Kon Hyong Kim, Steve Trettel, Dennis Adderton

Environment

AlloSphere, 2nd floor

The geometry of curved three dimensional spaces plays an important role in modern geometric topology. However, as the notion of "straight-line" becomes more complex when space itself curves, it becomes difficult to visualize these spaces. Instead, mathematicians typically sacrifice visual accuracy for computational simplicity and work in a highly distorted model of the space.

One of the chief difficulties of learning the mathematics of curved spaces is that they defy our intuition, intuition that was built out of day-to-day experiences in the flat space we live in. If we had access to an immersive, three dimensional environment which accurately modeled the curved space of interest we could retrain our intuition and hopefully refocus our mind on the properties and questions relevant to that world.

In *Three-Space*, we focus on producing perspectively correct models of the interior of the two simplest curved three dimensional spaces - the three dimensional sphere and hyperbolic space, using the AlloSphere. Immersive views of the Hopf fibration and the hyperbolic honeycomb are used as a demonstration of the experimental mathematics capability of the AlloSphere as well as showing the beauties of geometries relevant to the geometrization program, which led to the proof of the Poincare Conjecture.

Immersive models for these spaces could not only act as an introduction to the beauties of high dimensional geometry but lead to some novel mathematical work, including the accurate modeling of the intrinsic geometry of the spaces known as Nil, Sol and SL(2,R).