NON-EUCLIDEAN GEOMETRY

study of points, lines, planes, and space through the use of assumptions other than those forming the basis of Euclidean geometry. Figure 1: Rays having a common perpendicular in hyperbolic and elliptic geometries (see text). Non-Euclidean geometry results from making certain assumptions and then drawing conclusions generally consistent with one's spatial intuition concerning objects of moderate size and yet rich in certain relationships that affront the intuition, particularly relationships concerning the concept of parallelism extended to large distances. For instance, similar figures (the same shape) are necessarily congruent (the same shape and size): no plan or model or map can be truly accurate. The two principal types of such a geometry are vividly distinguished by referring to the following imaginary construction. Two rays in a plane are drawn from points A and B, perpendicular to and on the same side of the line that connects A and B (Figure 1). Instead of remaining equidistant they become farther apart or closer together. In the former case, when the rays diverge, the non-Euclidean geometry is said to be hyperbolic (from the Greek hyperballein, to throw beyond). In the latter case, when the rays converge and ultimately intersect, the geometry is said to be elliptic (from elleipein, to fall short). Because it is impossible in practice to measure how far apart the rays will be when extended millions of miles, it is quite conceivable that man is living in a non-Euclidean universe. (Because intuition is developed from relatively limited observations, it is not to be trusted in this regard.) In such a world, railroad tracks can still be equidistant, but then they will not be perfectly straight. Figure 2: Illustration of Euclid's fifth postulate (see text). In the language of axiomatic mathematics, non-Euclidean geometry satisfies all of Euclid's axioms except either the fifth or the second. The axioms of Euclid are stated fully in Euclidean geometry: Euclid's work. The second axiom states that an interval can be prolonged indefinitely. The fifth states that, if a line meets two other lines so as to make the angles a and b on one side of it together less than two right angles, the other lines, if prolonged indefinitely, will meet on this side (in Figure 2, the right side). Figure 1: Rays having a common perpendicular in hyperbolic and elliptic geometries (see text). Figure 3: Ultraparallel lines (see text). In hyperbolic geometry, axiom 5 is denied, because, if the ray from A in Figure 1 is replaced by one making a very slightly smaller angle with AB, the new ray from A and the old one from B may converge at first, attain a minimal distance, and then diverge (Figure 3). In elliptic geometry, axiom 5 is satisfied trivially, but axiom 2 (interpreted as giving the line an infinite length) is denied, because now the line is closed, like a circle. Additional reading Richard J. Trudeau, The Non-Euclidean Revolution (1987), presents the development of non-Euclidean geometry and its impact on geometry and natural philosophy. B.A. Rosenfeld (B.A. Rozenfeld), A History of Non-Euclidean Geometry (1988; originally published in Russian, 1976), is also of interest. D.M.Y. Sommerville, The Elements of Non-Euclidean Geometry (1914, reprinted 1958), is an elementary exposition. Julian Lowell Coolidge, A Treatise on the Circle and the Sphere (1916, reissued 1971), offers an early account of inversive geometry. H.S.M. Coxeter, Non-Euclidean Geometry, 5th ed. (1965), centres on the projective approach and includes an extensive bibliography. Patrick J. Ryan, Euclidean and Non-Euclidean Geometry (1986), is a development based on plane analytic geometry. Herbert Meschkowski, Noneuclidean Geometry (1964; originally published in German, 2nd ed., 1961), is a readable introduction to the Poincar model. Patrick du Val, Homographies, Quaternions, and Rotations (1964), explores spherical space with the aid of quaternions. L. Fejes Tth, Regular Figures (1964), includes good drawings of Euclidean and non-Euclidean tessellations. Hans Schwerdtfeger, Geometry of Complex Numbers (1962, reissued 1979), covers the analytic approach to inversive geometry. J.B. Wilker, Inversive Geometry, in Chandler Davis, Branko Grnbaum, and F.A. Sherk (eds.), The Geometric Vein (1981), pp. 379442, focuses on the Mbius group. H.S. MacDonald Coxeter The Editors of the Encyclopdia Britannica