Calculating the effective resistance of an electrical network is a common problem in introductory physics courses. Such calculations are typically restricted to two-dimensional networks, though even such networks can become increasingly complex, leading to several studies on their properties. Furthermore, several authors have used advanced techniques (graph theory, superposition of equipotential planes, and Green's functions) to perform theoretical calculations for three-dimensional networks, particularly focusing on the five Platonic solids due to their symmetry. However, these techniques are typically beyond the mathematical level of an undergraduate or advanced high school student. In this article, we outline techniques for analyzing these systems that are accessible to an introductory physics student. We also test these results experimentally using standard laboratory equipment.