Thurston Sphere Eversion

About

This video shows a sphere eversion using Thurston's corrugation method, as described in the video Outside In by the Geometry Center (Bill Thurston, Silvio Levy, Delle Maxwell, Tamara Munzner, Nathaniel Thurston, David Ben-Zvi, Matt Headrick, et al.), 1994, and the accompanying booklet Silvio Levy: Making Waves — A Guide to the Ideas Behind Outside In, A K Peters, Wellesley, MA, 1995.

An eversion turns the sphere inside-out in a smooth manner. Creases, pinch points, holes, etc. may not occur. However, the sphere may intersect itself during the eversion. In mathematical terms, the eversion is a regular homotopy between the sphere and the sphere point reflected at its center. To indicate the two sides of the sphere, its outside is colored gold and its inside purple.

The video has four parts. In the first part, an eversion of the complete sphere is shown. Conceptually, the sphere consists of eight spherical lunes that all deform in the same manner. First, the sphere is corrugated. This is necessary to add enough pliability for the following steps. Second, the northern and southern parts of the sphere are pushed through each other. This causes loops in the central part of the sphere. Third, the northern and southern part of the sphere are rotated by 180° in opposite directions. At the same time, the central part is twisted to remove the loops. Fourth, the central parts of the sphere are pushed through the vertical axis of the sphere, which moves the outside parts of the center of the sphere to the inside. Fifth, the corrugations are removed, resulting in the everted sphere.

In the second part of the video, the eversion is reversed. In this part, only a single hemisphere is shown. This allows to see the complex self-intersections that occur during the eversion, especially in the third and fourth steps. The eversion possesses a 180° rotational symmetry around an axis through the equator between the two hemispheres. Therefore, the deformation of the two hemispheres is identical.

The third part of the video shows the eversion of a single lune. The lune can be imagined as a belt that starts out with both ends in the same direction. Conceptually, the second step of the eversion can be visualized as moving the belt's ends past each other while the belt's ends maintain their orientation. This creates a loop in the belt that resembles a mirrored “α” (the Greek letter alpha). If the belt were straightened, it would contain a 360° twist. The third step removes this twist by rotating each end of the belt by 180° in opposite directions. This creates a belt that has the shape of a “Ω” (the Greek letter Omega), rotated by 90°. The fourth step pushes the center of the belt to the opposite side, which deforms the “Ω” into a rotated “U” and moves the outside of the belt (lune) to the inside of the sphere. Hence, this part of the video visualizes that the main deformation that occurs for each lune corresponds to the motion of creating a loop in a belt and then removing the loop by twisting the belt. The corrugations at the beginning and the end would not be necessary to evert a single lune. However, they are essential to be able to evert all eight lunes (the entire sphere) simultaneously.

The fourth part of the video shows a reverse eversion of the entire sphere, but with the inside painted with an image of the earth by night and the outside painted with an image of earth by day. This shows that the eversion actually effects a point reflection that reverses the orientation of the sphere. Note that, for example, Australia is upside-down and mirror-reversed (visible, for example, by the position of New Zealand with respect to Australia). After the eversion has been completed, the antipodal points of Australia are shown (North and South America).

Note that the video can be played in a loop using a suitable video player.

Archive

Wikimedia Commons, in Wikimedia Commons

Licence or status

CC BY-SA 4.0, as the Wikimedia Commons states it.

Rights statement

Creative Commons Attribution-Share Alike 4.0

Credit

Carsten Steger

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