Chapter 5

Hyperbolic Planes

[To son János:] For God's sake, please give it up. Fear it no less than the sensual passion, because it, too, may take up all your time and deprive you of your health, peace of mind and happiness in life.

[Bolyai's father urging him to give up work on non-Euclidean geometry.]

Wolfgang Bolyai (1775-1856)

[SE: Davis and Hersch, page 220

Hyperbolic geometry, discovered more than 170 years ago by C.F. Gauss (1777-1855, German), János Bolyai (1802-1860, Hungarian), and N.I. Lobatchevsky (1792-1856, Russian), is special from a formal axiomatic point of view because it satisfies all the axioms of Euclidean geometry except for the parallel postulate. In hyperbolic geometry there are infinitely many straight lines through a given point that do not intersect a given line.

Hyperbolic geometry and non-Euclidean geometry are considered in many books as being synonymous, but as we have seen there are other non-Euclidean geometries, particularly spherical geometry. It is also not accurate to say (as many books do) that non-Euclidean geometry was discovered about 170 years ago. Spherical geometry (which is clearly not Euclidean) was in existence and studied by at least the ancient Babylonians, Indians, and Greeks more than 2,000 years ago. Spherical geometry was of importance for astronomical observations and astrological calculations. Even Euclid in his Phaenomena [AT: Euclid] (a work on astronomy) discusses propositions of spherical geometry. Menelaus, a Greek of the first century, published a book Sphaerica, which contains many theorems about spherical triangles and compares them to triangles on the Euclidean plane. (Sphaerica survives only in an Arabic version. For a discussion see [NE: Kline, page 120].)

Most texts and popular books introduce hyperbolic geometry either axiomatically or via "models" of the hyperbolic geometry in the Euclidean plane. These models are like our familiar map projections of the earth and (like these maps of the earth) intrinsic straight lines on the hyperbolic plane (surface of the earth) are not, in general, straight in the model (map) and the model (map) also, in general, distorts distances and angles. We will return to the subject of projection, maps, and models in Chapter 16.

In this chapter we will introduce the geometry of the hyperbolic plane as the intrinsic geometry of a particular surface in 3-space, in much the same way that we introduced spherical geometry by looking at the intrinsic geometry of the sphere in 3-space. Such a surface is called an isometric embedding of the hyperbolic plane into 3-space. We will construct such a surface in the next section. Nevertheless, many texts and popular books say that David Hilbert (1862-1943, German) proved in 1901 that it is not possible to have an isometric embedding of the hyperbolic plane onto a closed subset of Euclidean 3-space. These authors miss what Hilbert actually proved. In fact, Hilbert [NE: Hilbert] proved that there is no real analytic isometry (that is, an isometry defined by real-valued functions which have convergent power series), but his arguments work in class C4 (that is, functions whose derivatives exist and are continuous up to the fourth derivative). Moreover, in 1955, N. Kuiper [NE: Kuiper] showed that there is a C1 isometric embedding onto a closed subset of 3-space, and in 1972 Tilla Milnor [NE: Milnor] proved that a C2 isometric embedding was not possible. The construction used here was shown to the author by William Thurston (1946- , American) in 19781; and it is not defined by equations at all, since it has no definite embedding in Euclidean space.

1 The idea for this construction is also included in Thurston's recent book Three-Dimensional Geometry and Topology, Vol. 1 (Princeton University Press, 1997, pages 49 and 50), and is also discussed in the recent book by the author, Differential Geometry: A Geometric Introduction (Prentice-Hall, 1998, page 31) where it is proved that the construction actually results in the hyperbolic plane (as d ® 0).