CS 39: Symmetry & Topology
Lecture #8 -- Mon. 4/1, 2019.

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Preparation:

Prepare six small objects that demonstrate these symmetries: D1=C2; D2; S2; D1d=C2h; C1h=C1v; D1h=C2v.
Send me an up-dated course project proposal.

Warm-up:

Discuss your symmetry models with your neighbors.
Check whether your models have these properties:
D1=C2;    -- only one C2-axis.
D2;         -- two perpendicular C2-axes.
S2;            -- only one S2-axis.
D1d=C2h; -- one C2-axis + mirror plane perpendicular to it.
C1h=C1v; -- one single mirror plane.
D1h=C2v; -- two orthogonal mirror planes + a C2-axis at their intersection.

Chart of Cylinder Symmetries

Review of Graph Theory

Elements of a graph (G):
a set of Vertices (Points, Nodes): V(G) = {U, V, W, X, Y, Z ...}. [these are their names].
a set of Edges (Links, Connections between Vertices): E(G) = {(U,V), (V,W), (X,Y), (X,Z) ... }. [defined by their end points].
-- Loops are OK, e.g.: (U,U).
-- Multiple edges between vertices are OK, e.g.: {... (V,W), (V,W), ...}.

Some Special Graphs:
A planar graph is one that can be drawn crossing-free into the plane.
A complete graph, K_n, has exactly one edge between every possible pair of n vertices. {K_n has n(n-1)/2 edges}.
The Bipartite Graph K_3,3 has 3 red nodes and 3 blue nodes and exactly one edge between every differently colored pair of nodes.

Think of a few things in your domain of interest where a graph-representation may be useful.
What would be represented by the vertices/nodes and by the edges/links ?

An Illustrating Exercise:
Can you draw the "Bipartite Graph K_3,3" aka "Utility Graph" without any crossings?

Introduction to Mathematical Knots

See posters on the wall!

What is a Knot? by Numberphile

Prime Knots by Numberphile

What is (and what isn't) a mathematical knot ?
How are knots named and tabulated ?
When can we be sure that two knots are the same ?
When do we know that two knots are different ?
Let's simplify some knots until we know for sure what they are . . .
KNOT_X KNOT_Y KNOT_Z

What are prime-knots ?
Where might mathematical knots and knot theory be useful ?

How do Topology, Graph Theory, and Knot Theory relate to one another?

One important connection:
What is the handle-body of lowest genus on which a particular given graph or knot can be “embedded”, i.e., drawn without any crossings?
==>> Homework exercises: (1) Draw the Utility Graph onto a torus; (2) Draw the Trefoil Knot onto a torus.

Hand back QUIZ !

Comments on THE QUIZ . . .

General Comments about Good Study Habits and Success at the University.

University courses introduce many new concepts, definitions, and algorithms,
and it is definitely challenging to absorb them all.
But there are techniques that make this easier!

Learning happens in small increments (Piaget).
To absorb a new piece of information and install it successfully in your brain,
the brain needs to be prepared and already contain a foundation on which the new item can be installed.
In a good course, one lecture provides the foundation for the material of the subsequent lecture.
This assumes the students' understanding is up-to-date and
that they have internalized the material of all previous lectures.
This then makes understanding of the new material easier.
If there are small gaps in a seamless construction of the new knowledge,
then questions should be asked by (properly prepared) students.

Understanding the above facts then leads to some good study habits:
As soon as possible after having heard a lecture,
go over all its material and make sure that you understand it all.
Do not wait until the evening before the mid-term exam to cram it all into your head!
Immediately tackle new homework within 24 hours after it has been assigned,
and make this the forcing function to test, whether you really understand the material of the last lecture.
This eliminates the stress of getting the homework finished and handed in before any stated deadline.

New Homework Assignments:

Due: Monday, April 8, 2019.

"Correct" your Quiz (using a different color pen)
-- and catch up with your understanding of the course material.

Second try to make six small 3D objects that demonstrate these symmetries:
D1=C2; D2; S2; D1d=C2h; C1h=C1v; D1h=C2v.

Draw the Utility Graph and the Trefoil Knot onto a torus.

Solidify your Course Project Proposal (send me e-mail).