CS 39R: Symmetry & Topology
Lecture #8 -- Mon. 4/1, 2013.

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Preparation:

Design a Highway 'Cloverleaf' for a Crossing of Three Highways.
Think about ways in which to evaluate your design with respect to the trade-off between efficiency and costs.

Warm-up:
Some objective measures are needed to judge the quality and efficiency of a design as well as its realization costs:

In your design, how many extra "connecting arcs" did you add? -- how many bridges?
In the worst case, through how many connecting arcs/bridges does a car have to travel to find the desired direction?

Discuss your cloverleaf layout and your design parameters with your neighbors!

Highway Cloverleaf Design

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.

Crossing-free graph-embeddings on a 2-manifold of suitable genus lies right at the heart of topology!

Follow-up: On the Embedding of the Complete Graph K8 in a genus-2 surface:

The math literature says that it is doable! (- even though the references I found don't actually give the solution).
If you choose to work on this puzzle over the Spring break, keep track of the various approaches that you are pursuing, and which one of them look more promising.

Discussion of some more or less promising approaches . . .

Diagrams of my own struggles with this non-trivial graph-embedding problem.

Wrap-up: Simple topological 2-manifolds, -- orientable and non-orientable

Last time we discussed: PPT presentation.

Classification of 2-Manifolds

Each closed surface can be constructed from an oriented polygon with an even number of sides,
called a fundamental polygon of the surface, by pairwise identification of its edges.
Restricting ourselves to rectangles, in square below, attaching the sides with matching labels (A with A, B with B),
so that the arrows point in the same direction, yields the indicated surface (as detailed in the PPT presentation above):

Classification Theorem of Closed Surfaces:
Any connected, closed (h=0, no punctures) surface is "homeomorphic" to some member of one of the following three families:

The SPHERE: 2-sided, X=2;

A connected sum of g TORI, (for g >= 1) -> "Handles on a sphere": 2-sided, genus=g, X= 2 - 2g;

A connected sum of k PROJECTIVE PLANES, (for k >= 1) -> "Cross-caps on a sphere": Single-sided, genus=k, X= 2-k.

Surfaces with Holes and Boundaries

If we allow surfaces to have "punctures" or "holes" -- which then have "boundaries" 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)?

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

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

Regular Homotopies (a more specific classification)

Which surfaces are transformable into one another through a "Regular Homotopy",
i.e., a deformation that allows surface regions to pass through one another,
but does not allow any cuts, or tears, or formation of creases or other singular points with infinite curvature.
(With this definition, it is possible to turn a sphere or a torus inside out -- but it is not easy!).

Next class will begin with some preparatory exercises:
-- Simplify the double 8 curve . . .
-- Try to turn a circle inside-out . . .

In preparation for that class, look at some of these movies:

Torus eversion by Cheritat (cut open, to see inside);

Turning a sphere inside out by Max;

Turning a sphere outside in by Thurston (more details Levy, Maxwell, Munzner);

Energetically optimal sphere eversion by Sullivan, Francis, Levy.

New Homework Assignments: Due April 8, 2013, before noon:

Give me a brief update on your projects: One paragraph summarizing your findings so far.
Please sen me e-mail before noon on Monday.