Body. Updated version is published in Mathematical Intelligencer, Vol. 23, No. 2, pp. 17-28, Spring 2001. David W. Henderson Department of Mathematics, Cornell University, Ithaca, NY, USA, dwh2@cornell.edu Daina Taimiða Department of Mathematics, Cornell University, dtaimina@math.cornell.edu For God's sake, please give it up. . Wolfgang Bolyai urging his son János Bolyai to give up work on hyperbolic geometry. In June of 1997, Daina was in a workshop watching the leader of the workshop, David, helping the participants study ideas of hyperbolic geometry using a paper and tape surface in much the same way that one can study ideas of spherical geometry by using the surface of a physical ball. But, Wait! Constructions of Hyperbolic Planes We will describe three different isometric constructions of the hyperbolic plane (or approximations to the hyperbolic plane) as surfaces in 3-space. 1. This is the paper and tape surface that David learned from William Thruston.
Figure 1. 2. Figure 2. Figure 3. 3. 1. Applications Of Non-Euclidean Geometry. Non-Euclidean Geometry. In three dimensions, there are three classes of constant curvature geometries. All are based on the first four of Euclid's postulates, but each uses its own version of the parallel postulate. The "flat" geometry of everyday intuition is called Euclidean geometry (or parabolic geometry), and the non-Euclidean geometries are called hyperbolic geometry (or Lobachevsky-Bolyai-Gauss geometry) and elliptic geometry (or Riemannian geometry). Spherical geometry is a non-Euclidean two-dimensional geometry. It was not until 1868 that Beltrami proved that non-Euclidean geometries were as logically consistent as Euclidean geometry. Non-Euclidean geometry. Version for printing In about 300 BC Euclid wrote The Elements, a book which was to become one of the most famous books ever written. Euclid stated five postulates on which he based all his theorems: To draw a straight line from any point to any other.
To produce a finite straight line continuously in a straight line. Proclus (410-485) wrote a commentary on The Elements where he comments on attempted proofs to deduce the fifth postulate from the other four, in particular he notes that Ptolemy had produced a false 'proof'. Playfair's Axiom:- Given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line. Although known from the time of Proclus, this became known as Playfair's Axiom after John Playfair wrote a famous commentary on Euclid in 1795 in which he proposed replacing Euclid's fifth postulate by this axiom.
To each triangle, there exists a similar triangle of arbitrary magnitude. Here is the Saccheri quadrilateral. Non-Euclidean geometry. Behavior of lines with a common perpendicular in each of the three types of geometry In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises when either the metric requirement is relaxed, or the parallel postulate is set aside. In the latter case one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the planar algebras which give rise to kinematic geometries that have also been called non-Euclidean geometry. Another way to describe the differences between these geometries is to consider two straight lines indefinitely extended in a two-dimensional plane that are both perpendicular to a third line: History[edit] Early history[edit] Terminology[edit]