Hyperbolic.m: A general-purpose hyperbolic geometry package

Article: 131 of geometry.college
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From: sander@geom.umn.edu (Evelyn Sander)
Subject: Hyperbolic.m: hyperbolic geometry software
Organization: University of Minnesota, Twin Cities
Date: Tue, 1 Feb 1994 20:58:29 GMT

"The Elements," a 325 BC publication of Euclid, records Euclid's principles of geometry. For many centuries, these principles remained the fundamental basis of the study of geometry. There are five postulates of particular importance:

1. A straight line may be drawn from any point to any other point.

2. The straight line may be produced to any length.

3. Around any point as a center, a circle of any radius may be described.

4. Any two right angles are equal.

5. Given a line and a point not on the line, there is exactly one line through the given point not intersecting the given line.[B]

This fifth postulate, also called the parallel postulate, receives special attention. Mathematicians did not like it because it possesses a different character from the first four, being global rather than local in nature. Since intersection of lines can take place at an arbitrary distance, there is no way to actually check whether lines intersect. Thus starting as soon as the 4th century AD, there were attempts to prove that 5 follows from 1-4. All attempts to prove this failed, since in fact the parallel postulate is independent of 1-4. It was not until the middle of the 19th century that Bolyai and Lobachevsky separately showed that in fact it is possible to construct a consistent geometry in which the parallel postulate does not hold. In this geometry, there are infinitely many non-intersecting lines through a given point. It is called hyperbolic geometry.

Mathematicians found it difficult to accept hyperbolic geometry. In the middle of the 19th century, most were unable to imagine or accept any geometric system which did not fit with Euclid's principles. As an indication, even Bolyai's father, a mathematician himself, was unable to believe or understand the work of his son because of his belief in the absolute truth of Euclidean geometry.[B]

Soon after the work of Bolyai and Lobachevsky, others constructed models of hyperbolic geometry within Euclidean space. By renaming certain geometric objects from Euclidean space as the lines and circles, people developed systems in which all of Euclid's postulates hold except that there were infinitely many non-intersecting lines, violating the parallel postulate. These models, all isometric, verify the existence of hyperbolic geometry. As Courant and Robbins state in "What is Mathematics," "This must, eo ipso, be just as consistent as the original Euclidean geometry, because it is presented to us, seen from another point of view and described with other words, as a body of facts of ordinary Euclidean geometry."[C]

The modern approach to geometry is somewhat different from that of "The Elements." However, a (geodesically complete Riemannian) geometry on a surface is essentially a structure on the surface obeying the modernized versions of postulates 1-4. Each surface has a unique associated geometry locally isometric to the sphere, the plane, or the hyperbolic plane. The sign of a special constant called the Euler characteristic determines which of the three kinds of geometries a surface possesses. There is a complete classification of all compact connected surfaces. By computing the Euler characteristic of all these surfaces, one can draw an amazing conclusion; of these infinite number of possible surfaces, only seven have non-negative curvature, implying that all but seven are hyperbolic. For example, all n-holed tori with n greater than 1 have an associated hyperbolic geometry. Among the quite common class of compact connected surfaces, a geometric structure that took over a millennium to discover is overwhelmingly the most frequent.

Since every surface with a hyperbolic geometry is locally isometric to the hyperbolic plane, the modern study of hyperbolic geometry involves studying models for the hyperbolic plane which are embedded in Euclidean space. I describe the properties of the Klein, Poincare, and upper half plane models with accompanying figures in the article Models of the Hyperbolic Plane.

Oliver Goodman, a postdoc at the Geometry Center, has written a general-purpose hyperbolic geometry software package called Hyperbolic.m. Operating within Mathematica, Hyperbolic.m allows the user to manipulate hyperbolic objects (in any dimensional hyperbolic space) such as points, lines, vectors, polyhedra, and other structures. The user enters these structures within any of five hyperbolic models. The package calculates angles and lengths for vectors and distances between points, convert between different models, and apply isometries. It also computes triangle groups. Hyperbolic.m allows the user to display the results of manipulations using Mathematica graphics. See the description of models referenced above, which contains figures made with Hyperbolic.m.

Although there are several programs which, when combined, do what Hyperbolic.m does, there are none that do everything. The package's versatility allows the user to use only one program, and to take advantage of other functions within Mathematica.

If you want a copy of this program, click to download hyperbolic.tar. Hyperbolic.m is also available by anonymous ftp from ftp.geom.umn.edu in the file hyperbolic.tar.Z in the pub/software directory. Any questions or comments on the package should be sent to Goodman (oag@geom.umn.edu).

This article was written based on interviews with Goodman, as well as material from the following references:

[B] Janos Bolyai, "Appendix," Originally published in 1832, North Holland, 1987, Introduction by Ferenc Karteszi.

[C] Courant/Robbins, "What is Mathematics?" Oxford University Press, London, 1941.

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