  While the interpretation of high-dimensional datasets has become a necessity
in most industries, and is supported by continuous advances in data science and
machine learning, 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.
