CS 39R: Symmetry & Topology
Lecture #10 -- Mon. 4/18, 2016.

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Warm-up:

Try to accomplish the following curve-shape-changes in the plane
using a continuous smooth deformation ("regular homotopy").
Draw a sequence of smooth key-frame shapes that would make up a continuous movie.

Change this right-arm "Klein-bottle profile" into a left-arm "Klein-bottle profile".	Simplify this "double-8" curve as much as possible.	Try to turn a circle inside-out, (reversing arrow direction).

Solution Solution NO Solution: incompatible turning numbers

Smooth Topological Deformations: "Regular Homotopies"

These are continuous smooth deformation that allow surface regions to pass through one another,
but do not allow any cuts, or tears, or formation of creases or other singular points with infinite curvature.

==> In the 2D plane, two smooth closed curves can be transformed into one another,
if the have the same "turning number"
(the number of full turns the headlights of a car would make while driving along the curve).

Derive the turning numbers of all the curves above...

In 3D space, these are some of the surfaces that can be turned outside-in:

Torus eversion by Cheritat (cut open, to see inside);

Earliest approach to: Turning a sphere inside out by Max;

Turning a sphere outside in by Thurston (more details Levy, Maxwell, Munzner);

Energetically optimal sphere eversion by Sullivan, Francis, Levy.

Additional "Elevator Speeches"

Project Presentations:

Done (4/11)
schedule 4-11

==> Today (4/18)
schedule 4-18

Next week (4/25)
schedule 4-25

Homework Assignments:

Due: April, 25, 2016

Refine Your Course Project Presentations.

In preparation for the next class you may want to watch this:
Perfect Shapes in Higher Dimensions -- Numberphile