While the interpretation of high-dimensional datasets has become a necessity in most industries, the spatial visualization of higher-dimensional geometry has mostly remained a niche research topic for mathematicians and physicists. Intermittent contributions to this field date back more than a century, and have had a non-negligible influence on contemporary art and philosophy. However, most contributions have focused on the understanding of specific mathematical shapes, with few concrete applications. In this work, we attempt to revive the community's interest in visualizing higher dimensional geometry by shifting the focus from the visualization of abstract shapes to the design of a broader hyper-universe concept, wherein 3D and 4D objects can coexist and interact with each other. Specifically, we discuss the content definition, authoring patterns, and technical implementations associated with the process of extending standard 3D applications as to support 4D mechanics. We operationalize our ideas through the introduction of a new hybrid 3D/4D videogame called Across Dimensions, which we developed in Unity3D through the integration of our own 4D plugin.