CS 39: Symmetry & Topology
Lecture #11 -- Mon. 4/22, 2019.

PREVIOUS <- - - - > CS 39R HOME < - - - - > CURRENT < - - - - > NEXT

Preparation:

Watch: Perfect Shapes in Higher Dimensions

Warm-up: An Exercise in 3D Visual Thinking:

Consider a cube, suspended on a thread attached to one of its corners.
Slowly lower the cube into a pool of water.
From the moment when the cube first touches the water surface,
until it is eventually completely immersed,
visualize the shape of the intersection line between the cube and the water.

Draw about 5 to 10 sketches of consecutive intersection profiles.

(If this seems hard, first try to visualize the 1D intersections
between a 1D horizon line and a 2D square suspended from one corner.)

Getting familiar with higher dimensions !

What is the 4th Dimension? Does it exist? -->> Conceptually, there are infinitely many spatial dimensions
How many mutually orthogonal coordinate axes are there in 4D space? -->> Four: x, y, z, w.
How many mutually orthogonal coordinate planes are there in 4D space? -->> Six: xy, xz, xw, yz, yw, zw.

Perfect Shapes in Higher Dimensions -- Numberphile

Discussion:

What makes a valid 4D "corner"?

Can a 4D polytope be seen as a 3D polyhedron evolving over time?

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.

Understanding higher dimensions step by step -- one dimension at a time. (Extrusion Pantomime)

Regular Polytopes in (Four and) Higher Dimensions (second half: slide 28)

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

Frequently Asked Questions about the 4th Dimension:

Is there a practical application for the regular 4D polytopes?

A good example for training your abstract thinking and reasoning by analogy.
(Consider going from 2D to 3D, and then use this to go from 3D to 4D).

Check the web for possible responses, e.g.,: Applications of Higher Dimension Formulas

The regular 4D polytopes are a piece of pure mathematics.
Often it takes several decades before a mathematical construct finds an application.
Minkowski Spaces (a combination of 3D Euclidean space & time) were found to be a good model for General Relativity.

Mathematics can also be "beautiful"!

By strict logical reasoning one can derive new mathematical insights
that are irrefutably true and not subject to later change.

By analogy from 1-, 2-, and 3-dimensional spaces, we can derive 4- and higher-dimensional spaces.
And then we can show unambiguously that in 4D-space there are exactly six totally regular polytopes.
No science or religion has this kind of internal consistency and permanence.

Wallpaper Symmetries:

Back to Two Dimensions, but with Lots of Symmetry!

How to find the elements of Wallpaper Symmetries ?

Annotate the worksheet of different wall-paper patterns.

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

First: Your PPT presentation:
-- Pick an interesting, relevant story.
-- Be clear about the main message that you want to get across to your audience:
    > Early on, tell the audience what your key contribution will be;
> then build up the needed knowledge, so your listeners can understand an appreciate what you are telling them;
    > Summarize, what it is that you would like your audience to remember.
-- Use simple, clear slides:
    > Only ONE concept per slide;
    > All text should be easily readable (font > 14pt).
    > About one slide for every 30-60 seconds of your talk.

The first presentation will be given next Monday (4/29) by Sophie Chen on Chinese Architecture.
(Any other volunteers?)

New Homework Assignments:

Due: Monday, April 29, 2019.

Annotate the worksheet of different wall-paper patterns.

Prepare your Course Project presentations:
>>> Send me a final title !