Constructions by Straightedge and Compass in the Hyperbolic Plane

Abstract

As in the Euclidean case, questions of constructibility by straightedge and compass involve consideration of certain algebraic number fields called quadratic-surd fields. They are obtained by starting with the rational field and adjoining square roots in succession. Roughly speaking, a point with polar coordinates r, θ can be constructed in the Euclidean case if r and sin θ are in such a field (in which case cos θ and tan θ are also in such a field), while in the hyperbolic case, the point can be constructed if tanh r and sin θ are in such a field. The first two sections give the proof (which is not short) that if the point can be constructed, then tanh r and sin θ are in a quadratic-surd field. Section 11.6 contains the proof (also not short) that every such point can be constructed. As a result, there are certain differences between Euclidean and the hyperbolic cases. In the hyperbolic case, a segment cannot in general be trisected, while some circles can be squared (that depends on the radius). In the general theory, the constructions start with a single given line and a single given point on that line, but no segment of unit length, because a unit of length is implied by the hyperbolic axioms. Certain construction problems are considered in which other things are given in advance, for example constructing the angle of parallelism II (y), when a segment of length y is given. Since any two ideal points in the hyperbolic plane determine a unique line, it is assumed that the straightedge can draw the line between any two ideal points, but it is shown in Section 11.9 that that is not really necessary. All constructions can be made (with more trouble) if the straightedge is only assumed to be able to draw a line between any two finite points. The purpose of the last section is to show that constructibility is not merely a matter of interest for engineering or draftsmanship, but also throws light on the geometry itself. The set of constructible points (it is a countable set!) constitutes a geometry that satisfies all the axioms of the hyperbolic plane, except that there is no axiom of completeness, either for the real number system ℝ or for the points on a line. (Those axioms are of course not categorical, since, in particular, the ordinary hyperbolic plane satisfies them.)