For more than a millennium, Islamic artists and craftsmen have used geometric patterns to decorate buildings, cloth, pottery, and other artifacts. Many of these patterns were "wallpaper" patterns - they were planar patterns that repeated in two different directions. Recently related patterns have also been drawn on the Platonic solids, which can conceptually be projected outward onto their circumscribing spheres, thus utilizing a second of the three "classical geometries". We extend this process by exhibiting repeating Islamic patterns in hyperbolic geometry, the third classical geometry.