CS 39R: Symmetry & Topology
Lecture #6 -- Mon. 3/11, 2019.

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Preparation:

Fill in the blanks in Work-sheet on Surface Classification

Construct a physical model of a single-sided surface of genus 2,
and then draw onto this surface TWO non-intersecting closed loops that together still leave the surface fully connected (just ONE domain).

Warm-up:

Compare your homework solutions with your neighbors.

Find out where you may have made mistakes -- and why...

Also show off your genus-2 surface and see,
whether your neighbors agree with your solution.

The Fundamentals of Topology

Topology is the science of interconnectedness.
("It can prove that you cannot take of your pants over your head." :-)
(Don't confuse this with "Topography": -- "the arrangement of the natural and artificial physical features of an area."

Revisiting Genus, EC, and Surface Classification:

Surfaces with Holes and Boundaries -- Review of some Key Concepts:
If we allow surfaces to have "punctures" or "holes" -- which then have "borders" or "rims"
-- things get a little more complicated.
But a topologist can still classify all the possible surfaces of that kind by only three characteristics:

ORIENTABILITY: Is the surface two-side (orientable) or single-sided (non-orientable)?

NUMBER OF BORDERS: How many "disks" have been removed from a closed surface;
or, how many individual rims or hole contours are there?

EULER CHARACTERISTIC, EC (or alternatively, its GENUS, g): How "connected" is the surface?
EC = #Vertices - #Edges + #Facets of a mesh approximating the surface.

Euler Characteristic and Genus: PPT

Discuss last Homework Assignment: Work-sheet on Surface Classification

Your single-sided, genus-2 surface models:
Paper-strip constructions of ever more complex surfaces . . .
How to make such a surface single-sided . . .

ECa

More Single-Sided, Non-Orientable 2-Manifolds: PPT (Second half)

Non-orientable surfaces come in many different forms: Moebius bands, Klein bottles, . . .
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.

OPTIONAL: If you are really interested in such weird surfaces, then peruse the following PPT:

"Cross-Caps -- Boy Caps -- Boy Cups"
(Some of this is a repetition of things I have discussed, but there are more extensive examples of what you can do with a Boy surface)

Prepare for an In-Class Quiz on Monday, March 18, 2019.

Next lecture we will have an in-class Quiz.
Participating in this quiz is one of the mandatory requisites to get a P-grade this class.

(The other one is to do a Project in the last few weeks of the course.

To prepare for the Quiz, look through all the course material and write a condensed fact sheet:
One two-sided, letter-sized sheet of paper, which you may then use to assist your memory during the Quiz.

(I have kept such sheets for many years after finishing some courses,
and they have come in handy often, when I quickly wanted to refresh my memory of some of the things I had learned in that course.)

New Homework Assignments:

Due: March 18, 2019.
1.) Prepare for the In-Class Quiz on Monday, March 18, 2019:
Review all course materials.
Prepare the one-page, double-sided fact sheet.

2.) Think about your individual Course Project:
Give me two proposals (just one line each) of what you might want to study concerning:
"The role of symmetry and/or topology in the field of <your special interest>"
Here is a list of possible titles to jump-start your imagination . . .
Send your proposals to me by e-mail before 9am on Monday, March 18, 2019.

Also: Make a second attempt at constructing a single-sided surface of genus 2
(with the two closed curves included to show that it is genus 2).