"Historically, hyperbolic geometry was first developed on an axiomatic basis. It arose as a result of efforts to prove the axiom of parallels from the other axioms. Doubt persisted for a long time as to whether this axiom could be deduced from the remaining axioms of Euclidean geometry. In their attempts to prove this axiom, mathematicians used the method of 'proof by contradiction' i.e., they assumed that the axiom of parallels was false and tried, on the basis of this assumption, to obtain a contradiction. All of these attempts were fruitless. True, the theorems obtained by negating the axiom of parallels appeared strange, but they did not contradict one another. The issue was resolved when C. F. Gauss, N. I. Lobachevski and J. Bolyai first stated explicitly that by negating the axiom of parallels one arrives at a new geometry, just as consistent as the usual (Euclidean) geometry." (Isaak Yaglom, "Geometric Transformations", 1973)
"Hyperbolic geometry is exceptional among non-Euclidean geometries because it satisfies all the axioms of Euclidean geometry except for the axiom of parallels. Other non-Euclidean geometries differ more radically from Euclidean geometry; in some of them a line segment cannot be produced indefinitely in both directions, and in others, two points cannot always be joined by a line." (Isaak Yaglom, "Geometric Transformations", 1973)
"We shall now compare the non-Euclidean geometry of Lobachevski-Bolyai with the geometry of Euclid studied in high school. One is immediately struck by how much the two geometries have in common. In both geometries two points determine a unique line, and two lines can have at most one point in common (this follows from the fact that lines of hyperbolic geometry are segments of lines in the plane). Further, in both geometries it is possible to carry a point and a ray issuing from it, by a motion, into any other point and a preassigned ray issuing from the latter point. The hyperbolic length of a segment and magnitude of an angle share many properties with their Euclidean counterparts; for example, in both geometries the length of the sum of two segments is the sum of their lengths, and the measure of the sum of two angles is equal to the sum of their measures." (Isaak Yaglom, "Geometric Transformations", 1973)
"Together with plane Euclidean geometry, spherical and hyperbolic geometry are 2-dimensional geometries with the following properties: (1) distance, lines and angles are defined and invariant under motions; (2) the motions act transitively on points and directions at a point; (3) locally, incidence properties are as in plane Euclidean geometry." (Miles Reid & Balazs Szendröi, "Geometry and Topology", 2005)
"When real numbers are used as coordinates, the number of coordinates is the dimension of the geometry. This is why we call the plane two-dimensional and space three-dimensional. However, one can also expect complex numbers to be useful, knowing their geometric properties […] What is remarkable is that complex numbers are if anything more appropriate for spherical and hyperbolic geometry than for Euclidean geometry. With hindsight, it is even possible to see hyperbolic geometry in properties of complex numbers that were studied as early as 1800, long before hyperbolic geometry was discussed by anyone." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)
"[...] the use of complex numbers reveals a connection between the exponential, or power function and the seemingly unrelated trigonometric functions. Without passing through the portal offered by the square root of minus one, the connection may be glimpsed, but not understood. The so-called hyperbolic functions arise from taking what are known as the even and odd parts of the exponential function." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)
"There are highly symmetric tiling patterns in hyperbolic geometry. For each of them, we can construct complex functions that repeat the same values on every tile. These are known as modular functions, and they are natural generalisations of elliptic functions. Hyperbolic geometry is a very rich subject, and the range of tiling patterns is much more extensive than it is for the Euclidean plane. So complex analysts started thinking seriously about non-Euclidean geometry. A profound link between analysis and number theory then appeared. Modular functions do for elliptic curves what trigonometric functions do for the circle." (Ian Stewart, "Visions of Infinity", 2013)
"When you encounter the classical wave equation, it’s likely to be accompanied by some or all of the words 'linear, homogeneous, second-order partial differential equation'. You may also see the word 'hyperbolic' included in the list of adjectives. Each of these terms has a very specific mathematical meaning that’s an important property of the classical wave equation. But there are versions of the wave equation to which some of these words don’t apply, so it’s useful to spend some time understanding them." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)
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