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.

Ink drop
Albert Chern, pencil, 9/1, 2015

Ink drop in water.

Egg
Albert Chern, pencil, 2/1, 2014

Sunny side up.

Glass of water
Albert Chern, pencil, 11/6, 2013

Glass of water, and the drawing tools.

Pouring
Albert Chern, pencil, 10/18, 2013

Water splash.

Moon
Albert Chern, pencil, 9/19, 2013

The moon.

Gyroid
Albert Chern, pencil, 1/7, 2013

A triply periodic minimal surface.

Hopf fibration
Albert Chern, chalk, 2/10, 2016

Hopf fibration. Every pair of circles are interlinked.

Moon (chalk)
Albert Chern, chalk, 11/7, 2014

The moon.

Riemann surface
Albert Chern, chalk, 10/17, 2014

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

Glass of water (chalk)
Albert Chern, chalk, 10/24, 2014

Glass of water.

Helicoid and Catenoid
Albert Chern, chalk, 2011

Conjugate minimal surfaces.

Boy's surface
Albert Chern, chalk, 2011

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

3D illusion
Albert Chern, chalk, 2011

An attempt of 3D illusion drawing.

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.

Julia canyon
Albert Chern, Houdini, 5/2, 2016

The Julia set.

Mandelbrot lake
Albert Chern, Houdini, 4/30, 2016

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.

Prime spiral
Albert Chern, Houdini, 4/30, 2016

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

Chain
Albert Chern, Houdini, 4/28, 2016

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.

Sudanese Möbius band
Albert Chern, Houdini, 9/23, 2015

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

Hairy torus
Albert Chern, Houdini, 10/3, 2014

A Houdini render of a hairy torus.