Project Five

December 2019
Jinlin Xu

This project re-implements and visualizes five Morin-Apery homotopies as web-based real-time interactive computer animations. They are the Cuboctahedron and Gastrula eversions of the sphere, the eversion of the cylindrical Neck of the Gastrula, and a homotopy between two forms of the Halfway Model in which pinch point pairs form and disappear, which is illustrated by Juno and the Pinchpoint cancelling animation. Smale (1958) proved that it is mathematically possible to turn a sphere inside-out without introducing pinch points anywhere. This means there is a regular homotopy from the standard embedding of the 2-sphere in Euclidean three-space to the mirror-reflection embedding such that at every stage in the homotopy, the sphere is being immersed in Euclidean space. Morin's eversion produced explicit algebraic equations describing the process. I have translated and re-implemented the C/IrisGL code for Gastrula, Juno, and Neck in JavaScript/WebGL, and I have developed visualization for Pinch based on its polynomial parametrization. Users of this real-time interactive computer animation can explore these five morphing surfaces that were created at the dawn of experimental mathematics done with computer graphics.

Project Narrative


Outside In developed by the Geometry Center in 1994


Thurston's eversion developed by Zack Reizner in 2013


Five homotopies developed by Jinlin Xu in 2019