Topology is essentially the study of the qualitative relations within different objects. The idea is that if two objects share some of the same basic defining qualities, they are equivalent in a way. There are two main types of equivalences in Topography. The first is metamorphism. Many describe this as if you can make one object look like the other without having to break, cut, or glue anything together then the objects are homeographic. I remember this by thinking about them morphing into each other by molding one into the other. The other type of equivalence is homotopy. Two objects are homotopy equivalent if they can be made from squishing some other object.
For example, look at the GIF to the right.. The idea here is that a doughnut and a coffee mug share some of these basic properties so that if you manipulate a doughnut you can shape it into a coffee mug. You would have to enlarge one half of the doughnut, then create a depression in the middle to form the mug. Notice that there is no disconnecting or reconnecting involved, so these two objects are in the same topographical class. Also, since one is reshaped into the other without cutting or gluing, the two are homeomorphic. |
For Topology often focuses on the number of holes and object has and the connections within the object. And these holes and connections are often used to split objects into homeomorhpic and homotopy equivalence classes. One example of this is the English alphabet. To split the alphabet into homeomorphic classes, letter are thought of in terms of how many holes and how many 'tails' they have. If letters have the same number of holes and tails, then they can be transformed into one another with any breaking, cutting, or gluing so they are homeomorphic to one another. For homotopic classes, the tails can all be 'squished' back into a point so the classes contain more letters and the defining characteristics are the number of holes the letters have. It is important to note that the classes can change depending on what font the letters are typed in.
Homeomorphism classes are:
| Homotopy classes are larger, because the tails can be squished down to a point. They are:
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Another cool object studied in Topology is the Mobius strip. A mobius strip has only one side and one edge. What? Seems impossible, right? It is defined as having one side and one edge because if you were to start walking on a Mobius strip, you would walk the entire length of the strip on both sides without crossing over an edge before returning back to your original position. It is easy to make a Mobius strip of your own by taking a rectangular strip of paper, giving it a half-twist and attaching the ends.
It is interesting tho, that if you give it a full twist, it is no longer a Mobius strip. Try it! Can you explain why a full twist makes the stirip lose it's Mobius strip-ness? (Hint: what would happen if you tried to walk along the strip like before?)
It is interesting tho, that if you give it a full twist, it is no longer a Mobius strip. Try it! Can you explain why a full twist makes the stirip lose it's Mobius strip-ness? (Hint: what would happen if you tried to walk along the strip like before?)
Now it's time for the cool Topology puzzle I promised you! This puzzle is called Tangled Hearts, and I found it at AIMS Education Foundation. (Spoiler Alert: It shows the solution, I encourage you to try it yourself before looking at the solution!) There are also many other Topology puzzles there if you are interested.
To assemble your puzzle (it's super easy and takes just a minute or two) you need:
To assemble your puzzle (it's super easy and takes just a minute or two) you need:
- heart cutout patterns
- 80-ish cm of yarn or string
- a hole puncher
- tape
- 2 pennies or other small weighted item
1. Print and cut out the large and small heart. Use a hole punch to punch holes where the circles are printed.
2. Cut approximately 80 cm or string.
3. Place the string on top of the large heart as shown.
2. Cut approximately 80 cm or string.
3. Place the string on top of the large heart as shown.
4. Pull both ends of the string through the bottom hole of the large heart from underneath.
5. Pull both ends of the string through the hole in the middle of the small heart, then put one end of the string through each hole at the top of the large heart.
6. Tape one penny to each end of the string.
ASSEMBLY COMPLETED!
THE CHALLENGE: Untangle the small heart without cutting the string or taking the tape off the ends.
BONUS ROUND! Can you put it back on?
BONUS ROUND! Can you put it back on?
I did this puzzle for one of my daily assignments and it was pretty mind blowing to me since it seems impossible at first. Here is my finished Product.
Sources:
https://en.wikipedia.org/wiki/Topology
http://blog.aimsedu.org/2014/03/24/tangled-hearts/
https://en.wikipedia.org/wiki/Topology
http://blog.aimsedu.org/2014/03/24/tangled-hearts/