The aim of this bachelor's thesis has been to investigate surfaces that are the main contributions to scattering amplitudes in a type of string theory. These are constant mean curvature surfaces in hyperbolic 3-space. Classically the way to find such surfaces has been to solve a non-linear partial differential equation. In many spaces constant mean curvature surfaces are intimately connected to certain harmonic maps, known as the Gauss maps. In 1995 Dorfmeister, Pedit, and Wu established a method for constructing harmonic maps into so-called symmetric spaces. I investigate a generalization of this method that can be applied to find constant mean curvature surfaces in hyperbolic 3-space by using the intimate connection between these surfaces and harmonic maps. This method relies on a factorization of a Lie-group valued map. I show an explicit method for finding the factorization in terms of what is known as the Birkhoff factorization. Because approximation methods for the Birkhoff factorization are known, this allowed me to use the method constructively to find constant mean curvature surfaces in hyperbolic 3-space.