Steve Trettel

MediumComputer generated using a custom-built shader material and THREEjs.
The ArtA flat torus embedded in the 3-torus, viewed as it would appear intrinsically, if light traveled to our eyes along geodesics of the 3-sphere metric. The torus is parameterized by drawing a closed curve on the 2-sphere, and then taking the preimage under the Hopf map.
The MathThe 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.
CategoriesDifferential Geometry, Computer Graphics, Math Illustration, Stereographic Projection