The Hyperbolic Geometry Exhibit (original) (raw)

Up: Colleen Robles

Welcome to the exciting world of hyperbolic geometry! Hyperbolic geometry is one of the most important examples of a "non-Euclidean" geometry, with far reaching applications in math and science, including special relativity. Moreover, by approaching hyperbolic geometry through analogies and models, even the novice can enjoy the elegance and surprising intricacy of a deep mathematical theory.

This exhibit presents an introduction to hyperbolic geometry with the assistance of graphics, animations and interactive applications. Enjoy the exhibit!

Euclidean Geometry

• Historically, hyperbolic geometry was discovered as a consequence of questions about the parallel postulate. Appearing in Euclid's original treatise, the parallel postulate provoked two millenia of mathematical investigation about the nature of logic, proof, and geometry.
• Once non-Euclidean geometries became known, it became clear that a good way to understand geometry is to consider its isometries, or "rigid motions".
• As an example of the way in which the underlying geometry and isometries are related, in Euclidean geometry, all isometries are products of reflections. The significance of this curious fact is that it allows one to think of all isometries as built up out of simple building blocks. When other kinds of basic building block are substituted for reflection, isometries for non-Euclidean geometries result.

Hyperbolic Geometry

• In hyperbolic geometry, lines, points, distance and motion all behave differently than their Euclidean counterparts. In order to work with these abstract concepts, we introduce concrete models of hyperbolic geometry. In particular, the Poincaré Disk, Upper Half Plane, Klein-Beltrami and Minkowski models are presented,together with expository text, appropiate equations, illustrative graphics, animations and interactive applications.
• Different models of hyperbolic space illustrate different features of the geometry. To fully understand a concept of hyperbolic geometry, it is usually best to look at it in all of the models. To do that, we consider the problem of converting between models, via both equations and animations. (These pages were contributed by John Hartman.)
• As in Euclidean geometry, the isometries of Hyperbolic 2-Space are at the heart of the geometry. We take a look at them in the various models, with text, graphics, animation and interactive applications.
• Another way to compare hyperbolic and Euclidean geometry is to compare trigonometry in both. A few relavent equations of hyperbolic trigonometry are included here.

Special Relativity

One of the most interesting contexts in which hyperbolic geometry plays an important role is special relativity. The relativistic concept of spacetime unifies both Euclidean geometry and the Minkowski model of hyperbolic space. In a way, hyperbolic geometry can be thought of as the geometry of a universe in which things travel faster than the speed of light.(These pages were contributed by John Hartman.)

Additional Information:

• Table of Contents
• Bibliography

Up: Colleen Robles

Created: Jul 15 1996 --- Last modified: Mon Jul 15 09:37:16 1996