Hyperbolic geometry

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Introduction

The Euclidean geometry that we all know and love is based on five simple axioms:

1. Each pair of points can be joined together by one, and only one, straight line;
2. Any straight line segment can be indefinitely extended in both directions;
3. There is exactly one circle of any given radius with any given centre;
4. All right angles are congruent to one-another; and, finally,
5. Given a line L and a point P not on L, there is a unique line through P that does not intersect L.

The fifth is called the parallel postulate. This axiom was always looked at as clumsy: it is less intuitive and harder to explain than the others. People figured that it must be the case that the first four imply the last. Hundreds of years of near-proofs came about, but no one managed to produce a true proof that the fifth was implied by the other four. Then suddenly, someone had a brilliant idea: what if it were not true? What if we assumed the parallel postulate was false? Well, once they did this, things started opening up big time. It was discovered that a false parallel postulate led to two different geometries that behaved much differently than the traditional Euclidean geometry.

There are two simple ways to make the parallel postulate false. First, we could change it to: « Given a line L and a point P not on L, every line through P intersects L. » This produces elliptical (or projective) geometry. The other way it can be made false is: « Given a line L and a point P not on L, there are infinitely many lines that pass through P and do not intersect L. » This produces hyperbolic geometry. These are both non-Euclidean geometries. (A given space might in fact be elliptical in some regions and hyperbolic in others (and Euclidean at the peripheries)).

General

Hyperbolic geometry was initially explored by Saccheri in the 1700s, who nevertheless believed that it was inconsistent, and later by Bolyai, Gauss, and Lobachevsky, after whom it is sometimes named. (See article on non-Euclidean geometry for more history.)

In Hyperbolic geometry (also called saddle geometry or Lobachevskian geometry) the term parallel only applies to lines that don't intersect in the hyperbolic plane but intersect at the circle at infinity. Lines that neither intersect in the hyperbolic plane nor the circle at infinity are called ultraparallel. One remarkable property of the hyperbolic plane is that there is a unique common perpendicular for each pair of ultraparallel lines (see Ultraparallel theorem).

There are four models commonly used for hyperbolic geometry: the Klein model, the Poincaré disc model, the Poincaré half-plane model and the Lorentz model.

The Klein model, also known as the projective disc model and Beltrami-Klein model, uses the interior of a circle for the hyperbolic plane, and chords of the circle as lines. This model has the advantage of simplicity, but the disadvantage that angles in the hyperbolic plane are distorted.

The Poincaré disc model, also known as the conformal disc model, also employs the interior of a circle, but lines are represented by arcs of circles that are orthogonal to the boundary circle, plus diameters of the boundary circle. The Poincaré half-plane model takes one-half of the Euclidean plane, as determined by a Euclidean line B, to be the hyperbolic plane (B itself is not included). Hyperbolic lines are then either half-circles orthogonal to B or rays perpendicular to B.

Both Poincaré models preserve hyperbolic angles, and are thereby conformal. All isometries within these models are therefore Möbius transformations.

A fourth model is the Lorentz model or hyperboloid model, which employs a 2-dimensional hyperboloid of revolution (of two sheets, but using one) embedded in 3-dimensional Minkowski space. This model is generally credited to Poincaré, but Reynolds (see below) says that Wilhelm Killing and Karl Weierstrass used this model from 1872.

Hyperbolic geometry has many properties foreign to Euclidean geometry, all of which are consequences of the hyperbolic postulate.

A physical model of hyperbolic geometry is Einstein's Special Theory of Relativity. For example, using the Poincare Disk model above, set up a polar coordinate system. Then any point on the disk can be identified with a rapidity vector in two dimensional space. (The point (2, 30), for example, could represent an object travelling on a plane with a uniform rapidity of 2 in the direction of 30 degrees north of the polar axis.) The Poincare distance between two points can be identified with the relative speed between two objects travelling in uniform motion on the plane. So every theorem in hyperbolic geometry can be translated into a true statement in special relativity.

Visualizing hyperbolic geometry

The famous circle limit III [1] and IV [2] drawings of M. C. Escher illustrate the unit disc version of the model quite well. In both one can see the geodesics (in III the white lines are not geodesics, but they run alongside them). It is also possible to see quite plainly the negative curvature of the hyperbolic plane, via its effect on the sum of angles in triangles and squares.

For example, in III every vertex is the intersection of three triangles and three squares. In normal Euclidean plane, this would sum up to 450°, leading to a contradiction. Hence we see that the sum of angles of a triangle in the hyperbolic plane must be smaller than 180°. Another visible property is the fact that the hyperbolic plane has exponential growth. In IV, for example, one can see that the number of angles with a distance of n from the center rises exponentially. The angles have equal hyperbolic area, so the area of a ball of radius n must rise exponentially in n.

Another fun thing to do, when things are slow at the office, is to cut a couple of sheets of paper into a few dozen identically sized squares, and tape them together putting five squares at each corner. Then note how two "rows" of squares which are next to each other at on point will diverge until they are arbitrarily far apart.

Relationship to Riemann surfaces

Two-dimensional hyperbolic surfaces can also be understood according to the language of Riemann surfaces. According to the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic. Most hyperbolic surfaces have a non-trivial fundamental group π₁ = Γ, known as the Fuchsian group. The quotient space H/Γ of the upper half-plane modulo the fundamental group is known as the Fuchsian model of the hyperbolic surface. The Poincaré half plane is also hyperbolic, but is simply connected and noncompact. It is the universal cover of the other hyperbolic surfaces.

The analogous construction for three-dimensional hyperbolic surfaces is the Kleinian model.

See also

• angle of parallelism
• Hyperbolic space
• Hyperbolic structure
• Fuchsian group
• Fuchsian model
• Hyperbolic 3-manifold
• Kleinian group
• Kleinian model
• Poincaré metric
• Riemann surface

References

• Reynolds, William F. (1993) "Hyperbolic Geometry on a Hyperboloid", American Mathematical Monthly 100:442-455.
• Stillwell, John (1996) Sources in Hyperbolic Geometry, volume 10 in AMS/LMS series History of Mathematics.de:Hyperbolische Geometrie

he:גאומטריה היפרבולית it:Geometria iperbolica pl:Geometria hiperboliczna zh:双曲几何