CS 39R: Symmetry & Topology
Lecture #9 -- Mon. 10/30, 2017.

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Preparation:

Work on Your 10minute Course Project Presentations!

Warm-up:

Review what you know about knots:

How are knots classified? What are prime knots?

Can you knot your arms?

Try to simplify and classify the knot on the right ==>

Mathematical Knots

(Hang posters on wall)

A wonderful, "must-have" resource: Colin C. Adams: "The Knot Book", W. H. Freeman and Co., New York, 1994.

Knot Tables
Simplifying knots: Reidemeister moves
Beautifying knots. . . -- Symmetry? Torus Knots
Beyond knots: Links! Link Tables

Applications in Nano Technology

The Beauty of Knots

Parameterized Generation of High-Genus Surfaces (2nd half)

"Sculpture Generator I" for Scherk-Collins Toroids
-- where symmetry, topology, and art come together...

From static shapes to dynamically deforming geometry:

Smooth Topological Deformations: "Regular Homotopies"

Which curves in 2D or surfaces in 3D are transformable into one another through a "Regular Homotopy",
i.e., a deformation that allows surface regions to pass through one another,
but does not allow any cuts, or tears, or formation of creases or other singular points with infinite curvature.
(With this definition, it is possible to turn a sphere or a torus inside out -- but it is not easy!).

Try to do these preparatory exercises:
-- Simplify the double-8 curve . . .
-- Reverse the Klein-bottle cross section . . .
-- Try to turn a circle inside-out . . .
Remember: You are not allowed to ever make any sharp creases with very high curvature!

In preparation for next class, look at some of these movies:

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.