The Shape of Space: The Beginning of Non-Euclidean Geometry

Overview

During the nineteenth century several mathematicians realized, at about the same time, that a geometry based on Euclid's classic system was not the only possibility. A non-Euclidean universe was too strange for many to accept at first. Yet it was non-Euclidean geometry that paved the way for Albert Einstein's theory of general relativity in the early 1900s and the modern understanding of space-time.

Background

Geometry is one of the oldest branches of mathematics. Because it deals with the properties of objects and space, such as lengths, areas, volumes, and angles, it was of immediate practical significance. Ancient artists, artisans, engineers, architects, warriors, navigators, and astronomers all had objects to measure and build, courses to plot, or trajectories to predict.

The familiar geometry of planes and solids that most modern students learn in high school was first formalized in Euclid's (c. 330-260 b.c.) Elements in about 300 b.c. The Elements were based on ten statements called axioms, or postulates. These assertions were used as starting points and accepted as given. From the ten postulates, several hundred theorems were then proven by deductive logic. This axiomatic-deductive method has become one of the standard ways of doing mathematics.

Euclid's fifth postulate is also called the parallel postulate. It states that one and only one line can be drawn parallel to a given line through a point not on the line. This is a more complex statement than is usual for a postulate, and in fact Euclid himself seemed rather reluctant to introduce it as such. He proved 28 theorems from the first 4 postulates alone, a group that has since become known as absolute geometry. As other mathematicians worked with the Elements, the number of theorems in this group continued to grow. Eventually, many assumed, the fifth postulate would be derived as a theorem from the first four.

After Euclid's time, many Greek, Islamic, and later European mathematicians tried to accomplish this feat. None ever succeeded, although some erroneously convinced themselves that they had. In most cases what they had actually done was to replace the parallel postulate with an equivalent but differently worded statement that they regarded as self-evident, such as "two parallel lines are equidistant." However, the mathematicians were correct to recognize the significance of postulate five. It was the point at which two great branches of geometry were to diverge.

Impact

Like many mathematicians before him, the Hungarian Farkas Bolyai (1775-1856) was obsessed with trying to prove the parallel postulate, and he passed this preoccupation along to his son János (1802-1860). The younger man eventually concluded that a proof was impossible and recognized that absolute geometry branched into two groups of cases: one group in which the parallel postulate was accepted, and one group in which it was not. He developed a geometry without it.

János Bolyai's creation was a form of hyperbolic geometry, in which the fifth postulate was replaced by one allowing more than one parallel line through the fixed point. In 1823 he described his work in a paper he called "Appendix Scientiam Spatii Absolute Veram Exhibens" ("Appendix Explaining the Absolutely True Science of Space").

Farkas Bolyai was an old friend of Carl Friedrich Gauss (1777-1855), and he proudly sent a draft of his son's paper to the eminent mathematician. Gauss replied that he had in fact worked such a system out himself decades before. Although he had never published his work, and therefore had no claim to precedence, his response stunned János Bolyai. The paper was published in 1832 as an appendix to a textbook authored by his father, but Bolyai himself faded into relative obscurity.

Nikolai Lobachevsky (1792-1856), a professor at the University of Kazan on the fringes of Siberia, was not to be so easily discouraged. His non-Euclidean geometry was the first in print, appearing in the Kazan Messenger in 1829. It was very similar to that of Bolyai, who was completely unknown to Lobachevsky. What Gauss might have thought of decades before but declined to publish did not concern him. Lobachevsky wrote several expositions of his system, culminating in the book Pangeometrie in 1855. However, his fame also grew mainly after his death, when the implications of non-Euclidean geometry were better understood.

It was Georg Friedrich Bernhard Riemann (1826-1866) who first began shedding light on those implications. Riemann's classic paper was written to secure his admission to the University of Göttingen as a Privatdozent, or unpaid lecturer dependent on student fees. It was entitled "Über die Hypothesen, welche der Geometrie zu Grunde Liegen" ("On the Hypotheses that Form the Foundations of Geometry"), and included another formulation of a non-Euclidean geometry.

Riemann's system differed not only from Euclid's but from those of Bolyai and Lobachevsky as well. Euclid had postulated only one parallel line through a point not on the line. Bolyai and Lobachevsky had allowed more than one. Riemann's elliptical geometry was built on the postulate that there are no lines parallel to a given line through a point not on it. Furthermore, all lines are of finite length. This can be visualized by considering a globe (since a sphere is just an especially symmetrical ellipsoid), on which all the meridians, or lines of longitude, meet at the poles. In his paper Riemann posed questions about what type of geometry represented that of real space. Thus began the idea that non-Euclidean geometry might have physical meaning.

In 1872 Felix Klein (1849-1925) published two papers entitled "On the So-called non-Euclidean Geometry." Klein's major contribution to this field was the idea that both Euclidean geometry and the non-Euclidean geometries of Lobachevsky and Riemann are special cases of a more general discipline called projective geometry.

Geometries may be classified by the type of transformations, or mathematical manipulations, that can be performed without contradicting any of their theorems. The more transformations under which the geometry is invariant (unchanging), the more general it is. Projective geometry is concerned mainly with properties such as when points lie on the same plane and when a set of lines meet in a single point. These properties are invariant under a larger group of transformations than the congruence, or equality of lengths, angles, and areas central to Euclidean geometry.

Hermann Minkowski (1864-1909) developed a geometry encompassing the usual three dimensions of space and adding time as a fourth dimension. This geometrical system has since come to be called Minkowski space. It was in the language of non-Euclidean geometries like Minkowski space that Albert Einstein (1879-1955) was able to frame his general theory of relativity. In general relativity gravity is described in terms of the curvature of spacetime. For example, imagine a sheet of rubber with grid lines like graph paper, suspended horizontally so that it forms a flat surface. With no weight on it, the grid has straight lines and right angles, corresponding to the "flat space" of Euclidean geometry.

If you place a ball on the surface, the rubber sheet stretches around it. The curvature of the grid increases as it gets closer to the ball. This corresponds to the curvature of space-time near a massive object. The description of this curved space was made possible by a few nineteenth-century mathematicians, who were willing to consider the idea that there were geometries beyond that of Euclid.

SHERRI CHASIN CALVO