Mathematics educators often despair at math's austere, "abstract" reputation. This paper describes recent work in developing an application named "HyperGami," which is designed to integrate both the abstract and"real-world" aspects of mathematics by allowing children to design and construct polyhedral models and sculptures. Children use formal reasoning for computation and for creating real mathematical objects in paper. HyperGami allows students to design not only standard polyhedra, but also customized variants of those shapes. They can decorate the shapes, have the program convert three-dimensional shapes into two-dimensional shapes, and then print the decorated shape for folding into a three-dimensional paper sculpture. The paper describes the construction of a sample HyperGami sculpture, a penguin, and presents observations during a year of pilot studies with 11 volunteer elementary and middle school students. Related, current, and future work on this topic is also discussed. (Contains 12 references.) (Author/SWC)