We have developed an experimental platform for immersive visualization of the Classical Non-Euclidean Spaces using real-time ray tracing.

The project includes the design and implementation of an extensive framework for creating interactive experiences in landscapes of three-dimensional Manifolds / Orbifolds having their geometric structure modeled by one of the classical tridimensional geometries, i.e., Flat (E3), Hyperbolic (H3), and Spherical (S3).

Classical 3-Manifolds

Geometric manifolds are abstract spaces spaces locally similar to the Euclidean space. We present the three classical examples of such spaces: Euclidean, hyperbolic, and spherical spaces.

Tridimensional Torus

The 3-dimensional torus T3 is generated by the action of the group of translations in the Euclidean space which coves T3, explaining thus the copies pattern.

Seifert-Weber Dodecahedral Space

Considering the dodecahedron embedded in the hyperbolic space, a special clockwise identification of its opposite faces gives rise to Seifert-Weber dodecahedral space. Its geometric structure is modelled by H3.

Poincaré Dodecahedron Space

Discovered by Poincare, is obtained by gluing the opposite faces of a dodecahedron embedded in the spherical space. Its geometric structure is modelled by S3.

Simple Orbifolds

Such spaces are modeled locally by quotients of a model geometry by discrete groups. We present two simple orbifold examples : the mirrored cube, and the mirrored dodecahedron.

Mirrored Cube

The mirrored cube is an example of a non-manifold with the geometric structure modeled by E3 through a special group of reflections.

Mirrored Dodecahedron

For an example of a non-manifold with geometric structure modeled by the hyperbolic space, consider the dodecahedron embedded in H3. Let T be the group of reflections generated by the dodecahedral faces. The quotient H3/T is the mirrored dodecahedral space.