Zhang, Yongheng

An homage to the 1994 movie Outside In

This animation shows a regular homotopy from the identity inclusion to the antipodal inclusion of the standard sphere in the three dimensional Euclidean space, using a corrugation technique worked out by William Thurston in 1974. The result demonstrates that a sphere can be turned inside out (or outside in) in a nice way♥, if it's made of a magic material which can pass through itself. Some material scientists have started to imagine such materials.

♥ This means partial derivatives exist and are continuous at all points and all time, and for each time instance, linearly independent tangent vectors at each point remain linearly independent, and throughout the whole movie, the tangent planes collectively turn continuously. 

(This is a large .gif file. If it appears choppy, refresh the browser after it is loaded once.)

The following is a 72-frame pdf file, about 92MB, which you can print out to make a page-flip animation. It may cost a few dollars at UPS/CVS..., but the file itself is free to download. It's prepared for faculty and students at Central Connecticut State University, but I thought it would also be useful to others.

Frame 0: standard embedding. Frame 1-10: corrugation. Frame 11-20: pole-to-pole. Frame 21-40: twist-and-turn. Frame 41-60: side-to-side. Frame 61-70: decorrugation. Frame 71: ingredients and recipe.

Q: A moon revolves around a planet as the planet revolves around a star. Assume that both orbits are circular and they are on the same plane, then which of the following curves could be the orbit of the moon around the star?

A: All are possible. A student would be able to give a precise analytical explanation within the first three weeks of taking multivariable calculus (using derivative, cross product and imagination), but if you'd like to see a quick proof without using algebra/calculus, click on the picture below.

We are also integrating the "boundary" (the differential) of A over B. This is the principle behind the generalized Stokes' Theorem, which put the Fundamental Theorem of Calculus, the Fundamental Theorem of Line Integrals, Green's Theorem, Stokes' Theorem and Gauss's (Divergence) Theorem all under its aegis. Curiously, some mechanical devices also employ this principle. In the following picture, if you move the wheel at the end of the long arm along the green curve once while keeping the cart on the black track, the number you would get after multiplying the total wheel mileage with the length of the arm is the area of the region bounded by the green curve! Indeed, how it works can be explained via Green's Theorem. You can learn it and much more (for example, a detective story in the last section, and a mathematical duality in the spirit of Appendix A of V. I. Arnol'd's book Huygens & Barrow, Newton & Hooke) from Prof. Tanya Leise's article linked below.

Leise, Tanya As the planimeter's wheel turns: planimeter proofs for calculus class. College Math. J. 38 (2007), no.1, 24-31.

The mechanical device shown in the above picture is one of the simplest types of planimeter, yet it can measure area of a variety of regions, as long as the following three conditions hold.

• We can fit the region into an infinitely long strip with width double the length of the measuring arm.
• The boundary of the region is piecewise smooth.
• The arm can only rotate in the forward 180 degrees range so that it never passes its shoulder. The reason is to get the abstract vector field associated to the measuring wheel always well-defined. One could also explain it using topological winding number, as already hinted in Tanya Leise's article.

The following animation shows how the device moves in order to measure the area of a disk with largest possible radius. Notice that the cart has to move during the second half of the trip in order to keep the third promise above.

In practice, one has to move the location of the measuring wheel or to change this subsystem to other recording mechanism, but the above version is simplest to understand if you want to lay out all the details, which a student would be able to  do within the last three weeks of taking multivaraible calculus.

I thought about this problem shortly after I moved from West Lafayette to Amherst around July 3 (?) of 2015. Click on the picture below to see a method, which is the result of solving a system of differential equations with constrained inputs, sprinkled with a little analysis, some elementary geometry and a bit of numerial computation. Dimensions are based on real data from U-Haul's website.