A new method for building icosahedral quasicrystal structures out of paper is explored and described. Three-dimensional quasicrystal patterns arranged in spherical formations can be modeled using sets of folded paper strips that are joined into loops. Each strip consists of a sequence of golden rhombi, separated by parallel mountain and valley folds that may be pre-scored. The loops overlap and are connected by tape such that corresponding folds in different strips reinforce one another to form a rigid structure. These sphere-shaped polyhedra are nonconvex but share similarities with zonohedra. We demonstrate how the strips appear both in their two-dimensional unrolled state and in their completed construction. Interesting patterns can be observed by comparing icosahedrally symmetric designs with those that are less regular. These differences are related to the underlying choice of tiling units that are used to generate the packing and resulting surface design.