CS 39: Symmetry & Topology
Lecture #10 -- Mon. 4/15, 2019.

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Preparation:

Watch the eversion movies listed above.

Warm-up:

Another Regular Homotopy Exercise:

Simplify these two curves as much as possible!

Remember: You are not allowed to make sharp kinks with high curvature!

Regular Homotopy
This transforms curves in 2D or surfaces in 3D into one another.
Curve segments or surface regions are allowed to pass through one another;
but it is not allowed to make cuts, or tears, or creases, or other singular points with infinitely high curvature.

Last week's examples, involving curves:

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

==> In the 2D plane, two smooth closed curves can be transformed into one another,
if the have the same "turning number". What are they for the above curves?

(With a Regular Homotopy, it is possible to turn a sphere or a torus inside out -- but it is not easy!).

These movies show regular homotopies on surfaces in 3D space:

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 } Details Levy, Maxwell, Munzner;

Energetically optimal sphere eversion by Sullivan, Francis, Levy.

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

Snowsculpting ("Turning a Snowball Inside-Out").

Any questions on one of these movies?

Stepping into the 4th dimension !

What is the 4th Dimension? Does it exist? -->> Conceptually, there are infinitely many spatial dimensions
How many mutually orthogonal coordinate axes are there in 4D space? -->> Four: x, y, z, w.
How many mutually orthogonal coordinate planes are there in 4D space? -->> Six: xy, xz, xw, yz, yw, zw.

Regular Polytopes in Four and Higher Dimensions

Perfect Shapes in Higher Dimensions -- Numberphile

New Homework Assignments:

Due: Monday, April 22, 2019.

Watch this movie: Perfect Shapes in Higher Dimensions

Work on your Course Project presentations.