Exam 3

Using the edge identification diagram of the Mobius band, prove that it has only one edge.

Because it goes around infinitely many times on the same edge

How do the Borromean rings differ from the Olympic rings?

Borromean rings can be dismantled by removing one ring, not true for the olympic rings.

Use a point removal argument to show that the letter O is topologically different from I, and the letter T is topologically different .from I.

If you remove any interior point from the letter "O," you still have one piece. If you remove any interior point from "I," you have 2 pieces. And if you remove the intersectng piece from the letter "T," you get 3 pieces (which, then, is different from letter "I")

Suppose we drill a hole in a silver dollar. Would that coin with a hole be equivalent by distortion to a straw? Explain

Yes, the coin with a hole in the middle is the same distortion as a straw. They both have a hole in the middle. If you take the straw and squash it down, it will look like the coin, round with a hole in the middle. Same if you did the coin, stretch it up, it will then have a hole like the straw.

Use the edge identification diagram for a Möbius band to argue that when you cut the band down the middle, you end up with a single band that is made from a strip of paper twice as long as the original

You end up with a larger band double the size when cutting a mobius strip down the middle because the strip is infinite.

What is an Euler path?

starts and ends at different vertices.

Explain why the number 0 is called an attractor and the number 1 is called a repeller.

Numbers between 0 and 1 will attract to 0 when squared and detract from 1 when squared 1 x (.1)2 = .01

Explain and illustrate why a torus and a sphere are not topologically equivalent.

A torus, like a donut, has a hole in the center of it. But a sphere, like a ball, does not have a hole in its center. Topology is the mathematical study of the properties that are preserved through deformations, twistings, and stretchings of objects. You cannot put a hole in a sphere, nor can you fill the hole in a torus without doing anything beyond these constraints.

What are the two most important features of a fractal?

1. Self-similarity 2. A pattern that repeats infinitely many times on smaller and smaller scales

Why are heavy industrial conveyor belts often built like a Möbius band?

Since the *only* side of the belt gets used, there's less wear. Also, since it's twice as long, it lasts longer.

At stage 0 the Sierpinski triangle consists of a single filled in triangle. At stage 1, there are three smaller, filled in triangles. How many at stage 2? How many at stage n?

Stage 2 = 9 triangles Stage 3 = 27 Stage n = 3n

Describe the collage method for generating a fractal image and illustrate it with the Koch curve.

Start with a line segment. Replace this segment with 4 segments of equal length, each one ⅓ as long as the original. Then, do this continuously— replace each segment with 4 segments, each ⅓ the length of the previous segments.

Euler's Circuit Theorem says that a graph has a Euler circuit if and only if it is connected and every vertex has even degree. Why must the graph be connected, and why must every vertex have an even degree?

The graph must be connected, because an Euler circuit has to start and stop at the same point. Also, each vertex must have an even degree, because each point has to have an exit for every entrance of the circuit since it has to hit every side exactly once.

What does it mean to Eulerize a graph?

The process of adding duplicate edges to a graph so that the resulting graph has no vertices of odd degree (and thus does have an Euler Circuit) is called EULERIZATION.

What does it mean to say that two things are equivalent by distortion?

Two things are said to be equivalent by distortion if one object can be stretched, bent, shrunk or twisted into the shape of the other, without cutting or distorting either `shape. When determining if two shapes are equivalent by distortion, assume that neither object is rigid—that is, each object is super stretchy, like a rubber band.

What is an Euler circuit?

uses every edge of a graph at least once.