Hilbert's arithmetic of ends
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In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, specifically in the area of hyperbolic geometry
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...

, Hilbert's arithmetic of ends is an algebraic construction introduced by German mathematician David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...

.

Given a hyperbolic plane
Hyperbolic space
In mathematics, hyperbolic space is a type of non-Euclidean geometry. Whereas spherical geometry has a constant positive curvature, hyperbolic geometry has a negative curvature: every point in hyperbolic space is a saddle point...

, Hilbert's construction yields a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

 with the ideal points or ends as elements of the field. (Note here that this usage of end is slightly different from that of a topological end.)

Introduction

In a hyperbolic plane, one can define an ideal point or end to be an equivalence class of parallel
Parallel (geometry)
Parallelism is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. The assumed existence and properties of parallel lines are the basis of Euclid's parallel postulate. Two lines in a plane that do not...

 rays. The set of ends can then be topologized in a natural way and forms a circle. This is most easily seen in the Poincaré disk model
Poincaré disk model
In geometry, the Poincaré disk model, also called the conformal disk model, is a model of n-dimensional hyperbolic geometry in which the points of the geometry are in an n-dimensional disk, or unit ball, and the straight lines of the hyperbolic geometry are segments of circles contained in the disk...

 or Klein model
Klein model
In geometry, the Beltrami–Klein model, also called the projective model, Klein disk model, and the Cayley–Klein model, is a model of n-dimensional hyperbolic geometry in which points are represented by the points in the interior of the n-dimensional unit ball and lines are represented by the...

 of hyperbolic geometry, where every ray intersects the limit circle (also called the circle at infinity) in a unique point
Point (geometry)
In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...

. One thing worthy of note is that these points are not part of the hyperbolic plane itself.

A line
Line (geometry)
The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...

 will have exactly two ends, and for each two distinct ends there is a unique line that is a representative of these two ends. For the purpose of Hilbert's arithmetic, it is expedient to denote a line as an ordered pair (ab) of its (necessarily distinct) ends.

We fix one arbitrary hyperbolic line and label its ends 0 and , calling it . The set is now defined to be the set of all ends in the plane different from , and , so that is the set of all ends of the plane.

We will now define an addition and a multiplication on that makes it into a field.

Addition of ends

Given two ends α, β not equal to . Suppose is a point on the line . Let be its hyperbolic reflection in the line and let be its hyperbolic reflection in the line . Then α + β is defined to be the end of the perpendicular bisector of other than .


This definition works because the perpendicular bisector under consideration always has as an end again. The definition becomes more intelligible when one sees that we have the equation


where for any end , denotes the hyperbolic reflection in the line . This also makes it clear that the addition is independent of the choice of the point , whence it is well-defined
Well-defined
In mathematics, well-definition is a mathematical or logical definition of a certain concept or object which uses a set of base axioms in an entirely unambiguous way and satisfies the properties it is required to satisfy. Usually definitions are stated unambiguously, and it is clear they satisfy...

. It makes the set (H, +) an abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

 with the element 0 as identity
Identity element
In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...

. The inverse of an element is the end other than of the line obtained by reflecting in .

The Three Reflections Theorem

The existence of this addition operation is based on a theorem on three hyperbolic reflections through parallel lines.

Theorem.
Given three lines a ,b, c in the hyperbolic plane with a common end ω, there exist a fourth line d with end ω such that reflection in d is equal to the product of the reflections in a, b, c:
where denotes the reflection in the line .


From this theorem it is clear that the addition defined above takes , and for a, b and c respectively. The line is then the line d whose existence is garantueed by the theorem.

Multiplication of ends

The multiplication over the field is defined by fixing a line perpendicular to the line where they meet at the point O, and label one of its ends 1, the other −1.

Definition.
Given ends , the lines and meet the line with right angles at A and B, respectively.
So C at where the line is perpendicular to , is the point which satisfies the relation,


where the point A' is the reflection of A respect to the point O.


In other words, the point C satisfies the relation OA + OB = OC, according to the euclidean segment addition. So the field has an additive multiplication over line segments. It makes an abelian group with identity 1.

Rigid motions

Let be a hyperbolic plane and H its field of ends, as introduced above. In the plane , we have rigid motions and their effects on ends as follows:
  • The reflection in sends to −x.


  • The reflection in (1, −1) gives,


  • Translation
    Translation
    Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. Whereas interpreting undoubtedly antedates writing, translation began only after the appearance of written literature; there exist partial translations of the Sumerian Epic of...

     along that sends 1 to any , a > 0 is represented by


  • For any , there is a rigid motion σ(1/2)a σ0, the composition of reflection in the line and reflection in the line , which is called rotation around is given by


  • The rotation
    Rotation
    A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...

    around the point O, which sends 0 to any given end , effects as


on ends. The rotation around O sending 0 to gives



For a more extensive treatment than this article can give, confer.
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