The focus of this paper is a class of symmetrically patterned spherical polyhedra we call “Twarock-Konevtsova” (T-K) tilings, which were first applied to the study of viruses. We start by defining the more general concept of a “wrapping paper pattern,” and T-K tilings (along with one other virus tiling) are outlined as a subset of them. T-K tilings are then considered as “spherical Penrose tilings” and a pair of algorithms for generating them is described. We believe they present good source material for mathematical sculpture, especially modular origami, having intriguing features such as consistent edge lengths and a limited selection of angles in their constituent faces. We conclude by looking at existing examples of T-K tilings in mathematical sculpture and consider different directions they could be taken in.