CS 39R: Symmetry & Topology
Lecture #5 -- Mon. 2/29, 2016.

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Preparation:

Design a "cubist" Moebius band and/or a "cubist" Klein-bottle.
Present your design as a physical model, using: paper, cardboard, plexi-glass, alu-foil, sheet-metal, or other similar thin, flat materials.
Try to find a design with a minimal number of individual facets, and (of secondary concerns) with maximal symmetry.

Warm-up:
Share your cubist model with your neighbors.
(Did you get any suggestions how it might be improved?)

What is the genus of the following objects:
-- A hollow sphere with 17 punctures in its shell ?
-- A fully connected graph with thick tubular edges connecting 6 nodes ?
-- and the following depicted objects:

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What might be the 2-manifold topology of the 2D-universe surrounding this island?
Determine the overall connectivity ("around the back-side") of this world.

Think of all possible (2-manifold) universes
that can be formed by connecting the edges
of a rectangle in different ways,
i.e., by identifying pairs of points on them
(assuming them to be connected) ?

Simple topological 2-manifolds, -- orientable and non-orientable.

Tori, Moebius bands, Klein bottles, Boy surface ...
PPT presentation.
Which ones 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!).

The Topology of Single-sided, Non-orientable 2-Manifolds: PPT

What is a Moebius band? >>> Done last lecture

What is a Klein bottle?
Klein bottles come in many different forms...

The Projective Plane and Boy's Model.

Two Moebius bands together make a Klein bottle: Limerik;
Demo with Cliff Stoll's zippered model.

How many different Klein bottles are there?

How to make a single-sided, non-orientable surface of genus g ?
==> Graft g cross-caps onto a sphere.

More extensive PPT talk: "Boy-cap, Cross-cap, Boy-cup"

The Surface Classification Theorem

All 2-manifolds can be characterized topologically by 3 parameters:
its sidedness (1 or 2); its # of borders (0 ... infinity); its genus (0 ... infinity).

New Homework Assignments:

Due: March 7, 2016

1.) Construct a physical model of a surface of genus 2,
and then draw onto this surface two closed loops which together still leave the surface fully connected (just ONE domain).

Bring this model to class on Monday, March 7, 2013.

2.) Think about your individual course project:
Give me two proposals of what you might want to study concerning a topic such as:
"The role of symmetry and/or topology in the field of <your special interest>"

Send your proposals to me by e-mail before noon on Monday, March 7, 2013.