Up: Colleen Robles
The HYPERBOLIC GEOMETRY Exhibit
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!
- 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.
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.)
- 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
- 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
- Another way to compare hyperbolic and Euclidean geometry is
to compare trigonometry in both. A few relavent equations of hyperbolic trigonometry are included here.
Up: Colleen Robles
Created: Jul 15 1996 ---
Last modified: Mon Jul 15 09:37:16 1996