CS 39R: Symmetry & Topology
Lecture #12 -- Mon. 11/20, 2017.

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Warm-up:

Find matching pairs with the same symmetries (rows 1, 2, 3; columns A, B, C, D, E).

For each pattern, make a list of all symmetries found in the linear friezes and in the hubcaps.
Compare those lists. Mark the corresponding rotation centers, mirror and glide lines in each
of the above patterns. That set of symmetry elements defines the overall symmetry class.

5 Project Presentations:

Learning three essential skills:
-- Giving an oral presentation with good visuals (PPT).
-- Writing an extended abstract (2-4 pages).
-- Preparing an "elevator speech" (60-90 seconds).

Some Clarification about the 4th Dimension !!

Hand back 4D Test.

What makes a valid 4D "corner"?
    -- Sum of dihedral angles along shared edge ≤ 360 degrees.
Can a 4D polytope be seen as a 3D polyhedron evolving over time?
    -- Yes! This corresponds to taking consecutive 3D slices.
>> How would we experience a 4D sphere passing through our 3D universe?
How can we best visualize 4D objects?
    -- By analogy! Use "Flatland" example to see how to understand 3D from a 2D world.

Going beyond the 4th dDimension !!

Regular Polytopes in Four and Higher Dimensions (second half)

If you are brave enough, look at: "A 10-Dimensional Jewel"

Wallpaper Symmetries

How to find the elements of Wallpaper Symmetries ?

Homework Assignments: Due: Nov. 27, 2017.

Annotate the worksheet of different wall-paper patterns.

Prepare Your Course Project Presentations.
Start working on your "90-seconds Elevator Speeches".
Start working on your final project reports.