CS 39R: Symmetry & Topology
Lecture #8 -- Mon. 4/4, 2016.

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Preparation:

1.) A real design task:

Design a Highway 'Cloverleaf' for a Crossing of Three Highways:
Consider that most of the traffic will probably go straight through the crossing;
but one must still be able to go from any direction to any other direction.
Bridges cost more than roads on the ground.
Use symmetry!

Warm-up:

Add up some rough cost estimate for your design:
-- How many loops did you have to add to the main freeways? Add up partial loops by counting quarter turns (90º).
-- How many bridges 1-story high, and how many bridges 2-story high, etc ... does your design include?

Evaluate the quality (user friendliness) of your solution:
-- Hand your design to your neighbor and let her/him figure out how to get from direction A to direction B,
e.g., from a red lane to a blue lane that is equivalent to a 120º left turn.
or, from a blue lane to a red lane that is equivalent to a 60º left turn.

Discuss what other criteria might be useful to select a best possible design.

Comparison and Discussion of Your Cloverleaf Designs . . .

Engineering Design typically has one or more clearly stated goals, often with different priorities.
Make sure you identify these goals and keep track of them throughout the whole Design process!

Engineering Projects also have constraints and limitations: budget; weight; space needed; material strength . . .
But some constraints are often more flexible than others (e.g., budget vs. gravity).

Some diagrams to back-up our in-class discussions.

Where we stand in our Course CS39R:

By now you should be familiar with:
Frieze symmetries, 2D rotational (hubcap) symmetries, symmetries of finite 3D objects;
2-manifolds and their classification;
graphs and their embedding in surfaces of appropriate genus.

Yet to come:
Mathematical knots;
Wall-paper symmetries; the regular 4D polyhedra;
Homotopies of 2-manifolds, turning surfaces outside-in;
==> Applications of Symmetry and Topology in various domains of interest.

Your Project Outlines:

Remember: Your presentation must be less than 10 Minutes !
(Your proposals are often suitable for a PhD thesis.)

Project Presentations: Creating a 1-minute "Elevator Speech"

Making a (good) Presentation: (an important "Meta-Skill"):

Prepare a brief story that you think will be of interest to your audience -- and fits into less than 10 minutes!
Prepare a 1-minute summary of the main point(s) of your presentation.
Ask yourself: What does the audience already know and what do you need to tell them so that they can understand my story?
What supporting material can make the presentation more interesting -- or easier to understand?

Structure your presentation:
Introduction: Outline the main focal point of your presentation.
Main Part: Tell your story; use your supporting props.
-- (I suggest: Aim for an 8-minute presentation with 8 -- 15 slides)
Wrap Up: Repeat the main point(s) that you want your audience to remember.
Knowing that you are well prepared: be confident in your presentation.
Speak slowly, loudly, clearly!

Mathematical Knots

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_A KNOT_B

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

New Homework Assignments:

Due: April 11, 2016

EVERYBODY: Prepare Your 5-10minute Course Project Presentations!