|TCCNS Course||MATH 1332: Contemporary Mathematics|
|UT Austin Course||M 302: Introduction to Mathematics|
Suggested Resources and Preparation
Materials and Technology
For the instructor: Physical objects that are or approximate the mathematical objects in this section will be useful: straws, coins, hollow balls, paper disks, plastic stacking rings, etc.
Chenille stems/pipe cleaners are extremely useful for modeling equivalence of 1D objects. You can put your finger down on a cut point to count how many pieces will be left.
Students should be familiar with the concepts of dimension and boundary as they are presented in Section 3.4.
Overview and Student Objectives
Students will understand that:
- In topology, two objects are equivalent by distortion if one can bend, stretch, shrink, or twist one into the other, with no cutting or gluing of the object.
Students will be able to:
- Describe or draw simple deformations of mathematical objects (e.g. a circle can be flattened into an oval by pushing down)
- Use simple topological invariants to argue/prove that two objects cannot be equivalent by distortion (e.g. a sphere cannot be equivalent to a circle because a sphere is 2D and a circle is 1D).
- Topology is the mathematical study of the properties that are preserved through deformations, twistings, and stretchings of objects. Tearing, however, is not allowed.1
- Two objects are equivalent by distortion if we can stretch, shrink, bend, or twist one object into the other without cutting or gluing.
Topologists care about the shapes of things, but they don't care about them in a rigid geometry kind of way. A figure 8 is not the same as a circle, but how do you say that with math? A donut has a hole in it and a sphere does not; how do you say that with math? It's important to "say it with math" because that math is key for computer programmers to, say, read an image and determine what shape is being shown. Geneticists care about topology, as well, because DNA coils itself into chromosomes so it can fit inside cells. Human DNA, when stretched out, is over six feet long!
Proving objects are equivalent by distortion
In order to show that objects are equivalent by distortion, you need to show motion. By that I mean you need to use words or pictures to explain how you would morph one object into the other without cutting or gluing.
Viewed as one-dimensional objects, the letters D and O in most sans-serif fonts are equivalent by distortion. You could express this through captioned or labeled drawings as shown:
Alternately, you could choose to describe the transformation in words: "Start with a letter D, push in the two corners to round them out, then pull out the flat side to round it out, then you will have a letter O."
Proving objects are NOT equivalent by distortion
In order to prove two objects are not equivalent, it's not good enough to say "you'd have to cut or glue." You have to prove it's literally impossible to do it without cutting or gluing. We've been here before; remember the proofs that the square root of 2 was irrational or the proof that the reals and naturals do not have the same cardinality?
In any case, we need a test called a topological invariant.
- A topological invariant is a quality about the object that doesn't change when you distort the object. We often think about topological invariants as tests that you perform on pairs of objects, where the only two possible results are "the objects are not equivalent" or "we don't know if they are equivalent."
- If the test gives you different results for your two objects, you know the objects are not equivalent by distortion.
- If the test gives you the same results for your two objects, you do not know if your objects are equivalent by distortion.
There are hundreds of tests out there, and topologists are inventing new ones all the time. You have to learn about three:
- The boundary test is where we count the number of pieces in the boundary of an object.
- The cutting test is where we count all the possible numbers of pieces we get after cutting an object.
- The dimension test is where we count how many dimensions an object has.
- The sphere is not equivalent by distortion to a disk. Both are 2D, but the sphere has no boundary at all while the disk has a one-piece boundary.
- The sphere is not equivalent by distortion to a torus. Both are 2D, and both have no boundary, but if you cut the sphere with scissors you always get two pieces and when you cut the torus with scissors you get either one piece or two, depending on where you cut.
Image credit: Krishnavedala
Letter X has 4 pieces.
Letter Y has 3 pieces.
Letter Z has 2 pieces.
2, 3, 6
No because the disk has a boundary and the sphere does not.
What does it mean to say that two things are equivalent by distortion?
- They have the same number of holes.
- They are exactly the same size and shape.
- One can be deformed into the other by stretching, shrinking, or bending; no cutting or gluing allowed.
- They look the same when you view them at the right angle.
What happens when you cut these three letters at the green spot?
Letter X has ____ pieces.
Letter Y has ____ pieces.
Letter Z has ____ pieces.
Which of the letters X, Y, Z below are equivalent by distortion?
- X and Z only.
- All of them
- X and Y only.
- None of them.
- Y and Z only.
What is a topological invariant?
- It is a test that can help us determine if two objects are topologically the same, or equivalent by distortion.
- It is a map that never changes.
- It is a test that can help tell us when two objects are topologically different, or not equivalent by distortion.
- It is a quality that never changes for any topological object.
Which of the following tests are good topological invariants? Select all that apply
- How many corners an object has
- Number of dimensions
- How many pieces you get when you make an n-1 dimensional cut in an n dimensional object
- How many edges an object has
- The size of the objects
- Existence of a boundary
Is a disk equivalent by distortion to a sphere?
- No because the disk is 2D and the sphere is 3D.
- No because the disk is 1D and the sphere is 2D.
- Yes because both objects are 2D.
- Yes because you can flatten the sphere and it becomes a disk.
- No because the disk has a boundary and the sphere does not.
We have to draw or describe how we would deform one object into the other without breaking the rules of topology.
We have to choose a topological invariant to use on both objects and show that we get different answers.
|Team||Letters in Team||# Boundary Points||Results of Cutting Test|
|1||CGIJLMNSUVWZ||2||always 2 pieces|
|2||AR||2||1 or 2 pieces|
|3||EFTY||3||2 or 3 pieces|
|6||HK||4||2 or 3 pieces|
|7||P||1||1 or 2 pieces|
|8||Q||2||1, 2, or 3 pieces|
|9||X||4||2 or 4 pieces|
If two teams are different in either column, they can't be equivalent by distortion. For example, teams 1 and 2 are not equivalent due to the cutting test, and teams 2 and 7 are not equivalent because they have different boundaries.
B vs. D and O is a separate case. We have to dig a little deeper to find out that B cannot be transformed into a D. If you delete two points from your letters instead of one point, you see that B can be cut into one or two pieces and the letter D will always fall into two pieces.
Yes. You can deform the object so that the top of the coin becomes the inside of the straw, the bottom of the coin becomes the outside of the straw, etc.
Yes. You can push the cup part into the handle, and then you have a solid torus.
If we think two objects are equivalent by distortion, what do we have to do to prove it?
If we think two objects are not equivalent by distortion, what do we have to do to prove it?
Sort all 26 letters of the English alphabet into groups of letters that are all equivalent by distortion to each other. Your justification will be choosing one letter from each group and showing why it is not equivalent to the representative letters from the other groups. Use this typeface:
I recommend you complete a table that looks like this, where each "team" is a group of letters that are equivalent to each other.
|Team||Letters in Team||# Boundary Points||Results of Cutting Test|
If I drill a hole in the center of a coin, is that object equivalent by distortion to a plastic straw? Why or why not?
Is a coffee mug equivalent by distortion to a doughnut? Why or why not? (Check out this GIF for help!)
Reflective writing exercise: Topology is frequently challenging for students who are used to the rules of geometry. Topology brings an entirely different set of rules to govern the objects around us.