Non Euclidean Geometries in Virtual Reality

Lattice in the Sol geometry

The Uniformization Theorem of complex analysis tells us that there are three important two-dimensional geometries: hyperbolic, euclidean and spherical. In contrast, the correct three-dimensional generalization, Thurston’s Geometrization Conjecture, singles out eight geometries fundamental to three-dimensional topology, namely $$\mathbf S^3,\ \mathbf E^3,\ \mathbf H^3,\ \mathbf S^2 \times \mathbf E,\ \mathbf H^2 \times \mathbf E,\ {\rm Nil},\ {\rm Sol},\ \widetilde{{\rm SL}(2,\mathbf R)}$$

The aim of this project is to develop accurate, real time, intrinsic, and mathematically useful illustrations of these geometries and more generally of homogeneous (pseudo)-riemannian spaces.

To see more pictures, videos and check out the simulation by yourself, visit the dedicated website . The papers below analyze some curious observation in these geometries