CS 39R: Symmetry & Topology
Lecture #10 -- Mon. 11/6, 2017.

PREVIOUS <- - - - > CS 39R HOME < - - - - > CURRENT < - - - - > NEXT

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 Solution

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...
Also, simplify the Klein-bottle profile as much as possible!

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 Nelson Max -- (in German!);

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

Energetically optimal sphere eversion by Sullivan, Francis, Levy.

A summary of the key steps in Morin's sphere eversion.

Snowsculpting.

Can a Boy-Surface be turned inside out with a regular homotopy ?

Project Presentation:

==> Today (11/06): Shawn Hua: "Symmetry in Tibetan Culture"
Discussion and critique.

Learning three essential skills:
-- Giving an oral presentation with good visuals (PPT).
-- Writing an extended abstract (2-4 pages).
-- Preparing an "elevator speech" (60-90 seconds).

Following weeks:

Next time: A step into the 4th dimension !

What is the 4th Dimension? Does it exist?
How many mutually orthogonal coordinate axes are there?
How many mutually orthogonal coordinate planes are there?

Homework Assignments: Due: Nov. 13, 2017.

In preparation for the next class, please watch this video:
Perfect Shapes in Higher Dimensions -- Numberphile

(There may be a short quiz at the beginning of next class.)

Prepare Your Course Project Presentations.