthrough for helping people understand hyperbolic geometry when she crocheted the hyperbolic plane. This was important because now people can touch and move the crocheted object to get a more intuitive understanding of what a straight line in hyperbolic space would look like.
Definition of Distance
Although it may be a bit surprising at first, the definition of metric, or measure of distance, is different for each type of geometry. In Euclidean geometry, the distance is given as
between two points and . Note that in euclidean geometry, the shortest distance between two points is a line (as postulated by Euclid). However, in hyperbolic space, the shortest distance between two points is the arc from point to point that lie on the unique semicircle centered at a point on the boundary as shown in Figure 5; if we parameterize the arc as ,
Now that we’ve gone over the founding of hyperbolic geometry and have some sense of what it is, we’ll talk about two more influential mathematicians in the field: Felix Klein and Henri Poincare.
Klein’s contribution to geometry is not only the famous Klein Bottle, but also his proof of the extension of the Cayley Measure on projective geometry to non-euclidean geometry. According to John Stillwell’s translation of On so-called Noneuclidean Geometry, Klein started by showing that each non-euclidean geometry is a special case of projective geometry; he then ended by concluding that non-euclidean geometry can be derived from projective geometry. Furthermore, projective geometry is independent from the theory of parallels; if not, the consistency of projective geometry would be questioned.
Projective geometry has simple origins from Renaissance artists who portrayed the rim of a cup as an ellipse in their paintings to show perspective. Projective geometry can be thought of in this way:
The theory of perspective regards the artist’s canvas as a transparent screen through which he looks from a fixed vantage point at the scene he is painting. Light rays coming from each point of the scene are imagined to enter his eye, and the totality of these lines is called a projection. 
From there, the idea eventually entered the academic mathematical community where it was incorporated into the study of linear algebra and other areas of mathematics. Projective geometry is the study of invariants on projections – properties of figures which are not modified in the process of projection . Projective geometry is more more general than both euclidean and hyperbolic geometries and this is what Klein uses to show that noneuclidean geometries are consistent.
Euclidean and hyperbolic geometry follows from projective geometry
Klein gives a general method of constructing length and angles in projective geometry, which he believed to be the fundamental concept of geometry. To borrow psychology terms, Klein’s approach is a top-down way to look at non-euclidean geometry while the upper-half plane, disk model and other models would be referred to as bottom-up methods. The bottom-up methods are easier to visualize and to deal with applications of hyperbolic geometry. This is remarkable because this top-down approach provides a “more comprehensive”  proof than those that had in a case by case manner. Moreover, projective geometry is consistent outside of geometry .
Distance in general
The different geometries we get from projective geometry come from the the projection of the fundamental conic. This idea is illustrated below in Figure 7.
First, Klein defines the general distance function as
Klein shows that the fundamental conic section “corresponds completely with the idea of hyperbolic geometry.” Now, the fundamental conic is the projection onto the plane with the appropriate metric; it “determines a measure on all basic figures of the first kind in the plane”. Here, the “first kind” is fuchsian groups , which are isometries of the hyperbolic plane. An isometry is a way of transforming a figure in a given space without changing its angles or lengths. Every isometry in hyperbolic space can be written as a linear fractional transformation in the following form:
where is the complex coordinate of a point. You can think of a complex coordinate in the same way that you think about on the euclidean plane where . The “basic figures” are the triangle, circle, and the square. So these isometries take triangles to triangles, circles to circles and squares to squares.
The fundamental conic that forms hyperbolic geometry is proper and real – but “we shall never reach the fundamental conic, let alone pass beyond it”. Recall, our visualizations of hyperbolic space using the upper-half plane model from Figure 4(A), then the fundamental conic is the real line and the fuchsian groups are the isometries acting on .
By using projective geometry to construct noneuclidean geometries, Klein has created a proof that is indisputable.
A discussion of the history of hyperbolic geometry cannot leave out Henri Poincare after which the Poincare disk model is named (See Figure 4 (B)). Originally, he began his studies in differential equations, then stumbled upon hyperbolic geometry naturally in his work. He wrote in his book Science and Methods that:
The incidents of travel made me forget my mathematical work. When we arrived at Countances, we got into a break to go for a drive, and, just as I put my foot on the step, the idea came to me, though nothing in my former thoughts seemed to have prepared me for it, that the transformations I had used to define Fuchsian functions were identical with those of non-euclidean geometry. 
In fact, he was studying the differential equation for a given function that resembles linear fractional transformations with real coefficients. When perturbed to have slightly complex coefficients, this yielded a group of linear fractional maps with complex coefficients that act on the Riemann Sphere. This is also related to chaos! He called the ones with real coefficients fuchsian groups after Fuchs and the ones with complex coefficients kleinian groups, though neither Fuchs nor Klein had much to do with the groups they are named after. Allegedly, Klein was offended when Poincare declared the fuchsian group, claiming it had been he, not Fuchs, who had thought about it. So to appease Klein, Poincare named the next group after him. Nonetheless, these are groups of isometries on the hyperbolic plane, .
There have been many other theorems (and still more though current research) in hyperbolic space but these have been instrumental in the creation of hyperbolic geometry. Despite its turbulent history surrounding the parallel postulate and its late debut in the mathematical world, hyperbolic geometry is gaining prominence and has many great applications to the real world.
Euclid’s Postulates :
1. A straight line segment can be drawn joining any two points.
2. Any straight line segment can be extended indefinitely in a straight line.
3. Given any straight lines segment, a circle can be drawn having the segment as radius and one endpoint as center.
4. All Right Angles are congruent.
5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two Right Angles, then the two lines inevitably must intersect each other on that side if extended far enough.
 Antonick, Gary. “The Non-Euclidean Geometry of Whales.” The New York Times. The New York Times, 08 Oct. 2012. Web. 04 Dec. 2016.
 Bonahon, Francis. Low-dimensional Geometry: From Euclidean Surfaces to Hyperbolic Knots. Providence, RI: American Mathematical Society, 2009. Print.
 Creates, Cris. “Crochet Mathematics: Hyperbolic Crochet.” Crochet Mathematics: Hyperbolic Crochet. N.p., 12 Jan. 2012. Web. 04 Dec. 2016.
 “Daina Taimina.” Daina Taimina | Crochet Coral Reef. N.p., n.d. Web. 04 Dec. 2016.
 Henderson, David W. Experiencing Geometry: In Euclidean, Spherical, and Hyperbolic Spaces. Contributer: Daina Taimina. 2nd ed. Upper Saddle River, NJ: Prentice Hall, 2001. Print.
 Lewis, Florence P. “History of the Parallel Postulate.” The American Mathematical Monthly, vol. 27, no. 1, 1920, pp. 16–23. www.jstor.org/stable/2973238.
 McMullen, Curtis T. “Euclid’s Postulates.” Euclid’s Postulates. Harvard University, 1997. Web. 07 Dec. 2016.
 Nicoguaro. Gaussian Curvature. Digital image. Wikicommons, 9 Jan. 2016. Web. 6 Dec. 2016.
 Poincaré, Henri. Science and Methods. Trans. Francis Maitland. London: T Nelson, 1914. Archive.org.
 Stillwell, John. “Introduction and Translation to Klein’s On the So-called Noneuclidean Geometry.” Sources of Hyperbolic Geometry. Vol. 10. Providence, RI: American Mathematical Society, 1996. 63-111. Print. History of Mathematics.
 Wylie, C. R. “Introduction to Projective Geometry.” Google Books. Courier Corporation, 2011. Web. 08 Dec. 2016.