The Amazing One-Sided Piece of Paper
A generalization of the math we already know
Ex: we know how to add & multiply natural numbers {0,1,2,3,…}
fractions & rational numbers
decimals & irrational numbers
each type of number has different rules associated with it – adding fractions is different from adding integers What about telling time – how do we add numbers on a clock?
In college algebra courses, you take the information you know about sets of numbers with certain properties and extend these to other sets of numbers. We ask the question: if two sets A & B have certain common characteristics, and we know how to add and multiply on set A, what does this say about addition and multiplication on set B. With the clock this is easy: we say the clock works like our natural numbers combined with the rule that 24=0, 25=24+1=1, and so on. Generalized algebra is called “abstract algebra.”
We’d also like to do this with geometry. So how do we generalize
geometry?
The geometry you already know: called EUCLIDEAN
R = the line = 
rays, line segments
R2 = the plane = 
circles, squares, triangles,…
R3 = 3 dimensional Euclidean space
cubes, spheres, cylinders,…
R4: What might a four-dimensional square look like?
What space is the floor of this room?
So, we would like to do math (add vectors, do calculus,…) on any space, not just Euclidean.
Generalized geometry is called TOPOLOGY.
The sphere (earth) looks like R2 locally (in small regions, really close up) – since we know how geometry works in R2, this is good.
A mathematical object that looks like Rn locally is called a MANIFOLD.
What about the circle and the sphere: they both locally look like Rn,
how do they differ?
Now let’s make a loop.
Imagine you’re an ant crawling along your loop. Trace around the middle.
If you were to cut along the line you just traced, what would you get?
Do it.
Now a moebius strip: a loop with a twist.
If you were to cut it in half as you just did the loop, what would you expect?
Now trace along the middle.
Do your expectations change? Why?
Cut your moebius strip in half.
Why do you get what you get?
What would happen if your loop had 2 or 3 or more loops?
Topologically the two objects are the same if they can be stretched into one another, as if they were made of play-doh (cutting and pasting are not allowed).
That’s why the loop and the moebius strip are different. One cannot be stretched into the other because they have different numbers of faces.
Spaces that can be stretched into one another share common rules of addition and multiplication (and other types of math).
The Alphabet: which letters of the alphabet are topologically equivalent?
This has to do with the properties of each letter: how many vertices, edges, faces, and holes it has.
In three dimensions:
Face: a bounded flat surface, like a square
Edge: a line segment at which two faces join
Vertex: a point at which edges and faces join
What are the numbers of faces, edges, and vertices for some common geometric shapes.
What do you notice?
Euler number for solids is 2.
What is the difference between the torus and the other objects?
Vertices are extremely important:
Even vertex:
Odd vertex:
Network Theorem: if a network (a two dimensional geometry) has either 2 odd vertices or no odd vertices it can be traced without lifting your pencil or retracing a line. How about these networks?
Extra: Topology can tell us about many properties of spaces, like when we cut the moebius strip apart we get one loop and not two.
Maps are two dimensional geometric objects. What can we say about them?
There is a minimum number of colors you can color a map with without having any two neighboring countries the same color. What is this minimum number? Try it.
Acme Klein Bottle (http://www.kleinbottle.com/) -- ordering information for a really interesting topological shape ( a kind of 4D moebius strip)
Gallery of Interactive Geometry (http://www.geom.umn.edu/apps/gallery.html) – pictures and interactive games
The Optiverse (http://new.math.uiuc.edu/optiverse/) -- a neat video turning the sphere inside out
Exploring the Topology of the Cosmos (http://www.britannica.com/bcom/magazine/article/0,5744,90414,00.html?query=topology) – a neat article about teams of cosmologists and topologists trying to discover the structure of the universe
History of Topology (http://www.math.utsa.edu/ecz/l_ht.html) – a brief history of topology
Worksheet #1: Triangular Prism, Tetrahedron, and Torus cutout patterns
Worksheet #2: Cube, Octahedron, and Torus cutout patterns
