Waterfall in the 3-sphere

Albert Chern, pencil, 6/4, 2016

A realization of MC Escher's Waterfall in the 3-sphere, visualized in \({\Bbb R}^3\) through stereographic projection. The gravity/vertical directions are along the Hopf fibers. The horizontal directions are those orthogonal to the Hopf fibers, with metric induced from \({\Bbb S}^3\subset{\Bbb R}^4\). Such vertical/horizontal splitting gives rise to a connection for the fiber bundle that has monodromy: walking horizontally along a loop returns to a different height on the starting fiber.

Penrose-Escher stairs in the 3-sphere

Albert Chern, pencil, 5/26, 2016

A realization of MC Escher's Penrose stairs in the 3-sphere, visualized in \({\Bbb R}^3\) through stereographic projection. The gravity/vertical directions are along the Hopf fibers. The horizontal directions are those orthogonal to the Hopf fibers, with metric induced from \({\Bbb S}^3\subset{\Bbb R}^4\). Such vertical/horizontal splitting gives rise to a connection for the fiber bundle that has monodromy: walking horizontally along a loop returns to a different height on the starting fiber.

Boy's Surface double-covered

Albert Chern, Houdini, 5/9, 2016

This is an immersion of \({\Bbb R}{\Bbb P}^2\) in \({\Bbb R}^3\) that is also a Willmore surface. The \({\Bbb R}{\Bbb P}^2\) is double-covered by \({\Bbb S}^2\) (identifying antipodal points of \({\Bbb S}^2\)), which gives this two-sided texture for \({\Bbb R}{\Bbb P}^2\) from NASA's satellite images.