Hyperbolic geometry

Lines through a given point P and hyperparallel to line l.

A triangle immersed in a saddle-shape plane (a hyperbolic paraboloid), as well as two diverging parallel lines.

In mathematics, Hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is rejected. The parallel postulate in Euclidean geometry states (for two dimensions) that given a line l and a point P not on l, there is exactly one line through P that does not intersect l. In hyperbolic geometry, this postulate is false because there are at least two distinct lines through P which do not intersect l. Upon assuming this, we can prove an interesting property of hyperbolic geometry: that there are two classes of non-intersecting lines. Let B be the point on l such that the line PB is perpendicular to l. Consider the line x through P such that x does not intersect l, and the angle theta between PB and x (counterclockwise from PB) is as small as possible (i.e., any smaller an angle will force the line to intersect l). This is called a hyperparallel line (or simply parallel line) in hyperbolic geometry. Similarly, the line y that forms the same angle theta between PB and itself but clockwise from PB will also be hyper-parallel, but there can be no others. All other lines through P not intersecting l form angles greater than theta with PB, and are called ultraparallel (or disjointly parallel) lines. Notice that since there are an infinite number of possible angles between theta and 90 degrees, and each one will determine two lines through P and disjointly parallel to l, we have an infinite number of ultraparallel lines.

Thus we have this modified form of the parallel postulate: In Hyperbolic Geometry, given any line l, and point P not on l, there are exactly two lines through P which are hyperparallel to l, and infinitely many lines through P ultraparallel to l.

The differences between these types of lines can also be looked at in the following way: the distance between hyper parallel lines goes to 0 as you move on to infinity. However, the distance between ultraparallel lines does not go to 0 as you move to infinity.

The angle of parallelism in Euclidean geometry is a constant, that is, any length BP will yield an angle of parallelism equal to 90°. In hyperbolic geometry, the angle of parallelism varies with what is called the Πぱい(p) function. This function, described by Nikolai Ivanovich Lobachevsky produced a unique angle of parallelism for each given length BP. As the length BP gets shorter, the angle of parallelism will approach 90°. As the length BP increases without bound, the angle of parallelism will approach 0°. Notice that due to this fact, as distances get smaller, the hyperbolic plane behaves more and more like Euclidean geometry. So on the small scale, an observer within the hyperbolic plane would have a hard time determining they are not in a Euclidean plane.

History

Hyperbolic geometry was initially explored by Omar Khayyám and later Giovanni Gerolamo Saccheri, in an attempt to prove it inconsistent and thereby prove the parallel postulate. In the nineteenth century it was fully explored by János Bolyai, Karl Friedrich Gauss, and Nikolai Ivanovich Lobachevsky, after whom it sometimes is named. Eugenio Beltrami then provided models of it, and used this to prove that hyperbolic geometry was consistent if Euclidean geometry was. (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 pairs of lines that don't intersect in the hyperbolic plane but intersect at the circle at infinity. Pairs of 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).

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

Models of the hyperbolic plane

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, or hyperboloid model. These models define a real hyperbolic space which satisfies the axioms of a hyperbolic geometry.

Poincaré disc model of great rhombitruncated {3,7} tiling

1. 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.
2. 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.
3. 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.
4. 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.
   • This model has direct application to special relativity, as Minkowski 3-space is a model for spacetime, suppressing one spatial dimension. One can take the hyperboloid to represent the events that various moving observers, radiating outward in a spatial plane from a single point, will reach in a fixed proper time. The hyperbolic distance between two points on the hyperboloid can then be identified with the relative rapidity between the two corresponding observers.

Visualizing hyperbolic geometry

M. C. Escher's famous prints Circle Limit III and Circle Limit IV illustrate the conformal disc model quite well. In both one can see the geodesics (in III the white lines are not geodesics, but hypercycles, which run alongside them). It is also possible to see quite plainly the negative curvature of the hyperbolic plane, through its effect on the sum of angles in triangles and squares.

For example, in Circle Limit III every vertex belongs to three triangles and three squares. In the Euclidean plane, their angles would sum to 450°, i.e. a circle and a quarter. From this we see that the sum of angles of a triangle in the hyperbolic plane must be smaller than 180°. Another visible property is exponential growth. In Circle Limit IV, for example, one can see that the number of angels within a distance of n from the center rises exponentially. The angels have equal hyperbolic area, so the area of a ball of radius n must rise exponentially in n.

There are several ways to physically realize a hyperbolic plane (or approximation thereof). A particularly well-known paper model based on the pseudosphere is due to William Thurston. The art of crochet has been used to demonstrate hyperbolic planes with the first being made by Daina Taimina. In 2000, Keith Henderson demonstrated a quick-to-make paper model dubbed the "Hyperbolic soccerball."

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 $\pi _{1}=\Gamma ,$ known as the Fuchsian group. The quotient space H/Γがんま 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
• Elliptic geometry
• Hyperbolic space
• Hyperbolic structure
• Hyperboloid model
• Hyperboloid structure
• Fuchsian group
• Fuchsian model
• Hjelmslev transformation
• Hyperbolic 3-manifold
• Klein model
• Kleinian group
• Kleinian model
• Poincaré disk model
• Poincaré half-plane model
• Poincaré metric
• Riemann surface
• Khayyam-Saccheri quadrilateral
• Special relativity
• Ultraparallel theorem

External links

• Visions of Infinity: Tiling a hyperbolic floor inspires both mathematics and art Science News: Dec. 23, 2000; Vol. 158, No. 26/27, p. 408

• Java freeware for creating sketches in both the Poincaré Disk and the Upper Half-Plane Models of Hyperbolic Geometry University of New Mexico

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.
• Samuels, David. (March 2006) "Knit Theory" Discover Magazine, volume 27, Number 3.