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.

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.

Ink drop in water.

Sunny side up.

Glass of water, and the drawing tools.

Water splash.

The moon.

A triply periodic minimal surface.

Hopf fibration. Every pair of circles are interlinked.

The moon.

The Riemann surface for the complex function \(f(z)=\sqrt{1-z^2}\).

Glass of water.

Conjugate minimal surfaces.

A smooth immersion of \(\mathbb{RP}^2\) in \({\Bbb R}^3\).

An attempt of 3D illusion drawing.

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.

The Julia set.

The depth of the terrain at \(c\in{\Bbb C}\) is approximately the logarithm of the number of iteration for \(z_{n+1}={z_n}^2+c\) to have \(z_n\) exceed 2. The resulting graph resembles a natural terrain.

When the positive real line is rolled up into a spiral, the prime numbers form this pattern.

This is an exercise of computing a parallel frame for a given curve, according to which one could lock the metal rings with a consistent angle.

This is a Willmore surface with a Möbius band topology and a round-circular boundary.

A Houdini render of a hairy torus.