This is an expository article on the visualization of C¹ flat tori in three dimensions. It is very surprising that a torus of curvature zero can be embedded is

This is an expository article on the visualization of C¹ flat tori in three dimensions. It is very surprising that a torus of curvature zero can be embedded isometrically in three-dimensional space with any regularity, and C¹ is the highest possible regularity (if second derivatives were continuous, that would be enough to carry the curvature form, giving a C² curvature-zero closed surface in 3D, which is impossible). The visualization is quite impressive and comes with a beautiful video at https://www.youtube.com/watch?v=RYH_KXhF1SY https://www.youtube.com/watch?v=RYH_KXhF1SY and 3D-printed models like https://i.stack.imgur.com/7EcHp.png https://i.stack.imgur.com/7EcHp.png . This feat rests on convex integration and full details are provided in this paper : https://www.emis.de/journals/em/images/pdf/em_24.pdf https://www.emis.de/journals/em/images/pdf/em_24.pdf