We have developed a Java-based teaching tool, “Appreciating Hubble at Hyper-speed” (“AHaH”), intended for use by students and instructors in beginning astronomy and cosmology courses, which we have made available online. This tool lets the user hypothetically traverse the Hubble Ultra Deep Field (HUDF) in three dimensions at over ∼500×1012 times the speed of light, from redshifts z = 0 today to z = 6, about 1 Gyr after the Big Bang. Users may also view the Universe in various cosmology configurations and two different geometry modes — standard geometry that includes expansion of the Universe, and a static pseudo-Euclidean geometry for comparison. In this paper we detail the mathematical formulae underlying the functions of this Java application, and provide justification for the use of these particular formulae. These include the manner in which the angular sizes of objects are calculated in various cosmologies, as well as how the application’s coordinate system is defined in relativistically expanding cosmologies. We also briefly discuss the methods used to select and prepare the images in the application, the data used to measure the redshifts of the galaxies, and the qualitative implications of the visualization — that is, what exactly users see when they “move” the virtual telescope through the simulation. Finally, we conduct a study of the effectiveness in this teaching tool in the classroom, the results of which show the efficacy of the tool and provide justification for its further use in a classroom setting.