# Hopf_torus

Steve Trettel

Medium Computer generated using a custom-built shader material and THREEjs. A flat torus embedded in the 3-torus, viewed under stereographic projection. The torus is parameterized by drawing a closed curve on the 2-sphere, and then taking the preimage under the Hopf map. The torus admits a flat metric (one can be constructed abstractly by identifying opposing sides of a square sheet of paper), but such flat tori cannot be constructed in 3-dimensional space. However, a simple construction, due to Ulrich Pinkall, shows that all flat tori can be embedded in 4-dimensional space, on the surface of the 3-sphere. Differential Geometry, Computer Graphics, Math Illustration, Stereographic Projection