Constructible numbers in double elliptic geometry.

Abstract

Constructible numbers serve to connect the areas of algebra and geometry. In the Euclidean plane, the straight edge and compass can be used to create line segments with the length of any finite combination of arithmetic operations or square roots. This set of numbers has the algebraic properties of an Archimed an ordered field. Double elliptic geometry differs from Euclidean geometry in that it requires that all lines meet and are hence finite in length. Constructions in this geometry are isomorphic to constructions on a sphere. The constructions presented in Euclid's Elements can be redone on the sphere to demonstrate the differences between the two geometries. When this is done, the line segments that are constructed are not the same lengths as the line segments in Euclidean geometry. Instead, this set of segments has properties based upon trigonometric identities, specifically ones involving the cosine function. Though the numbers re different, theorem concerning the number resemble theorems concerning the numbers in Euclidean geometry. This resemblance between the theorems can be generalized to other geometries as well.