Hyperbolic geometry
Hyperbolic geometry is a
non-Euclidean geometry, meaning that the
parallel postulate of
Euclidean geometry is rejected. The parallel postulate in Euclidean geometry states that given a line
l and a point
P not on
l, there is a unique line through
P that does not intersect
l. In hyperbolic geometry, this postulate is false in the following way: 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.
Encyclopedia
Hyperbolic geometry is a
non-Euclidean geometry, meaning that the
parallel postulate of
Euclidean geometry is rejected. The parallel postulate in Euclidean geometry states that given a line
l and a point
P not on
l, there is a unique line through
P that does not intersect
l. In hyperbolic geometry, this postulate is false in the following way: 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 is as small as possible . This is called a
hyperparallel 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 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 ? 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 Lobachevsky, after whom it is sometimes named. Eugenio Beltrami then provided models of it, and used this to prove that hyperbolic geometry was consistent if Euclidean geometry was.
In
Hyperbolic 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 .
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.
- 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 .
- 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 transformation
...
s.
- A fourth model is the Lorentz model or hyperboloid model, which employs a 2-dimensional hyperboloid of revolution embedded in 3-dimensional Minkowski space. This model is generally credited to Poincaré, but Reynolds says that Wilhelm Killing and Karl Weierstrass used this model from 1872. One can take the hyperboloid to represent the events that various moving observers radiating outward from a single point will reach in a fixed proper time.
There are alternative ways to set up a physical model of hyperbolic geometry in Einstein's Special Theory of Relativity. For example, set up a polar coordinate system on the hyperbolic plane. Then any point can be identified with a uniform motion on a relativistic plane. The hyperbolic point , 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 hyperbolic distance between two points of the hyperbolic plane can be identified with the relative speed between two objects travelling in the corresponding uniform motion on the relativistic plane. So every theorem in hyperbolic geometry can be translated into a true statement in special relativity.
Visualizing hyperbolic geometry
M. C. Escher's famous prints and
illustrate the conformal disc model quite well. In both one can see the geodesics . 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 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 . A particularly well-known paper model based on the
pseudosphere is due to
William Thurston. In 1997,
Daina Taimina crocheted a hyperbolic plane based on Thurston's models. 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 known as the Fuchsian group. The
quotient space H/G 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
References
- Reynolds, William F. "Hyperbolic Geometry on a Hyperboloid", American Mathematical Monthly 100:442-455.
- Stillwell, John. Sources in Hyperbolic Geometry, volume 10 in AMS/LMS series History of Mathematics.
- Samuels, David. "Knit Theory" Discover Magazine, volume 27, Number 3.
